﻿ Pizza Seminar -- Previous Abstracts

## Abstracts from Previous Pizza Seminars (in reverse chronological order):

### Spring 2016

• Date: April 29 Justin Semonsen 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Learning and VC Dimension As graduate students, our primary job is to learn, as well as to teach others. Although our students sometimes seem incapable of learning, is that their fault? Today we discover what can be learned, and what is impossible to learn, beginning with pizza of course!
• Date: April 22 Yonah Biers-Ariel 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center It's a Pirate's Life for Me! Before the advent of the modern university system, many young mathematicians, having completed their education, set out upon the high seas as pirates, capturing wayward sailors and posing fiendish mathematical riddles (or so my calculus teacher told me). In this talk, we will examine two (or maybe three) such problems. In the first one, the captured sailors can use their knowledge of group theory to dramatically improve their odds of survival, and in the second one, the pirates can lower these odds by taking the right number of captives. Even if you never become a pirate, they make for good parlor tricks.
• Date: April 15 Nathan Fox 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Why is this talk different from all other talks? What is time? This is a deep philosophical question that we will not address in this talk. But, we will discuss perhaps the most important human aspect of this question: how do we measure time? From seconds to days to months to years, humans have had different methods for measuring these from culture to culture and from age to age. In this talk, we will give a history of many of these different clocks and calendars, and we will try to put them in an astronomical and mathematical context. Disclaimer: A large portion of this talk will be about the Hebrew Calendar, since it is the one I know the most about, and it is extremely quirky.
• Date: April 08 Professor Lieberman et al. 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Panel Discussion: Teaching Mathematics We will have a panel composed of professors Lieberman, Rosenstein and Weingart and our own Pat Devlin to discuss topics pertaining to the teaching of mathematics, especially at a college level. Come prepared with your questions..
• Date: April 01 Keith Frankston 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Seemingly Paradoxical Results of Random Games Bud and Burak walk into 701. We seem to have a paradox. Find out for sure on Friday afternoon when we will also talk about a couple seemingly of surprising results of games played against probabilistic antagonist, who is both probably antagonistic and antagonizes according to a probability distribution. Generally, this is what we will cover, though unlike General Lee, we will not wage war against the union. We may fight one or two battles against the intersection, however. Send your appreciation and vitriol to Nathan for the puns.
• Date: March 25 Alexi Hoeft 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center How to Create and Prevent Traffic Jams We'll explore (vehicular) traffic flow patterns and phenomena like traffic waves and phantom traffic jams (which are all too real). Amazingly, it can take just one driver to create a traffic jam and it can take only one driver to prevent a traffic jam, and I'll teach you how to be each kind of driver.
• Date: March 11 Tim Naumovitz 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Ranking Algorithms and Tournament Formats Given n competitors, what is the best tournament structure to find a ranking of the competitors? Additionally, how can one design an accurate system of such rankings given tournament results? I'll talk about some ideas that have been used to attempt to address such questions, as well as some flaws with them that have been exploited.
• Date: March 04 Pedro Pontes 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Pretty Pictures in Complex Dynamics I will try to convince people of the beauty of complex dynamics by showing pretty pictures of fractals. The general setting will be the iteration of rational functions and the Fatou/Julia dichotomy. There will be many pretty pictures of Julia set fractals, and we will see how simply varying c in the quadratic polynomial z^2+c, the simplest of functions, can lead to dramatically different pictures and very interesting changes in dynamics. We will also study some general results on the classification of Fatou components for rational functions and, of course, see some nice pictures of those cases as well.
• Date: February 26 Katie McKeon 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Non-Euclidean Geometric Combinatorics We'll examine some oddly behaved (aka non-Euclidean) geometries and see if some of the major theorems in discrete geometry still apply to them.
• Date: February 19 Sam Braunfeld 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Formal Methods for Causation We know the mantra "correlation is not causation", so how can causation be established? The general standard is a randomized controlled experiment, but this is not always feasible or legal. However, under certain causal assumptions, it may be possible to determine causality without performing an experiment at all. The methods for doing so primarily involve graph representations of causal models (essentially Bayes nets, which were created by the developer of much of this theory). This talk will discuss how to handle these models, and consider some simple examples.
• Date: February 12 Mike Donders 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Pseudo Random Numbers Random numbers are something we often want to use but have a lot of trouble finding. Compiling a sequence of truly random numbers is at best slow and at worst impossible, so instead it's simpler to just come up with a list of numbers that seems random enough. We'll look at a few of the methods of creating pseudo random sequences and what criteria allows us to call a deterministic sequence 'random'.
• Date: February 05 Ed Karasiewicz 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Reciprocity Laws and Conjectures One aspect of the Langlands Program, known as the Langlands Reciprocity Conjecture, deals with a relationship between geometric objects, such as solutions of systems of polynomial equations, and analytic objects, such as Hecke characters and modular forms. We will begin with a detailed discussion of quadratic and cubic reciprocity in their classical forms and then transition into examples of reciprocity laws involving elliptic curves and modular forms.
• Date: January 29th, 2016 Martin Köberl 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center CH Depending on the audience's interest and background I will talk about various topics hopefully all illuminating Cantor's Continuum Hypothesis and its role in set theory now and then. We might see why cardinal addition and multiplication are boring while exponentiation is not. I will also mention the phenomenon of logical independence and the pioneering role played by CH. Time permitting I will relieve you from any worries about the problems caused by CH's independence for your mathematical work.

### Fall 2015

• Date: December 4th, 2015 Johannes Flake 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center How to Improve Images Images are all over the place. Good images are rare. People want good images. Surprisingly, this has promoted the practical use of some of the most basic, but highly abstract mathematical concepts. Maybe this will be explained in my presentation, but you should definitively come anyway, because the discussion will be a mathematically light-weight dessert for the pizza and there will be plenty of weird images. Oh, and a Playmate.
• Date: November 20th, 2015 Fei Qi 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Regular Functions of one Quaternionic Variable The richness of the theory of functions over the complex field makes it natural to look for a similar theory for the only other non-trivial real associative division algebra, namely the quaternions. However, it turns out that many of the notions in complex analysis does not naively carry over. In this talk I'll explain why we can't define differentiability for quaternionic variable functions as that in real or complex variable functions. Then I'll present two versions of regularity of quaternionic functions and briefly discuss properties they have.
• Date: November 13th, 2015 Érik Amorim 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Theorems as Random Number Generators We will present and somewhat prove three simple, cool and surprising theorems that should be more well known than they are, because of their randomness. What is meant by that is the fact that some unexpected, seemingly random number pops out of nowhere in their statements, but it makes sense after one sees the proofs. The three results are completely unrelated (one is Real Analysis, one is Complex Analysis, the other Point Set Topology), so this will actually be three talks in one, and I will even let the audience choose the order in which they want to see each one.
• Date: November 6th, 2015 Joel Clingempeel 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Morse Theory and Quantum Physics Given a real-valued function defined on a compact surface, it is natural to ask about the geometry of the level curves. Although the geometry varies continuously, the topology only changes when one passes a critical point. It turns out that (under mild technical assumptions) understanding the analytic behavior of the function near the critical points actually is sufficient to determine the topology of the surface. This sort of phenomenon can be studied in higher dimensions and serves as the basis for the subject of Morse theory. In 1982, Ed Witten wrote a spectacular paper establishing that this could in fact be understood in terms of an evolving quantum mechanical system which at "time zero" would naturally encode topological information and at "infinite time" would naturally encode information pertaining to the critical points. The goal of this talk is to explain this beautiful theory without presupposing any prior background in differential geometry or quantum mechanics.
• Date: October 30, 2015 Cole Franks 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Disturbing Images A surface, or a continuous image of a plane region, seems a very familiar and intuitive object. Intuition failed the mathematicians of the late 19th century, however, luring them to an nonsensical definition of surface area. We'll talk about this definition and how it was dismantled. Next we will look at Lebesgue's definition of surface area and some of its strange implications.
• Date: October 23, 2015 Chloe Urbanski 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Miracle of Modern Notation Imagine a world where numbers are not thought of as abstract objects in their own right, but are inseparable from the length of a line segment which they represent, where "algebra" is only done by manipulating geometric objects, where n and n^2 are thought of as different mathematical objects, because one represents a length while the other represents an area. This world is one in which much of the math we know today was done. Though we have all likely seen proofs or constructs of the concepts I will present, the way in which they were originally published is very different from the way we think about it now. I will show a few examples of such differences and compare the original proofs to the way one would present it in today's notation.
• Date: October 16, 2015 Professors Cakoni, Radziwill, Tumulka, Weingart 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Job Market and its Applications Four professors in our department will lead a panel discussion about jobs after graduation and how to apply for them. Come prepared with any questions you may have.
• Date: October 09, 2015 Matt Charnley 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center History of the Solutions of PDE's Over the course of the last two or three hundred years, people have been looking to solve Partial Differential Equations in a variety of situations. A lot of things have changed over this time frame. Not only have we been able to solve more of these equations (obviously), but our idea of what constitutes a solution has grown as well. This talk will give a summary of some of the different types of solutions to Partial Differential Equations, how they came about, and why we care.
• Date: October 02, 2015 Ross Berkowitz 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Claude Shannon and the Beginnings of Information Theory The capacity of abstracts for math talks is quite low, so instead I just have to urge you to come to my talk and find out about Claude Shannon and the beginnings of Information Theory.
• Date: September 25, 2015 Pat Devlin 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Same Old Theorems via Different and New Proofs The pizza seminar challenge (PSC) is to eat an entire pizza while presenting a talk. It has long been conjectured that the PSC is possible, and in this talk, we will provide the first proof of this conjecture. [The proof is probabilistic and non-constructive, and it will assume a generalization of the Reimann hypothesis.] Time permitting, we may also discuss the following math. (Optional math content) What's your third favorite proof that root 2 is irrational? What's the weirdest proof you know of the infinitude of primes? How can we prove the Pythagorean theorem using combinatorics? In this talk, we'll discuss these questions and more in a guided tour of interesting proofs of elementary facts.
• Date: September 18, 2015 Tom Sznigir 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Homicidal Chauffeur Suppose you find yourself in a large parking lot where a maniac is trying to run you over with his car. You are agile and capable of moving in any direction. Your opponent moves much faster, but his maneuverability is limited by his car's turning radius. Do you survive? The answer is provided by the theory of differential games. Analogously to discrete games, a differential game is played by two (or more) players who each try to influence the game state for their own benefit. Unlike the discrete case, players must continuously make their moves (in this case, one player runs for dear life while the other chases him). The evolution of the game state satisfies a set of differential equations, which is why such games are known as differential games.
• Date: September 11, 2015 Burak Kaya 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Axiom of Determinacy and Some of its Consequences Being bored of finite games, Jake and Matthew decide to play an infinite game! They are given a subset A of the unit interval. Jake starts the game by choosing some a_0 in {0,1}. Then Matthew chooses some a_1 in {0,1}. Choosing a_i in {0,1} for each i alternately, Jake and Matthew are slowly constructing a real number in the unit interval whose binary expansion is 0.a_0a_1a_2... After "infinite time" passes, they obtain a real number a in the unit interval. Jake wins the game if a is in A. Otherwise, Matthew wins. If there is a winning strategy for either Jake or Matthew, we call the set A determined. For example, if A is countable, then A is determined since Matthew can simply diagonalize and always wins. What if A is open? What about some arbitrary A? Does there exist a winning strategy for either one of players? It follows from the axiom of choice that not all subsets are determined. What if we reject the axiom of choice and add the assumption "all subsets A of [0,1] are determined" as an axiom to ZF? This assumption is called the Axiom of Determinacy (AD). It turns out that AD affects topological and measure theoretic behavior of subsets or real numbers. In this talk, we shall prove some nice consequences of AD on the behavior of sets of reals. If time permits, we shall also talk about to what extent subsets of reals are determined (under ZFC).

### Spring 2015

• Date: May 1, 2015 Kellen Myers 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Tragedies and Comedies of Mathematics We will discuss some anecdotes, scenarios, trivia, and apocrypha of mathematics in the mode of comedy -- and then that of tragedy. This talk is BYOC (bring your own chorus).
• Date: April 24, 2015 John Chiarelli 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Burden of Proof: The Events and Legends Behind the Evolution of Greek Mathematics Ancient Greece has long been hailed as the source of the finest and most ingenious mathematicians in the West before the modern era. How, though, did the Greeks begin their explorations into the abstract mathematics they are so well known for? In this talk, I will delve into the origins of Greek mathematics, and the traditions from older civilizations that helped spur their developments. Involved in this process will be investigating what histories ancient scholars espoused on such origins, and identifying the foundations of their interpretations of the past.
• Date: April 17, 2015 Panel Discussion 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Other Job Market This pizza seminar will be a panel discussion with three Rutgers Ph.D. graduates who are currently working in industry. Our panelists will be Sara Blight: 2010 Graduate under Dr. Iwaniecz. She currently works for the NSA. Nic Trainor: 2012 Graduate under Dr. Vogelius. He currently works for Numerix LLC, a derivatives pricing and risk analytics software company. Justin Bush: 2014 Graduate under Dr. Mischaikow. He currently works at Palantir, a software company specializing in data analysis. They will be here to answer any questions you have about making the transition to jobs outside of academia, or anything else you may be curious about. Come with any questions you have.
• Date: April 10, 2015 Matthew Welsh 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Eigenvalue Problems, Selberg's Trace Formula and Analytic Number Theory In his 1966 article "Can One Hear the Shape of a Drum?", Mark Kac asked if it was possible to recover the shape of a drum from its fundamental overtones -- i.e. whether or not two isospectral domains are necessarily isometric. This has since been answered in the negative, but we can still ask which aspects of a drum can be heard. This is a wonderful problem in that it's interesting to both mathematicians and physicists, prototypical of many problems in applied mathematics (including number theory), and can be easily explained to my parents. I will sketch Kac's proof of the most famous result (Weyl's law) in this direction and discuss the limitations of his method. Proceeding anachronically, I will then introduce Selberg's trace formula (generalized Poisson summation) as a tool to help understand the problem in the context of Riemannian surfaces. This will be motivated by observing the equivalence between the eigenvalue problem on a torus and Gauss' circle problem. After sketching the proof of the trace formula, I will note some structural similarities with explicit formulae that indicate why techniques from analytic number theory might be useful. If there happens to be any remaining time, I can discuss how Sunada used a cheap version of the trace formula to show how a group theoretic criterion implied isopectrality. This turns out the be the key idea for answering Kac's question in the negative.
• Date: April 3, 2015 Keith Frankston 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Topological Games and an Application of the Continuum Hypothesis This will actually be two distinct talks, each covering an interesting proof from Oxtoby's Measure and Category. The first part will be about the Banach-Mazur game. The Banach-Mazur is a topological game which is a type of perfect information positional game played on a topological space. Unlike finite perfect information positional games (think Hex), such infinite games can be non-deterministic (i.e. neither player has a winning strategy independent of the others strategy). We will see a brief proof that such a winning strategy always exists for one player or the other in the finite case and will show that the Banach-Mazur game is non-deterministic. The second part of the talk will be a beautiful proof by Ulam that any "nice" measure which is defined over every subset of an $\alpha_1$ cardinality space must be 0 everywhere. Specifically, if we assume the continuum hypothesis this gives a direct proof that there is no extension of the Lebesgue measure to all subsets of the real line without having to explicitly constrict Vitali sets.
• Date: March 27, 2015 Joel Clingempeel 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Topological Quantum Field Theory A topological quantum field theory (TQFT) is a rule for assigning invariants to nice manifolds of a fixed dimension such that given a decomposition of a manifold into pieces, one can easily compute its invariant in terms of the invariants of its pieces and how they are glued together. Specializing to the case of two dimensions, we see that one only needs four pieces to construct any surface, and from here one can show that all the information in a 2D TQFT is captured by a single algebraic structure. It turns out that a converse theorem holds: namely, given such a structure, one can engineer a 2D TQFT giving rise to it. We sketch a proof of these results using Morse theory which provides a way of decomposing a surface into pieces by looking at level sets of real valued functions - essentially a rigorous way of dunking surfaces into bathtubs or donuts into coffee. Though the notion of a TQFT was inspired by physics, we will take an axiomatic approach which requires no prior knowledge of physics.
• Date: March 13, 2015 Katie McKeon 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Behavior of Continued Fractions The simple continued fraction for Pi begins as [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, ...], which looks a bit random. The 'randomness' of Pi will not be addressed in this talk. Instead, we will use ergodic theory to outline the statistical regularity of continued fractions for almost every other number.
• Date: March 6, 2015 John Miller 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Riemann Zeta Function This talk will be an introduction to the Riemann zeta function, discussing its pre-Riemann origins, Riemann's contribution, and the link between prime numbers and the zeros of the zeta function.
• Date: February 27, 2015 Tim Naumovitz 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Mathematics of Auctions We'll discuss a few different types of auctions, looking at how well they satisfy various criteria that may be important to us. We'll also look at ways in which we can modify our framework and ask related questions, possibly leading us to one or two interesting thought experiments. Additionally, the expected number of auctions that will take place during this talk is greater than zero.
• Date: February 20, 2015 A bunch of people 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center 5 Minute Talks A series of short talks about a variety of different topics.
• Date: February 13, 2015 Pedro Pontes 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Mathematics for the Zombie Apocalypse We will address the extremely important subject of preparing for the inevitable zombie apocalypse, using what we can do best: mathematics. Following the original paper by Munz et al., we will develop a few systems of ODE's whose (numerical) solution is to model the event of a zombie outbreak. We will see how the populations of uninfected humans, infected, zombies, and dead will interact, and study a few strategies to stop the advance of the zombie disease. This will end in the sad conclusion of almost certain human extinction (spoiler!). After that, we will talk about the reception to this original paper, how others have improved on the work of Munz et al., and ways to improve the model in order to save mankind.
• Date: February 6, 2015 Ross Berkowitz 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Method of Moments and the Erdős Kac Theorem: Applying Probability to Number Theory. In 1940, Paul Erdős and Mark Kac published the (appropriately named) Erdős-Kac theorem characterizing the behavior of the number of prime divisors of a random integer. The result was a considerable extension of the earlier Hardy-Ramanujan Theorem, and rather amazingly it was proved using not the standard tools of number theory, but rather statistics. I will introduce the method of moments and show how Erdős and Kac modeled a seemingly complicated number theoretic problem with a basic probabilistic one.
• Date: January 30, 2015 Matt Charnley 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Orientability and the Möbius Strip Everyone knows that the Möbius strip is a non-orientable surface, but what exactly does that mean? In this talk, we will define what an orientation is on both a vector space and a manifold, covering enough of the basic manifold theory to understand all of the terms involved. It will then be proven that the Möbius strip is not orientable from these definitions. Time permitting, the necessity of orientability for integrating on manifolds will be discussed, along with the general statement of Stokes' Theorem.
• Date: January 23, 2015 Nathan Fox 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Kolmogorov Complexity How complicated is the number 3? (Answer: not very.) How about 1000000000, 123456789, or 142937721? Kolmogorov complexity is one way to quantify how complicated these numbers are. The Kolmogorov complexity of n is defined as the length of the shortest computer program that outputs the number n and then halts. This may sound simple, but Kolmogorov complexity is actually impossible to compute. We will discuss some properties of Kolmogorov complexity, and then we will use it to prove Gödel's Incompleteness Theorems.

### Fall 2014

• Date: December 12, 2014 Richard Voepel 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center One Big Bin - A Half-baked Presentation About Classification If you're like some mathematicians, there is a deep, dark corner of your soul that craves order -- and not just any kind of order! We're talking 0 Kelvin crystal lattices here. It's why we have classification theorems, whole books of them (5000 pages for simple groups). It's why we like canonical forms, or complete invariants. It's even how we can hope to make headway in automated theorem proving. But there is a catch; in practically every case, we exploit (often 'till we're blue in the face) some underlying algebraic structure. But this begs the question: can we classify the complex numbers, and what would that even look like? In this presentation we will answer that question, and attempt to make that answer understandable. Highlights include a brief introduction to the complex plane in terms of a game about transcendence, my "Top 10" list of complex numbers, a couple of theorems and black-boxes you have to trust me on, and time permitting, a very sloppy argument as to why the most complicated complex numbers have the best approximations.
• Date: December 5, 2014 Senia Sheydvasser 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Sifting for Gold(bach) Sieve theory has in some sense been around since the 3rd century BCE, yet it still has much to offer us: it was instrumental in proving the bounded gaps conjecture, as well as the large gaps problem. I will be giving an introduction to sieve methods, particularly the relatively simple Brun sieve, which is nevertheless powerful enough to prove the Schnirelmann-Goldbach theorem.
• Date: November 21, 2014 Ed Karasiewicz 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center A Family of Elliptic Curves and L-functions We will consider a particular family of elliptic curves, E: y^2 = x^3-Dx, and compute their L-functions using Jacobi sums. By relating the Jacobi sums to Hecke characters one can show that the elliptic curve L-function possesses an analytic continuation and a functional equation. Time permitting we will discuss the Birch and Swinnerton-Dyer conjecture.
• Date: November 14, 2014 Bud Coulson 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Non-Standard Calculus Ever wanted to rigorously do calculus the way Newton and Leibniz did it? Without any limits, epsilons, and deltas - just good old-fashioned infinitesimals? In this talk, we'll take a tour through the hyperreal number system, and live the dream of freshmen calculus students everywhere.
• Date: November 7, 2014 Sam Braunfeld 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Forcing via Boolean-Valued Models Continuing last week’s theme, I will be discussing an example of how to scare people with math. The invention of forcing revolutionized set theory, and is still the most important technique in much of the field. After introducing the technique, we will use it to more or less prove the independence of the continuum hypothesis. No special knowledge of set theory assumed.
• Date: October 31, 2014 Matthew Russell 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Mathematics of Fear Methods for using mathematics to scare people will be discussed.
• Date: October 24, 2014 Érik de Amorim 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center An Infinitely Fast Tasmanian Devil and his Special Peano Curve Porky Pig is a dimensionless point in the plane. He is initially inside a square whose length is 8 miles, and cannot move faster than 1mph. He is going to move along a continuous curve, trying to run away from Taz, who is also a dimensionless point initially in that same square, and who can move as fast as he wants, describing another continuous curve. However, Taz is blind and can never know Porky Pig's location. Is there a continuous path for Taz that guarantees he will catch Porky Pig, no matter how and where the latter chooses to run? This talk is justified simply by the fun nature of this problem, but if you need more than that, let's say it can also teach you some things about Peano curves!
• Date: October 17, 2014 Professors Han, Kontorovich, Lepowsky, and Tumulka 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center The Job Market and its Applications Four of our professors will lead a panel discussion about jobs after graduation and how to apply for them. Come prepared with any questions you may have.
• Date: October 10, 2014 Semeon Artamonov 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Hamilton Dynamics on Associative Algebras In this talk I will start with a brief introduction to Hamilton dynamics on manifolds. I will give the definition of a Poisson bracket, Hamilton flow and provide a relation to the first integrals of the corresponding system of ODE. Next, I will go beyond manifolds. Namely, I will explain how we can define a Hamilton flow on associative algebra (in general noncommutative). I will provide an explicit construction for some particular system of ODE recently proposed by Kontsevich.
• Date: October 3, 2014 Nathaniel Shar 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center 1/z We'll explore the geometry of the complex-valued function f(z) = 1/z and see how that geometry (and higher-dimensional analogues) can be used to give simple proofs of difficult theorems. We might also see how it can help you pass the written quals....
• Date: September 26, 2014 Fei Qi 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Theory of Quasideterminants and Applications According to Retakh, Gelfand had been asking about noncommutative determinants every semester. He finally got a satisfactory answer almost 3 decades later, with a paper published with Retakh proposing the theory of quasideterminants which would work over ANY division ring. And the definition is so natural that even a high school student knowing Gaussian elimination could play with it. As one might imagine, such generality could result in numerous applications. I'll probably cover two of them, namely the noncommutative Viete theorem and the main theorem of noncommutative symmetric polynomials. The latter is proved by Robert Wilson and refered as Wilson's theorem.
• Date: September 19, 2014 Pat Devlin 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Figure not to Scale (Drawing for the Classroom) A good picture is the most powerful part of any talk. In class, your drawings need to clarify and illuminate. They shouldn't be sloppy or confusing, and they must be carefully chosen. But above all, your drawings need to inspire! But for many of us, drawing just doesn't come naturally (and trying to draw those bizarre calculus shapes seems out of the question!). In this talk, we discuss how to draw anything you'll need from calculus to linear algebra so that you can take advantage of this outrageously important teaching tool. We'll discuss how to draw in math (from basics to the complicated) as well as what to draw (e.g., pictures that help with rank-nullity theorem or the FTC). Of course, no drawing background is assumed, and I promise instant improvement or it's free!
• Date: September 12, 2014 Burak Kaya 2:00 PM (Please note the different start time.) Graduate Student Lounge, 7th Floor, Hill Center Friedman's Borel diagonalization theorem or: Can you prove Cantor's theorem in a uniform way? One can view Cantor's diagonal argument as an "explicit" map taking a list of real numbers and producing a real number which is different than any real on the list. Of course, the real number we produce not only depends on the set of reals we are given but also on the ordering they have in the list. A natural question to ask is whether or not there is an "explicit" diagonalization procedure that is independent of the ordering. In other words, can you prove Cantor's theorem in a uniform way? Surprisingly, the answer to this question turns out to be negative as was proven by Harvey Friedman in 1981. There cannot be an "explicit" procedure that diagonalizes countable sets of reals in an order independent way. In this talk, we will prove Friedman's basic Borel diagonalization theorem. If time permits, we will also talk about its generalizations and how these relate to infinite games and determinacy of sets of reals. The talk will be almost self-contained (modulo Baire category theorem and basic facts about topological spaces) and no prior knowledge is needed. A small note for those who have not yet been converted to the Borelian religion by descriptive set theorists: EXPLICIT=BOREL!

### Spring 2014

• Date: May 2, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Fei Qi Young Symmetrizer, Irreps of S_n and Weyl Module Last year I gave a talk about symmetric functions, where I stated the result that irreps of S_n can be parametrized by Young diagrams. I saw skepticism from people's faces so now I am going to sketch the proof. I'll start by introducing the Young symmetrizer, then prove the statement by constructing the irreps explicitly. Depending on the interest of audience, I'll either present full details of the proof, or talk about further application of Young symmetrizer, which, under a slightly different context, also constructs Weyl module whose character is precisely the Schur function.
• Date: April 25, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Robert McRae Modular Invariance in Lattices and Codes I'll talk about how the theta functions of certain lattices (which encode the lengths of all the vectors in the lattice) satisfy a suitable invariance under an action of SL(2,Z), and I will discuss how a similar phenomenon arises in the simpler setting of binary linear codes. I will also show how to obtain lattices from codes, time permitting. Note: there will be some overlap between this talk and Neil Sloane's excellent Faculty Perspectives Seminar talk from Monday, but the two talks will be different.
• Date: April 18, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Rebecca Coulson An Introduction to Definability: What Can You Say? We've all been there – you're at some seminar, the speaker starts talking about Godel's incompleteness theorems or the axiom of choice, and you think, "Man, logic is cool! I wish I knew more about it." This talk is here to help you. We will provide a friendly and fun introduction to questions in logic via the topic of definability. We will see an interplay between logic and other areas of mathematics. Questions about definable sets – what can been defined in a language, what the definable sets as a whole look like – will be discussed, along with their implications.
• Date: April 11, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Shashank Kanade Some Gems from the Theory of Partitions Join us for a free trip to the beautiful land of integer partitions! Along the way, we will stop at the famous tourist destinations, and marvel at the beauty of some truly radiant gems from mathematics. Itinerary includes the Rogers-Ramanujan identities, the Rogers-Ramanujan continued fraction, generalizations of these identities and their relations to Lie theory and a few more topics if time permits. Skipping through the proofs and walking through some history is guaranteed!
• Date: April 4, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Anthony Zaleski Douglas Hofstadter's Godel, Escher, Bach I will discuss some of the fascinating ideas of Hofstadter's Pulitzer Prize-winning masterpiece, which connects such diverse fields as philosophy, music, art, math, metamath, and turtles. First I will describe some troubling self-referential paradoxes of math, and the attempts to banish such oddities with formal systems. Then I will (very informally) motivate Godel's Incompleteness Theorem, which essentially says those attempts are bound to fail. This will lead us into the largely uncharted waters of the nature of Truth, Zen, consciousness, and turtles. I will conclude the talk by showing you, once and for all, a picture of God.
• Date: March 28, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Ed Karasiewicz The Geometry and Arithmetic of Modular Forms Modular forms are functions on the upper half plane that satisfy certain holomorphy conditions and transform in a particular way with respect to certain discrete subgroups of SL(2,Z). We will discuss the geometry of modular forms and their connections with number theory. Specifically, some basic examples of modular forms will be presented and shown to have Fourier coefficients that are "arithmetically interesting". From there we will move on to the geometry of modular forms. This will involve studying various Riemann surfaces and will result in showing that the space of modular forms breaks up into finite dimensional subspaces. This finite dimensionality will be used to deduce various formulas for the representation of an integer n by a quadratic form. If time permits I will stand in silence and eat pizza.
• Date: March 14, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Tim Naumovitz The mathematics of pi(e) Is pi*e irrational? What about pi+e or pi-e? In my talk, I will refrain from addressing such foolish questions, and instead talk about several interesting problems, principles, anecdotes, etc dealing with pi or pie.
• Date: March 7, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Kellen Myers Analysis You Can Count On Human beings have used many tools for counting. Knots in string, notches on sticks, fingers, and even toes. Eventually, technology leads to the abacus, the Babbage machine, the lesser-known Cabbage machine, the Turing machine, and finally the iPad. Mathematics has also developed many tools for counting - bijections, group actions, generating functions, sieves, and so many other amazing combinatorial and algebraic tools. Analysis, being the other strong arm of mathematics, can also be used to count, and in this talk, I will provide some description of how Fourier analysis can be used to solve some enumerative questions. Historical context, general ideas, and a few simple examples will be presented.
• Date: February 28, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Joel Clingempeel From Calculus to Cohomology Classical problems in vector calculus such as determining whether or not a vector field defined on a region of R^3 is conservative lead to considerations about the topology of the underlying region. This motivates one to construct topological invariants for open subsets of R^3 using purely vector calculus, but unfortunately, this fails to generalize in any naive sense to higher dimensions. To extend this, we introduce the machinery of differential forms, show how it reduces to classical vector calculus in dimension 3, and then show how in direct analogy this can be used to define topological invariants for open subsets of R^n for any n. We then show how to further extend this to study the topology of smooth manifolds. Time permitting, further topics will be discussed.
• Date: February 21, 2014 1:35 PM Graduate Student Lounge, 7th Floor, Hill Center Burak Kaya The Measure Problem and Some Large Cardinals In his PhD thesis, Lebesgue asked whether it is possible to construct a non-trivial translation invariant countably additive measure on all subsets of the real line. Later, Vitali showed that under AC there exist non-Lebesgue measurable sets. Moreover, it is known that we need AC to construct non-measurable sets for such measures. But what if we drop the condition that the measure is translation invariant? Do non-measurable sets necessarily exist? Or is it possible to extend the Lebesgue measure to all subsets of reals if the extending measure is allowed to be not translation invariant? More generally, is it possible to have some non-empty infinite set S such that there exists a countably additive probability measure that takes value 0 on singletons and every subset of S is measurable? In this talk, we will try to solve this problem, explore its connections to certain large cardinals and prove that it is consistent with ZFC that there are no such measures (however, existence of such measures are currently not known to be inconsistent with ZFC, and many set theorists believe that they are consistent).
• Date: February 14, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Matthew Russell The Mathematics of Love Methods for using mathematics to improve your love life will be discussed.
• Date: February 7, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Matthew Charnley Navier-Stokes Equations: An Introduction The Navier-Stokes equations are some of the most studied partial differential equations because they are important to both theoretical and applied mathematics. Theoretical mathematicians are attempting to prove that these equations admit a unique solution for given sets of initial data, and applied mathematicians use them to model the flow of an incompressible fluid in a variety of situations. But where do these equations come from? In this talk, some of the history of the Navier-Stokes equations and a derivation of them from physical principles will be presented. Then, a few simple problems will be discussed to show how by making some assumptions (which may or may not be accurate), explicit solutions of these equations can be obtained. Finally, modern results will be shown to explain what mathematicians have so far in terms of proving the desired existence and uniqueness results.
• Date: January 31, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center John Chiarelli Alfred Kempe and the Five-Color Theorem When Kenneth Appel and Wolfgang Haken revealed their proof of the Four-Color Theorem in 1976, they were met with a lot of skepticism, and not just because of their use of computers. The Four-Color Theorem - which states that any planar graph can be colored with four colors such that no two adjacent vertices are the same color - has a long history of failed attempts at a solution. In this talk, I will look into the details of one particular failed attempt at proving the theorem, posited by Alfred Kempe in 1879. While it ultimately turned out to be incorrect, the methods used have notable applications. Accompanying this "proof" will be Percy James Heawood's modifications to Kempe's work to prove a weaker proof, the Five-Color Theorem.
• Date: January 24, 2014 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Sam Braunfeld Zero-Knowledge Proofs As mathematicians, we frequently read proofs not because we doubt the claim, but because we expect the proof itself to contain useful information. This talk will introduce zero-knowledge proofs, which are widely used in cryptography and give essentially no information beyond the truth of their claim. In the course of defining zero-knowledge proofs and giving examples, this talk will also touch on several other fundamental tools of modern cryptography.

### Fall 2013

• Date: December 6, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Richard Voepel A Brief Survey of Classical Theorems and Open Questions in Transcendence Theory Despite the fact that almost every number is transcendental, the basics facts and theorems of transcendental number theory are relatively esoteric to the average student, due in no small part to the myriad tools necessary to resolve even the most specific problems, such as the transcendence of e or pi. Moreover, even innocent looking problems like the transcendence of e+pi remain completely intractable in the face of the most advanced techniques of the field. While these and other questions are tantalizing to think about, in this talk we will focus on 5 theorems from classical Transcendence Theory, in the pursuit of proving the transcendence of three constants: Liouville's constant, e, and pi. We then mention two open questions and discuss the partial results or interesting implications that can be derived from them, and (time permitting) will briefly discuss the ways in which Transcendence Theory has evolved in two directions from its classical roots.
• Date: November 22, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Frank Seuffert A proof of the Sobolev Inequality, Its Optimizer, and a Generalization to Continuous Dimensions This talk will concentrate on the Sobolev Inequality for functions on R^N in the completion of smooth functions with compact support under the norm given by the square gradient, i.e. || grad f ||_{L^2}. The inequality that we will examine is that there exists a positive constant C such that || f ||_{L^2*} <= C || grad f ||_{L^2} for functions in the aforementioned space. A simple proof of this will be provided. Next, we will address the existence of optimizers to this inequality (proving that it is in fact sharp) by a change of variables as per a paper by Elliott Lieb. Finally, we will consider the notion of this inequality for radial functions in continuous dimension. Some ideas for this generalization will be mentioned, I will, however, try to keep things from getting too technical.
• Date: November 15, 2013 1:40 PM Seminar Room, 7th Floor, Hill Center Jane Gilman (Rutgers-Newark) Grant proposal writing and job opportunities at the National Science Foundation This talk will focus on opportunities for mathematicians at the National Science Foundation (the NSF). It will include a description of the various programs that fund mathematical and/or educational activities, mainly the Division of Mathematical Sciences (DMS) and Education and Human Resources (EHR) and their relative responsibilities. The talk will describe the processes in use now for reviewing and deciding on funding grant proposals. It will also include a discussion of proposal writing and how to consult with your program officer along with a description of how the DMS is staffed and job opportunities in various parts of the NSF. The talk will aim to provide information for graduate students and also post-docs, both grant-supported and non-tenure track faculty with very new PhDs. Questions will be most welcome. This panel discussion is made possible through partial funding from an I-cubed funded (minigrant) project to the GSNB Project AGER.
• Date: November 8, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Daniel Scheinerman Lattices and Cryptography A lattice is the integer linear span of a set of linearly independent vectors in R^n. Since the coefficients come from the integers and not the reals many questions that are trivial for vector spaces become challenging for lattices. These challenging problems can be harnessed to make interesting cryptosystems. We will see several of these as well as several cryptosystems which are lattice problems in disguise.
• Date: November 1, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Bud Coulson Summation Methods Do you feel that the usual notion of convergence of series is just too... restrictive? Ever want to evaluate a geometric series with r>1, even though "the man" says you can't? In this talk we will learn methods to assign values to many so-called divergent series, including: 1+1+1+1+... 1+2+3+4+... 1+2+4+8+... 1-2+3-4+... 1-1+2-6+24-120+...
• Date: October 25, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Mike Donders The p-adic Numbers Let's take the world of rational numbers and, instead of giving each number size based on its distance from zero, we instead give it a size based on how many, or rather how few, factors of p it has. That is, for some prime p, we take each rational number and say the more times p divides it, the smaller it is. Doing so establishes the p-adic numbers, an extension of the rationals unlike the real or complex numbers. We will discuss precisely what the p-adic numbers are, some of their properties and their relevance to an array of mathematical fields including analysis, algebra, number theory and topology.
• Date: October 18, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Ross Berkowitz Mathematics of Preferences So two events walk my way, and an economist demands that I choose which one I prefer. What do I do? The answer to this question has consistently bedeviled the economists who bother to think about it (not all of them do of course), and many solutions have been proposed. One particularly nice answer was given by John Von Neumann and Oskar Morgenstern. They proposed a simple and seemingly weak set of axioms on human preferences with the strong consequence that our preferences can be given by maximizing the expectation of a single utility function. We will discuss these axioms, their result, and the inevitable inaccuracies and "paradoxes" that others have used to criticize it.
• Date: October 11, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Nathan Fox Mobius Inversion If set A has 5 elements, set B has 8 elements, and their intersection has 3 elements, what is the size of their union? The solution to this problem involves an application of the Principle of Inclusion-Exclusion. In some similar settings we want to count objects where the notions of "intersection" and "union" are less clear; instead there are notions of "contains" and "contained in." The solutions to such problems lie in the Mobius Inversion Formula. In this talk, we will state and prove this formula in the general combinatorial setting. Then, we will use it to derive some interesting results, ranging from computing Euler's phi function to determining the minimum sizes of generating sets of certain groups.
• Date: October 4, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Zheng-Chao Han, Feng Luo, Shubhangi Saraf, Roderich Tumulka, Chris Woodward (and special guest Henryk Iwaniec) Panel Discussion: Job Applications A panel of speakers will discuss the job application process and answer your questions. Our discussion will cover issues involving research and teaching statements, letter of recommendation and other issues. This discussion is obviously directly relevant to students who are considering graduation next year, but it is never too early to get prepared. All are welcome.
• Date: September 27, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Nathaniel Shar Archimedes Archimedes may not have built a giant death ray, but he did build giant levers to lift attacking ships out of the water. What is less well known is that he also used the principle of the lever to do calculus. I'll explain what he did, and, time permitting, talk about some of his other great discoveries.
• Date: September 20, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Tom Sznigir The Dirac Delta Function The traditional definition of the Dirac delta is a function such that delta(x) = 0 when x != 0, and the integral of delta over the real line is one. While a mathematician may recoil in horror at such a definition, physicists had no such qualms. It quickly saw use in representing things like point masses or point charges. The myriad practical uses led mathematicians to seek a sound rigorous foundation for this "function". The result of this search was Laurent Schwartz's theory of distributions (or generalized functions), a vast extension of the notion of a function. One consequence is that every generalized function can be differentiated. The Dirac delta in this case is the derivative of a step function. These generalized functions have many applications to partial differential equations.
• Date: September 13, 2013 2:00 PM Graduate Student Lounge, 7th Floor, Hill Center Patrick Devlin Getting Your Zero-Dimensional Subspace Across: A Mathematician's Guide to Mathematical Self-Expression From pizza seminars to evenings with friends, a mathematician's life is filled with opportunities (and occasional obligations) to actually talk about what we do for a living. And yet, whether you're talking to your calculus student, to a colleague, or to your prospective father-in-law, there's always that awkward disconnect between what you'd like to say and what they're actually readily able to digest. In this talk, we discuss this and other such issues in order to discover how to better communicate mathematics in the classroom, at a conference, and (even!) on a date.

### Spring 2013

• Date: May 3, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center John Kim Why Static Networks Have More Disease Than Dynamic Networks Many epidemiological models rely on the assumption that static network approximations for dynamic networks provide a sufficient mathematical framework from which to predict disease epidemics. However, computational experiments have shown that static networks support more disease than dynamic networks, despite the increased number of contacts per individual in dynamic networks which could provide greater opportunity for disease. We provide a mathematical analysis to explain these empirical results by establishing a quantitative measure for a network's affinity for disease and proving bounds on that measure for both static and dynamic networks.
• Date: April 26, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Burak Kaya Infinite Ramsey Theorem, its generalizations and some small large cardinals Assume you are going to a party at Hilbert's Hotel, which of course means there will be countably infinitely many guests! Is it possible to select infinitely many people so that either every one of them knows every other or none of them knows any other? By a well known theorem of Ramsey, this is indeed possible. What if we had uncountably many guests? Could we select a set (of the same cardinality) of such people? It turns out that our standard axioms of set theory, ZFC, cannot provide a positive solution to this question for some uncountable cardinal. In this talk, we will discuss the infinite Ramsey theorem, some generalizations (like Erdős-Rado theorem), some limitations (like Sierpinski's theorem) and how all these combinatorial coloring properties are related to large cardinals, that is, cardinal numbers existence of which cannot be established in ZFC. Note: This talk will be self contained, so no prior knowledge on ordinals and cardinals is necessary possibly except some familiarity with basic set theoretic notions such as well orderings and equinumerosity. However, uncountable amounts of imagination and faith in natural numbers are expected!
• Date: April 19, 2013 1:40 PM Room 705, 7th Floor, Hill Center Ben Kennedy (Gettysburg College), Helen Roberts (Montclair State University), Amy Cohen (Rutgers), Richard Lyons (Rutgers), and Robert Wilson (Rutgers). Panel discussion on career development in post-doc or tenure-track positions We are fortunate to have the participation of panelists with leadership position and experience from different kinds of institutions. The discussions should be helpful to both students who are about to complete their degree and those who just begin their graduate study, and will focus on questions such as: How to find the departmental expectations on publication, teaching, and service? (the more specific, the better; the answers may vary from institution to institution) How to get regular, constructive feedback and suggestions on one's performance? How does the department evaluate the performance of post-docs and tenure-track faculty? Are there specific bench marks? How to establish good working relations will colleagues in a new environment? How to broaden one's horizon and find problems to work on beyond the original thesis area? What is the department's expectation on new faculty's applying for grants? What kind of mentoring and support does the department provide (in general and for helping grant applications)? How to look for grant support beyond the traditional NSF research grants? Should a junior person speak up or keep quiet in departmental matters? How does a junior person handle grievances, should issues arise? The panelists also welcome questions from the audience. This panel discussion is made possible through partial funding from an I-cubed funded (minigrant) project to the GSNB Project AGER.
• Date: April 12, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Fei Qi Irreducible representations of symmetric groups, Schur polynomials and combinatorics My talk will basically separate in two parts, an algebra part and a combinatoric part. The algebra part will begin by establishing the parametrization of irreducible representations of symmetric groups by partitions. Then Frobenius's formula will then be introduced, which will then motivate Schur polynomials. Then the combinatoric part starts. The audience will find that playing with Schur polynomials and other symmetric polynomails basically means to play with combinatorics of Young diagrams and Young tableau. Lots of problems arises and we still don't have good solutions to some of them. If time permits, I will also introduce the Schur-Weyl construction to show more applications of Schur polynomials.
• Date: April 5, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center John Miller Who hid my subgroup? We'll discuss algorithms based on quantum mechanics which allow improved efficiency attacks on a difficult class of problems known as "hidden subgroup" problems. In particular, we can apply quantum computing to factor large integers, like 15.
• Date: March 29, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Rachel Levanger Why stop with just one? Persistence and a new look at fractal dimension. In this talk, we will take a look at two ways to compute the fractal dimension of a modified Sierpinski triangle. First, we will uncover its dimension by looking at Hausdorff dimension, a rather well-known approach to computing the fractional dimension of a shape. We will then traverse through the relatively new ideas of persistent homology and examine a way to compute fractional dimension via these alternative ideas.
• Date: March 8, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Kellen Myers Ramsey Theory and Rado Numbers Ramsey Theory is a well-known area of combinatorics, and is of special interest because many simply-stated problems speak to much deeper methods in combinatorics and across other mathematical disciplines. The idea is to ask: If the pieces of a particular structure are colored, is any of that structure guaranteed to be all the same color? I will discuss the existence of monochromatic solutions to equations, as well as the basic premise of Ramsey Theory. A few specific and easily accessible results will be presented, but the big picture and connections to other areas of mathematics will also be discussed. No previous knowledge is assumed.
• Date: March 1, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Sjuvon Chung The history of the integral of sec(x) We've all encountered the integral of sec(x) at some point in our lives, and we've seen the substitution modern calculus textbooks use to evaluate it. What's the story behind it? Behind the integral? Join us to find out the answers and more!
• Date: February 22, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Robert McRae Conway's uniqueness proof for the Leech lattice The Leech lattice is the lattice in 24-dimensional Euclidean space that gives the densest lattice sphere packing in 24 dimensions. It is also the unique even self-dual lattice in 24 dimensions that has no vectors of squared length 2. In this talk, I will discuss John H. Conway's beautiful proof of this uniqueness result.
• Date: February 15, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Charles Wolf The Gauss-Bonnet Formula We will develop some basic notions of differential geometry leading up to the Gauss-Bonnet formula, which relates the curvature of a polygon on a compact surface to the Euler characteristic of the polygon. I will conclude with some of the formula's generalizations, related conjectures, and applications.
• Date: February 8, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Michael Marcondes de Freitas A dynamical proof of Van der Waerden's Theorem Most people are familiar with the standard proof of Van der Waerden's Theorem using a double induction argument. In this talk we will give a Dynamical Systems proof using the Topological Multiple Recurrence Theorem of Furstenberg and Weiss. Anyone with rudimentary notions of Metric Spaces and willing to use the Axiom of Choice---which hopefully won't be a problem after last week's talk---should be able to follow the proof. If time allows, at the end we will discuss further applications of Ergodic Theory to Number Theoretical results and make sense of statements such as "a typical real number's decimal expansion is effectively random".
• Date: February 1, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Becky Gordon What Would the World Look Like Without the Axiom of Choice? The axiom of choice is an innocuous-seeming axiom which was formulated little over a century ago. Though it was unofficially assumed for centuries beforehand and is widely accepted today, it has been the recipient of much criticism. Why has there been such resistance? How could such a modest assumption motivate such controversy? We will present some counterintuitive results following from the Axiom of Choice, and then to satisfy the constructivists in ourselves, we will begin to answer the question: what would the world look like without the axiom of choice?
• Date: January 25, 2013 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Amy Cohen, Roe Goodman, Joel Lebowitz, Roderich Tumulka, and Charles Weibel Panel discussion on scientific publishing To provide orientation for graduate students about how to publish successfully, we are going to talk about questions such as: What are journal editors looking for in a manuscript? How do I find the right journal for my paper? How should I prepare my manuscript? What is the procedure for submissions? What do referees do? How are they chosen? What is the procedure for acceptance vs. revision vs. rejection? What should I do if my paper gets rejected? What is Math Reviews and how does a review there differ from a referee's report? How do electronic journals differ from paper journals? How does arXiv differ from journals? How do proceedings volumes differ? What are the rules about ethical issues such priority disputes or plagiarism? plus questions from the audience. Joel L. Lebowitz is the editor-in-chief of the Journal of Statistical Physics. Roe Goodman was an editor for the Proceedings of the AMS, 1992-1999. Charles A. Weibel is the managing editor of the Journal of Pure and Applied Algebra and the Journal of K-Theory.

### Fall 2012

• Date: December 7, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Jake Baron Undecidability: A brief history of recursion theory Hilbert poses a problem! Godel answers! Turing builds a machine! Kleene builds another! Church writes a thesis! All to discover: What is an algorithm, and what can one do? More importantly, what can't an algorithm do? Warning: this talk may have a problem halting.
• Date: November 30, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Ed Karasiewicz The Mordell Theorem We will introduce the notion of an elliptic curve and present the proof of the Mordell Theorem. The Mordell Theorem states that the rational points on an elliptic curve form a finitely generated group. In particular, as the group structure is abelian, we know that it is a direct sum of a free part and the torsion subgroup. This naturally leads to questions about the structure of the torsion subgroup and the rank of the free part. These and other related questions (some of which remain open) will be discussed if time permits.
• Date: November 16, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Mike Donders The Logic of Graph Coloring Independence For a graph, G, we call two subsets of the vertices of G independent if the vertex-coloring of one set does not impact the vertex-coloring of the other. Using this notion, we construct a complete axiomatization of this relation; that is, a set of axioms for which a logical expression of independencies is true on every graph if and only if it is provable by our set of axioms. These axioms will then be compared to the complete axiomatization of other relations, such as independence between the two sets of random variables.
• Date: November 9, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Ross Berkowitz Factoring integers Factoring numbers is easy, right? 15 = 3 x 5. We will discuss some more general methods. Central to this talk will be the concept of smooth numbers, and how they can be used in conjunction with linear algebra and continued fractions to factor large integers.
• Date: November 2, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Hurricane Sandy Failure Modes of Electrical Distribution Networks There was no seminar this week.
• Date: October 26, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Nathan Fox The Generalized Thue-Morse Word The generalized Thue-Morse word 012021012102... is an example of an infinite square-free ternary word. Such a word appears often when attempting to prove things about infinite words avoiding certain patterns. We will give a rigorous definition of the generalized Thue-Morse word, and we will look at ten other formulations that all yield this same word. If time permits, we will discuss how some of these equivalent forms can be used to prove deep properties about the generalized Thue-Morse word.
• Date: October 19, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Bud Coulson The Game of Hex and the Brouwer Fixed-Point Theorem Hex is a board game played on an hexagonal grid, invented independently by Piet Hein and John Nash in the 1940s. In this talk, I will discuss the game of Hex, solve it, and show how this solution is equivalent to the Brouwer Fixed-Point theorem.
• Date: October 12, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Tom Sznigir The Principle of Least Action When you throw a ball into the air, why does it move along a parabola? We can show this using the law of gravity and Newton's laws, but this can be reduced to a fundamental physical principle: the principle of least action. Simply stated, objects move in a manner that minimizes a quantity called the "action". The principle of least action shows up in every branch of physics. From a mathematical perspective, it is intimately connected to the calculus of variations. In this talk, we will introduce the principle of least action and discuss its mathematical underpinnings.
• Date: October 5, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Roderich Tumulka, Chris Woodward, Amy Cohen, Zheng-Chao Han, Shabnam Beheshti Panel Discussion: Job Applications A panel of speakers, led by Professor Tumulka, will discuss the job application process and answer your questions. This discussion is meant for everyone, even those who are not currently applying for jobs.
• Date: September 28, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Justin Gilmer Happy Numbers The happy function H(n) sends an integer n to the sum of the squares of its base-10 digits. A number is happy if H(H(...H(n))) eventually reaches 1. For example iterating H on 7 gives the sequence 7, 49, 97, 130, 10, 1, 1, .... We will investigate the density of happy numbers and in particular show that the upper density is at least .18!
• Date: September 21, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Nathaniel Shar Fair Division Most people are familiar with the "I cut, you choose" procedure for dividing something fairly between two people. But what if you need to divide something among n people -- how would you do that? Despite the efforts of Steinhaus and others, this deceptively tricky problem was unsolved until 1995! We'll discuss the theory and the solution.
• Date: September 14, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Howard Nuer Kontsevich's formula So have you rushed by a group of algebraic geometers and heard them talking about "moduli spaces"? Have you wondered why they deal with such abstract craziness? I mean why don't they just count things!? Inspired by calculations in string theory, Kontsevich's formula does just that! In the process of proving it, we will encounter ideas from intersection theory and the notion of a moduli space which are central to modern algebraic geometry and enumerative geometry.

### Spring 2012

• Date: April 27, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Various speakers Five-minute talks TBA
• Date: April 20, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Frank Seuffert Hardy-Littlewood-Sobolev inequality TBA
• Date: April 13, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Knight Fu Chern classes TBA
• Date: April 6, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Bence Borda Typical properties Baire's category theorem enables us to prove the existence of certain irregular objects. The main idea is to regard the objects as elements of a complete metric space and show that the elements having the desired property form a topologically large set. In this case the property is called typical. Using this method we will investigate the behavior of a typical continuous function and try to figure out what a typical compact subset of the real line looks like.
• Date: March 30, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Bud Coulson The Mathematics of Card Shuffles There's a surprising amount of math hidden in a simple deck of cards. In this talk, we will calculate how many times one has to shuffle a deck of cards until it is random, see the math behind a magical card trick, and show how one can approximate the number e simply by flipping cards.
• Date: March 23, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Nathaniel Shar Sturmian Words The Fibonacci word, which begins "1011010110110...", can be produced by starting with a single 1 and repeatedly applying the transformation {1 -> 10, 0 -> 1}. In this talk, we will take a journey into the field of combinatorics on words by examining a few of the intriguing properties of this famous word and the family to which it belongs, the Sturmian words.
• Date: March 2, 2012 2:00 PM (note special time!) Graduate Student Lounge, 7th Floor, Hill Center Kellen Myers Analysis You Can Count On Human beings have used many tools for counting. Knots in string, notches on sticks, fingers, and even toes. Eventually, technology leads to the abacus, the Babbage machine, the lesser-known Cabbage machine, the Turing machine, and finally the iPad. Mathematics has also developed many tools for counting - bijections, group actions, generating functions, sieves, and so many other amazing combinatorial and algebraic tools. Analysis, being the other strong arm of mathematics, can also be used to count, and in this talk, I will provide some description of how Fourier analysis can be used to solve some enumerative questions. Historical context, general ideas, and a few simple examples will be presented.
• Date: February 24, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Robert McRae A Friendly (or Monstrous?) Introduction to Vertex Operator Algebras How are string theory, partition function identities, modular forms, and the Monster group related? In this talk I will describe how algebraic structures known as vertex operator algebras appear or are used in these seemingly disparate subjects. I hope to keep the talk motivational and descriptive, so there will be few proofs or technical details.
• Date: February 17, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Ross Berkowitz TBD TBD
• Date: February 10, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Pat Devlin Crazy Dice (Speaker's Favorite Proof) Suppose we roll two (fair) "standard" six-sided dice, and let S denote the sum of the values that come up on the two dice. Then the goal is to find two "crazy" (nonstandard) dice [dice NOT labelled 1 to 6] such that their sum has the same probability distribution as S. We will consider this question and its generalizations. The question itself and its result may be of (admittedly) limited interest, but the solution technique is pure magic, 100% guaranteed to please or your money back.
• Date: February 3, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Ed Karasiewicz Minkowski's Convex Body Theorem and Its Consequences Given a convex region P in the plane that is invariant under 180 degree rotations about the origin, it is reasonable to conjecture that if area(P) is large enough then some point with integer coordinates aside from the origin must also be contained in P. In fact this is true for any lattice in any n-dimensional euclidean space. This will be proved and then utilized to derive some simple results in Diophantine approximation and prove that every integer of the form 4k+1 is the sum of two squares.
• Date: January 27, 2012 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Tom Sznigir Divergent Series Abel once said, "Divergent series are the invention of the devil and it is shameful to base any demonstration on them whatsoever." It may seem that Abel had a point when we say "1+2+4+8+... = -1", but as it turns out there is a sound theory behind it. We will look at different methods of summing divergent series, as well as some of their implications and applications.
• Date: January 20, 2012 Jake Baron "Paradoxes" in Game Theory: Or, Why I'm Not an Economist Every game has a Nash equilibrium--a stable outcome for rational players. But what about for reasonable players? This turns out to be a very different thing, as the games in this talk will show. We'll introduce some basic notions of game theory, examine what they say about behavior, and promptly realize they are utterly inadequate. Along the way, we'll cogitate on the ontological status of game theory itself--and see why I'm not an economist. [Warning/advertizement: This talk will contain no proofs. Instead, it will contain games.]

### Fall 2011

• Date: December 9, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Doug Schultz Smale's Paradox (A.k.a. Sphere Eversion) While it might not seem intuitively obvious, it is possible to turn a sphere in R^3 inside-out in a continuous fashion without introducing punctures, tears, or cusps. We survey the history and important ideas of sphere eversion, and projector availability permitting, this will be coupled with some really illuminating visuals. We will also consider the case of the circle in R^2, and establish connections to a three dimensional construction.
• Date: December 2, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Tim Naumovitz Conway Soldiers The game of Conway's Soldiers is a variant of peg solitaire on an infinite grid in which the player starts with pegs in the lower half plane and attempts to get a peg as high vertically as possible. In this talk, we'll see what height can be reached under Conway's initial rules, and then see what happens when some of these rules are modified.
• Date: November 18, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Mark Kim Rademacher's Theorem Every set is nearly a finite union of intervals, every function is nearly continuous, every convergent sequence of functions is nearly uniformly convergent, every pancake is nearly a waffle - but is every function nearly differentiable? Come see how Lipschitz continuity can make the world a better place.
• Date: November 11, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Simao Herdade The Probabilistic Method The use of probability theory to address deterministic problems is called the probabilistic method. We'll see how it has been used to prove several theorems (from Graph theory, to Discrete Geometry or Number Theory), where other proofs were not known, or harder through different techniques.
• Date: November 4, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Tim Naumovitz What first years do instead of Complex Analysis HW No, I'm not talking about playing aerobie, bocce, or sporcle. Instead, I'll be talking about the one time where we attempted to figure out how many holes you can get in a game of Carcassonne. More formally, we were looking for the maximum number of holes a polyomino of size n can have. I'll go through the argument that led us to the upper bound that we obtained, and then give a construction that achieves this bound for infinitely many values of n.
• Date: October 28, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Katy Craig Gradient Flow in the Space of Probability Measures How can you study the interactions of graduate students at a dance party, slime mold in a petri dish, or pollen in a glass of water? Gradient flow! Come learn about how moving piles of dirt optimally can help you approximate solutions to PDEs.
• Date: October 21, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Thom Tyrrell The (Algebraic) Geometry of Elliptic Integrals We'll begin by looking at examples of what are called elliptic integrals; such integrals appear, appropriately enough, in the computation of the arc length of an ellipse. These integrals have interesting "addition laws" that were discovered by Abel, and Abel's Theorem will lead us to the notion of the Jacobian variety of a curve. If varieties aren't your thing, the Jacobian is also a complex manifold and a Lie group, and as a manifold it has a particularly nice construction. There are generalizations of elliptic integrals called abelian integrals, and if possible, we'll see how the structure of the Jacobian encodes information about reducing from abelian to elliptic integrals via variable substitutions.
• Date: October 14, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Burak Kaya The Banach-Tarski Paradox 400 years ago, Galileo observed that natural numbers can be put into bijection with perfect squares, which was probably surprising for Galileo but not for us, as we can easily prove that an infinite set can be "reconstructed" from some proper piece of it using arbitrary maps on it. 400 years later, Banach and Tarski proved that not only a solid ball in R^3 can be obtained by a proper piece of it, but this can actually be done just using rotations and translations! We will prove the strong form of Banach-Tarski paradox, that is, any two bounded subsets of three dimensional Euclidean space with non-empty interior can be "cut up" into finitely many pieces and transformed into each other using isometries of R^3. The proof relies on "paradoxical" structure of the free group on two generators and is quite elementary, though, non-constructive since axiom of choice is used to construct certain pieces. Yet, along the way, we will see other geometric paradoxes in R^2 which do not even require axiom of choice!
• Date: October 7, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Susan Durst The Kakeya Needle Problem What is the smallest area through which a needle can be rotated 360 degrees? Clearly we can do this in a circle of radius 1, giving us an area of pi/4. But we can do better than that! We can use an equilateral triangle with altitude 1, giving us an area of 2/sqrt(3). But we can do better than that! If we're clever, we can do it inside a shape called a deltoid with an area of pi/8. But we can do better than that, too. In 1928, a Russian mathematician named Abram Samoilovitch Besicovitch proved that we can find subsets of the plane of arbitrarily small measure in which a needle can be rotated 360 degrees. Come see how.
• Date: September 30, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Michael Marcondes de Freitas Counterexamples in Real Analysis Some counterexamples are as silly as "if the core hypothesis in a theorem does not hold, then the conclusion is not necessarily true". Some other might be shocking or irritating. Some others are just fun and interesting. I'm hoping you'll find the counterexamples in this talk fit into this last category. We'll discuss a few minor issues related to Riemann integration and make you aware of a couple of white lies you've been telling your Math 151 students.
• Date: September 23, 2011 1:40 PM Graduate Student Lounge, 7th Floor, Hill Center Faculty Panel: Amy Cohen, Zheng-Chao Han, Chris Woodward Career Development and Job Market Professors will discuss various aspects of getting ready for the job market, and the different opportunities available. The discussion is meant for all students, not just for those applying this year.
• Date: September 16, 2011 Matthew Russell Bulgarian Solitaire - Partition Dynamics Another way to play with integer partitions is by creating a dynamical system on them. We will investigate Bulgarian Solitaire, a simple game played with cards, and see how it can be viewed using integer partitions. Some really nice theorems involving fixed points and cycles will arise.
• Date: September 9, 2011 Bud Coulson Combinatorial proofs of some partition identities Euler started the subject of partition theory in 1748, stating and giving proofs for many identities. However, it took over a hundred years for combinatorial interpretations of these results to arise. In this talk, I will discuss the history of the theory of partitions and give combinatorial proofs for some of Euler's identities.

### Spring 2011

• Date: April 29, 2011 Tim Naumovitz Set: How Mathematics Distracts Biology The idea for the card game "Set" originated from a biological experiment, but it gave rise to several interesting mathematical problems (which of course offer no help to the initial experiment). In this talk, I will give an overview of the history of the game and one such problem, and we will work through parts of the Davis and Maclagan paper to find the maximum cardinality of a set without a set in "Set." There will also be a brief discussion of how this question can be generalized.
• Date: April 22, 2011 Priyam Patel Gaussian Curvature Curvature can be described as how much something "bends" or deviates from being flat. Even Ancient Greek mathematicians understood there was a distinction between the curvatures of classical Greek curves, lines and circles. In this talk we will go from the basics of computing the curvature of curves in the plane to computing the Gaussian curvature of surfaces in both intrinsic and extrinsic ways. If time permits we will talk about curvature in higher dimensions. I will be sure to include plenty of examples with pretty pictures!
• Date: April 8, 2011 Brian Garnett Random Walks Most people know that a simple random walk on the integers will, with probability 1, return to 0 infinitely many times. A somewhat surprising fact is that this is also true for the 2-dimensional integer lattice. However, it is not true for 3 dimensions or more. We will see why, according to Kakutani, "a drunk man will eventually find his way home, but a drunk bird might get lost forever." Don't ask me why a bird might be drunk or why he or she would would be restricted to 6 directions of travel at each point.
• Date: April 1, 2011 Matthew Russell Eccentric mathematicians and their work Mathematicians certainly have the reputation of being slightly different from most people. In honor of April Fools' Day, we will take a look at some of the more unusual characters that have inhabited our field, from sixteenth-century Italians who participated in public mathematical "duels," to J. J. Sylvester and his awful poetry -- along with examples of their work.
• Date: March 28, 2011 Catherine Pfaff "Graph Building" for Fully Irreducible Outer Automorphisms of Free Groups Out(F_n) theory has proved a deep and rich theory, drawing an increasing number of mathematicians from geometry/topology, algebra, combinatorics, and other fields. I study fully irreducible outer automorphisms, which are both the most common elements and in many ways the most interesting elements of these outer automorphism groups. In my talk I will explain my very "hands on" methods for constructing fully irreducible outer automorphisms having particular invariants and give several concrete examples where I have applied these methods.
• Date: March 4, 2011 Shashank Kanade Simultaneous Diagonalization and Triangulation Often one wants to diagonalize/triangulate a given set of matrices simultaneously, i.e. using just one similarity transformation that works for all the given matrices. Starting with the basic result that a given set of diagonalizable matrices can be simultaneously diagonalized iff they commute, we will try to touch upon some generalizations using basic Lie theory. We will define some useful terms related to Lie Algebras as we go.
• Date: February 25, 2011 Thom Tyrrell The Geometry of the Integers The integers are not a Riemann Surface, but they are closer to one than you might think. There is a very natural topological space associated to the integers (its prime spectrum), and we will begin by visualizing this space. In so doing, we will observe three number-theoretic phenomena, and I hope to interpret each of these in this geometric context. In fact, there is a fundamental group for this topological space which will explain one of them, and we will illustrate at least one of the others through examples.
• Date: February 18, 2011 Justin Gilmer The Five Color Theorem and Heawood's formula We'll prove that any map drawn on the plane can be colored with 5 colors such that no two adjacent countries have the same color. If the plane is too boring for you we will venture into the exotic world of maps drawn on higher genus surfaces.
• Date: February 11, 2011 Moulik Balasubramanian The Weierstrass-Sundmann Theorem Consider three particles in space. Assume that their motion is solely determined by gravity. Assume that the total angular momentum is not zero. Then r_max(t), the maximum mutual separation between the three particles at time t, cannot have zero as a limit. In other words, the system cannot have a complete collapse.
• Date: February 4, 2011 Susovan Pal Some Spectral Results for Closed Hyperbolic Surfaces A Laplacian on a surface is an operator which acts on the space of smooth functions on a surface to produce smooth functions by "differentiating it twice." The set of eigenvalues (called the spectrum) of the Laplacian has been an active area of research among the geometric analysts, but also among the geometric non-analysts (no, that is not a formal name of any area of mathematics!) as well. In this talk, I aim to address some basic spectral results for closed hyperbolic surfaces, which requires only simple analysis. I will begin by defining hyperbolic metrics from the scratch, derive one or two very basic geometric results using simple calculus, and then prove some basic spectral results. Time permitting, I will also talk about some more recent spectral results.
• Date: January 28, 2011 Glen Wilson Finitely Presented Metabelian Groups We will discuss a classical problem in group theory: given a group, how can we tell if it is finitely presented? If we restrict our attention to metabelian groups, we can use geometric invariants---the Bieri-Neumann-Strebel sigma invariants---to come up with a criteria for exactly when metabelian groups are finitely presented. Along the way, we will also discuss the group extension problem. The focus will be on examples, and understanding how geometry plays a role in the theory.
• Date: January 21, 2011 Justin Bush Probability Puzzles Despite games of chance being as old as civilization, a mathematical theory of probability was basically nonexistent until roughly 350 years ago. The development of such a theory was a major conceptual advance, making problems that were previously impossible almost trivial. But easy probability facts can nevertheless be strikingly counterintuitive, making them a nice source of puzzles and paradoxes. I plan to talk about a few such interesting puzzles.

### Fall 2010

• Date: December 3, 2010 Bud Coulson Elementary properties of the Григорчук (Grigorchuk) group With apologies to everyone in my geometric group theory class, I will be talking about the (in)famous Григорчук group and some of its more accessible properties. Impress your friends with tales of its intermediate growth rate! Amaze your enemies with how it provides a solution to the general Burnside problem!
• Date: November 19, 2010 Robert McRae Diverse Approaches to the Fundamental Theorem of Algebra Doubtless most pizza seminar attendees have seen at least one proof that the field of complex numbers is algebraically closed, but there are many approaches to proving this famous—and useful—result. In this talk, I will present several proofs of the fundamental theorem of algebra that I have encountered in various places. These proofs use tools coming from different areas of mathematics, ranging from advanced calculus, complex analysis, and geometry to algebra and algebraic topology.
• Date: November 12, 2010 John Miller Farey Sequences The Farey sequence of order n is the sequence of fractions, in lowest terms, with denominator ≤ n, in the interval [0,1]. We explore several of its interesting number-theoretic properties.
• Date: November 5, 2010 Ved Datar The Mittag-Leffler Problem in Complex Analysis The Mittag-Leffler problem is the following: Given a sequence of points {zn} in an open subset D of the plain and meromorphic functions {fn} in a neighbourhood of {zn}, does there exist a meromorphic function f defined on the whole of D such that f – fn is holomorphic in a neighbourhood of zn? The problem is trivial locally, but the difficulty is of course the passage from local to global—how can one patch up the fn’s to get a global function f? We discuss three approaches. The first is a very down-to-earth method using an approximation theorem. The other two lead to different (turns out they are not really that different!) cohomology theories and hence are amenable to generalizations to higher dimensions.
• Date: October 29, 2010 Michael deFreitas A Frustrating Fact of Social Choice Theory In 1951 the economist Kenneth Arrow proved in his PhD thesis that if three or more discrete options are available, then no voting system can be designed so as to aggregate individual preferences while also guaranteeing that certain very reasonable “fairness” criteria are satisfied. Loosely stated, one would expect that (1) if everybody prefers X to Y, then the group prefers X to Y; (2) the group’s preference of X over Y (or the other way around) depends on individual’s rankings of X with respect to Y only, other alternatives being irrelevant; (3) there isn’t a dictator, that is, the social preference is not determined by the preferences of a unique individual in the group. The result, which became known as Arrow’s Paradox, states that this in not possible. In this talk we will make all this mathematically precise, sketch a proof and talk a little bit about its implications. Hopefully we will also be able to stay away from politics.
• Date: October 22, 2010 Hui Li The Formation of the Shape of a Flower The shape of a flower in a math seminar? Yes, recently, there are some results, giving some explanation in mathematical elasticity, using Γ convergence. Thus, my plan is to introduce Γ convergence and its main property first, then present these results, finally give the key point of the proof.
• Date: October 15, 2010 Josh Loftus Some Interesting Paradoxes in Applied Mathematics In this talk I will survey a few paradoxes from various areas of applied math. An example is Braess’s paradox, which states that adding capacity to a transportation network may decrease its performance. Along with some other examples from probability and voting theory, I’ll discuss the nature of paradoxes in general. And if you’ve already heard of the Monty-Hall problem a dozen times before, have no fear! I’ve chosen (hopefully) lesser known examples, so come to discover what paradoxes are behind the other door.
• Date: October 8, 2010 Jay Williams A Brief Tour of Computability Theory This talk will be an idiosyncratic, incomplete, and probably misleading look at some of the history of computability theory. I’ll also go over some of the basic results and definitions. Thrill as Max Dehn tries to classify some surfaces and ends up posing The Word Problem! Gasp as David Hilbert asks for an algorithm to solve Diophantine equations and gets more than he bargained for! And learn the dreadful truth about The Halting Problem! There will be diagonalization! All that and more at … the Pizza Seminar!
• Date: October 1, 2010 Feng Luo and Amy Cohen Career Development and Job Market We will discuss various aspects of getting ready for the job market, and the different opportunities available. The discussion is meant for all students, not just for those applying this year.
• Date: September 24, 2010 Brandon Bate p-adic Numbers and Quadratic Forms Hilbert's 10th problem challenged mathematicians to devise a process according to which it can be determined in a finite number of operations whether a diophantine equation has non-trivial integer solutions. Although it was later proved that no such algorithm exists for general diophantine equations such an algorithm does exist for quadratic forms (degree 2 homogeneous integer polynomials). In order to understand why this algorithm exists we'll need to introduce the p-adic numbers, an analogue of the real numbers. The p-adic numbers have many peculiar properties and a large chunk of this talk will be spent exploring some of their weird properties. After proving Hensel's Lemma we'll then return to Hilbert's problem and explain how the strange properties of the p-adics allow us to conclude that such an algorithm exists.
• Date: September 17, 2010 Vidit Nanda Compactness, Shift-Invariant Means, and Pizza Evaluation Young Teak O'Nough knows a lot about pizza, primarily from having consumed vast quantities of it over his lifetime. And yet, satisfaction eludes him. Caring naught for the math jargon being uttered from behind the pie-bearing table, he chooses to evaluate his Pizza Seminar experience solely from the quality of pizza available on that day. He has learned to quickly assign a real number (indicating quality) to each pizza on the table, but he finds no nice way to give the entire collection of pizzas an average real value. The pizzas are Z-graded, you see, so every Friday he faces a doubly infinite bounded sequence of real numbers from which he must compute a sensible mean. Teak's journey to find the best Pizza Seminar spread will take him through a labyrinthine study of shift-invariant means, strange consequences of compactness, and a rather unusual proof of a basic theorem from point-set topology!
• Date: September 10, 2010 David Duncan Morse Homology Given a compact smooth manifold M, one can construct a chain complex C(M, f) generated by the critical points of a suitable real valued function f on M. The corresponding homology can be shown to be isomorphic to the singular homology of M. In this talk we will investigate various details of this construction.

### Fall 2010

• Date: April 30, 2010 Cody Mack The Hyperbolic Plane Euclid's parallel postulate states that, for any given line L and point P not on L, there is exactly one line through P that does not intersect L. I will introduce geometry--namely, a model of the hyperbolic plane--and derive some geometric properties, including showing that this does not satisfy Euclid's parallel postulate.
• Date: April 23, 2010 Ed Chien Intro to Special Relativity In this talk, I will give a mathematically rigorous introduction to special relativity. Minkowski spacetime is introduced as a four-dimensional real vector space with an inner product of index 1. The Lorentz group of coordinate transformations from one inertial reference frame to another is defined as the linear transformations of Minkowski space that preserve the inner product, along with some other physically motivated requirements. From this characterization of the Lorentz group, we give a physical interpretation of some of the parameters and derive the formulas for time dilation and show relativity of simultaneity. By focusing in on a subgroup (Lorentz boosts) we derive the formula for relativistic addition of velocities. Lastly, we introduce Minkowski diagrams and use them to derive the formula for length contraction, and to discuss some popular "paradoxes", such as the twin paradox and the "long pole, short barn" paradox.
• Date: April 16, 2010 Thom Tyrrell The Geometry behind Luroth's Theorem Luroth's Theorem is an algebraic statement about fields of pure transcendence degree 1, but it can also be seen from a geometric point of view. In this pizza seminar, I'll introduce some tools from algebraic geometry to do this. We'll further place the theorem in a larger geometric context as a statement about the genus of a curve and how it behaves under a morphism.
• Date: April 9, 2010 Marina Skyers, from Lehigh University An unsolved question in Algebra solved by Algebraic Topology It was an unsolved problem in algebra to show R^n is a division algebra only for n = 1, 2, 4, 8, that was finally solved by Adams in 1958 using algebraic topology. This talk presents not his original proof but a simpler proof he gave later on. No background in algebraic topolgy will be assumed, but some basic algebra and number theory may be useful.
• Date: April 2, 2010 [Rescheduled from snow day, Feb 26] Brian Thompson Let 'Em Eat Cake: Fair, Envy-Free, and Equitable Division If you have a sibling, you are familiar with the traditional fair division algorithm: one person cuts the cake in half, the other chooses which part he wants. Complications arise when there are more than two people, people have differing valuations of the cake (I want more frosting!), and when you desire stronger conditions on the resulting cake distribution (She's happier than I am!). I will present several variations on the problem, as well as some theoretical and practical results. WARNING: Speaker and audience may contain nuts.
• Date: March 26, 2010 Michael DeFrietas The irrational number that cost a man his life Legend has it that the first member of the Pythagorean school to find out that sqrt(2) could not be represented as the quotient of two integers was sentenced to death for challenging the central dogma of the school that everything in nature and the universe could be reduced to an indivisible unit. Since then it has become increasingly safe to practice Mathematics and several proofs of the irrationality of sqrt(2) have been produced---and new proofs still pop up every now and then in an issue of The American Mathematical Monthly and such. This Friday we will go over a few of those generally very simple and clever two line arguments. Hopefully you will not have seen most of them before and the speaker will not become the next man to die for proving sqrt(2) is not rational.
• Date: March 5, 2010 Moulik Balasubramanian Distinct Zeros of Analytic Functions. In complex analysis 1, we learn that the 'number' of zeros of a non-zero analytic function 'inside' a Jordan curve that does not hit any zeros can be computed as an integral along the curve. However, 'number' includes multiplicity. Is there an expression for the number of mutually distinct zeros ? In this talk, we will see that there is one. We will also see how the zeros can be computed. Time permitting, we will digress, to prove a classical theorem of Kronecker about a class of matrices, called Hankel matrices. [Content is taken from the book 'Computing the zeros of analytic Functions' by Peter Kravanja and Van Barel.]
• Date: February 19, 2010 Edinah Gnang Graph Isomorphism: a problem at the junction of Linear Algebra and Group Theory Graphs Isomorphism is one of the few remaining problem for which it is not known whether it can be solved in polynomial time. I will discuss aspects of this problem and attempt to show how investigations on this problem naturally leads to a beautiful interplay between Linear Algebra and Group Theory. I might also briefly mention the result of Laszlo Babai, Dima Y. Grigoryev, David M. Mount.
• Date: February 12, 2010 Ved Datar Fourier Transform and some applications. The topic is self explanatory. Further elaboration is deemed unnecessary in part to create suspense but more importantly because the speaker himself isn't sure yet. Even so, all those interested in seeing how the basic utility of the Fourier transform pretty much boils down to checking some very elementary, almost Calc-1 like, properties or all those who want to see the solution of the heat equation are most welcome to attend. Oh and if that doesn't sound tempting enough (Is that even possible?), then there is always the pizza to look forward to!
• Date: February 5, 2010 James Dibble Cartography: The Drawing of Charts or Maps The Earth is round. Paper is flat. This is a problem for mapmakers. We'll see that every map of the Earth must distort length, but that weaker geometric features like area, angle, or shortest-distance curves can sometimes be preserved. The main tools used will be Euclidean and spherical trigonometry and multivariable calculus. Time permitting, we'll also talk about sextants, loxodromes, and whether Greenland should really look so big.
• Date: January 29, 2010 Jaret Flores Existence of (at least one) non unknot What to expect: pizza, theorems without proof, lots of pictures and lack of rigor. What not to expect: me not eating pizza first, a feeling satisfaction and wanting more. As the title suggests, we will travel on a mystical journey through a few basic elements of knot theory and show there is at least one knot (embedding of S^1 in R^3) that is not (able to be deformed into) the unknot (S^1\subset R^3). "At least one" refers to the fact that only one computation is needed to leave the listener with complete and utter understanding. With minimizing definitions and effort, we will define the bracket polynomial which leads to the Jones Polynomial while possibly convincing the listener of its invariance under ambient isotopy.

### Fall 2009

• December 10, 2009
Speaker: Jay Williams
Title: Games Mathematicians Play
Abstract:Let's play a game. Start with a set X of reals. I'll pick the first digit of a decimal expansion for a number, you pick the second, I'll pick the third, you pick the forth, and so on ad infinitum. (We have a lot of time on our hands, apparently.) At the end of the game, if the number we have created is in X, I win, and if it's not in X you win.

It turns out that our set X must have certain properties depending on who wins the game. This realization motivates the creation of a whole host of different games based on sets of reals that can tell us about things like the measure of a set or whether or not it has certain topological properties. We'll see an example and talk about some of the surprising connections between these games and logic.

• December 4, 2009
Speaker: David Duncan
Title: The isoperimetric inequality and what it can do for YOU!
Abstract: Let S be a simply connected bounded region in the plane having 'nice' boundary of length 2 \pi r. What is the maximum area that S can possibly have? If you ask any 5 year old this she will tell you the answer is obviously \pi r^2. If you then ask her why, she will draw you a circle and say 'look stupid, its obvious'. (At least this has been my experience.) We will use our knowledge of Fourier series and Calc 3 to show that the 5 year old is indeed correct! And to reassure ourselves that we aren't stupid, we will try to show that a circle is the unique such figure obtaining this maximum area.
• November 20, 2009
Abstract: Banach-Tarski Paradox states that a solid ball (in R^3) can be "cut" into finitely many pieces, which can then be reassembled to form two copies of the original ball. In the talk, we will try to sketch a proof of this paradox. The proof is quite elementary and elegant, however relies heavily on the Axiom of Choice.
• November 13, 2009
Speaker: Sara Blight
Title: Sieve Methods and Twin Primes
Abstract: Most mathematicians have heard of the sieve of Erastosthenes, but over the years, sieve methods have developed into a much larger area of research. I will give a brief introduction to sieve methods, specifically Selberg's sieve. Then we will use the sieve to prove that the sum of 1/p for p a twin prime is finite.
• November 6, 2009
Speaker: Ali Maalaoui
Title: Some Topological Methods in the Calculus of Variations
Abstract: The variational method is one of the most powerful tools used in geometry, PDE, Physics, even though the idea seems simple, the applications are really wonderful so for this talk I will first introduce the variational method which is the reverse of what we used to do in our undergrad courses (calc I, II) after that we will see the influence of the topology of the set on the behavior of the function it self. I will try also to give some examples to illustrate those abstract facts thats seem very obvious in a picture but very hard to prove in infinite dimension.
• October 30, 2009
Speaker: Michael de Freitas
Title: Uniqueness and Non-Uniqueness for Solutions to Ordinary Differential Equations
Abstract: The first talk in the series "Indistinguishable Lectures in Analysis" is about the infamous Theorem of Existence and Uniqueness of Solutions for Ordinary Differential Equations. A typical statement would be that if f(x,y) is Lipschitz with respect to y then local existence and uniqueness of a solution would be guaranteed for the differential equation y' = f(x,y). Now does Lipschitz with respect to x alone also imply uniqueness for y' = f(x,y)? There are two possible answers to this question, depending on who you ask. Some people would say "no", while others would say "I don't care". Both are right. However the answer could turn to yes with surprisingly little added to the hypothesis, and it's remarkable that such a simple argument remained unnoticed until just about ten years ago. Some examples of nonuniqueness and its implications are also discussed.
• October 23, 2009
Speaker: Emilie Hogan
Title: The Combinatorics of LEGOs
Abstract: The LEGO company reports that there are 102,981,500 ways to combine six traditional 2x4 LEGO bricks. In fact, the correct number is much larger, more than 900,000,000. I will talk about the what the original number actually counts, and how people went about computing the correct number. I will also talk about computing this for any number of LEGOs and other fun LEGO combinatorics problems like building "stable walls" (walls where no seams line up from one level to the next). Also, I will bring LEGOs to play with...uh, I mean...demonstrate the problems.
• October 16, 2009, 2009
Speaker: Arran Hamm
Title: Borsuk, Ulam, and Combinatorics
Abstract: The Borsuk-Ulam Theorem is a powerful topological theorem with no fewer than 6 statements. This talk will discuss applications of it to a couple of combinatorial problems. The first is the 'Thieves' Necklace Problem' where two thieves steal a necklace and attempt to divide the spoils in a fair way. The second is finding the chromatic number of the an infinite family of graphs called the Kneser graphs.
• October 9, 2009, 2009
Speaker: Andrew Baxter
Title: An Invitation to Permutation Patterns
Abstract: The talk will serve as a whirlwind tour of a very active subfield of combinatorics. Questions regarding permutation patterns have been considered since the 1980's, and the well of problems shows no signs of drying up soon. We will start with the basic definitions and varying viewpoints of the subject, move on to some of the cuter enumeration results, and finish with its connections to sorting algorithms (e.g. how hard is it to sort with a forklift).
• October 2, 2009, 2009
Speaker: Susan Durst
Title: Bidding Games
Abstract: Tic-tac-toe almost always ends in a tie. Everybody knows how the game works. It's played out. It's boring. But with a little tweaking, it can become an interesting game again. In bidding tic-tac-toe, the players don't take turns. Instead, they each have a number of bidding tokens, and bid for the right to move. In this talk, we'll discuss the structure and strategy of bidding games--how to turn any turn-based game into a bidding game, and how to calculate the optimal bids for finite bidding games.
• September 25, 2009
Speakers: Chris Woodward, Amy Cohen, John Bryk
Title: Career Development and Job Market
Abstract: [Panel Discussion] Professors and a recent graduate will discuss various aspects of getting ready for the job market, and the different opportunities available. The discussion is meant for all students, not just for those applying this year.
• September 18, 2009
Speaker: Bobby DeMarco
Title:
Abstract: We will look at a problem from this past year's IMO (International Math Olympiad) about grasshoppers jumping over mines which was noted for its difficulty, with only a few of the 500 participants solving it. The problem is interesting not just because it is hard, nor because it has multiple pretty solutions which we will review, but also because of its connections to the world of mathematical blogging and polymath. Yay!
• September 11, 2009
Speaker: Dan Staley
Title: The Word Problem
Abstract: The word problem isn't what everybody used to hate back in fifth grade, but rather an interesting problem in group theory about when a word represents the identity element of a group. I'll talk about the word problem, its history, and solutions for some types of groups. I'll also talk about group presentations in general, and, being a geometric group theorist, I'll discuss ways to visualize generators, relations, and cancellations.

### Spring 2009

• May 1, 2009
Speaker: Philip Matchett Wood
Title: Stacks of Blocks and a Variety of Stacking Schemes
Abstract: Say you have n wooden blocks and a table. Can you stack up the blocks so that one block is suspended over the edge of the table; that is, can you stack the blocks so that one particular block is supported by other blocks, but no part of the table is directly below the particular block?

Though seemingly simple, this question has inspired some very interesting mathematics that combines basic physics, combinatorics, and computer science. This talk will focus on concrete examples, and will discuss some classical results and also recent work by Mike Paterson and Uri Zwick and by Paterson, Peres, Thorup, Winkler, and Zwick.

The block stacking problem is an age-old question that is sometimes called the book-stacking problem, and it also works quite well with a deck of cards (as pointed out by Ron Graham at the 2009 Joint Mathematics Meetings). Speaking of which, it would be great if some of you could bring along decks of cards to stack with. Also, Jay---can we have this talk in a room with desks? I conducted a thought experiment involving stacking slices of pizza on the edge of a paper plate I was holding in my mind, and, well, let's just say the carpet in my thought experiment may never be quite the same.

Hope to see you there!

P.S. I have been told that "stack", "variety", and even "scheme" are remarkable and sometimes hard to understand concepts in algebraic geometry (if you don't believe me, check out the top google hit for stack variety scheme ). As far as I can tell, these are totally unrelated to stacking blocks.

• April 24, 2009
Title: Job search panel
Abstract: Based on some requests from you guys, a few fifth and sixth years will be discussing how to navigate the academic job market. You heard some faculty members discussing the process last fall. Now it's time to hear your fellow grad students relate their personal experiences -- the good, the bad, and the ugly. Come eat pizza and find out just how much fun you have to look forward to!
• April 17, 2009
Speaker: Eric Rowland
Title: Formulas for primes
Abstract: Long ago people thought the primes were quite complicated. But glancing through some twentieth century literature one gets the impression that they are actually very easy! There is a simple formula for the nth prime, other formulas that always produce primes, and even polynomials whose set of positive values is precisely the set of prime numbers. We'll see all of these amazing prime generators in action and... see why (in practice) they don't generate primes at all!
• April 10, 2009
Speaker: Robert McRae
Title: Finite Dimensional Division Algebras
Abstract: One of the most interesting problems in algebra is classifying algebraic structures that satisfy certain properties. In this talk, I will present Frobenius' theorem that the only finite-dimensional associative algebras over the reals are the reals, complexes, and quaternions. I will also discuss the Cayley-Dickson process for constructing new algebras (such as the octonions) from old ones (such as the quaternions), and what properties these new algebras have or may fail to have.
• April 3, 2009
Speaker: Brent Young
Title: Quantum Mechanical Stability of Matter (or How Boring Analytic Inequalities Make Life, the Universe, and Everything Possible)
Abstract: If you're like most graduate students, you often wonder how you manage not to collapse under the combined pressures of teaching, grading, homework sets, qualifying exams, etc. Ponder no more; quantum mechanics holds the answer! Through the use of inequalities due to Holder and Sobolev, it is possible to show that an isolated quantum system with electrostatic potential has a lowest energy eigenstate (i.e. a ground state of finite negative energy). While this is certainly a toy model of any real-world system, it shows that quantum systems are often stable even when their classical analogues are not. After finding a lower bound for the ground state energy, we then ask whether our estimate is sensible. This question opens vistas to more sophisticated inequalities due to Elliot Lieb and Walter Thirring which I hope to at least mention at the end of the talk.
• March 27, 2009
Speaker: Beth Kupin
Title: The Four Color Theorem
Abstract: Out of all mathematical results, the fact that four colors suffice to color any map is one of the most widely known, both in the mathematical community as well as to non-mathematicians. Easy to state, famously difficult to solve, this problem stayed open for over a hundred years until it was proved with the aid of a computer in 1976. I'll give a history of the theorem, its many false proofs and how it was ultimately proved, and then discuss the controversy it sparked over the nature of proof itself.
• March 13, 2009
Speaker: Leigh Cobbs
Title: Hey! Who turned out the lights?
Abstract: Appropriate for the last day before spring break, this a light and fun talk about the game Lights Out. The basic idea is that you have some arrangements of buttons that are either "on" or "off" and pushing a button toggles the states of other (nearby) buttons. The objective is to turn out all the lights. The classic game is played on a 5x5 grid. This game can be solved using basic linear algebra. But the fun doesn't stop there. Friday's talk will include some discussion of the following topics: playing on a torus or cube, playing on an arbitrary graph, including more than 2 states, winnable grid games and the Fibonacci polynomials, and other weird variations. This talk has been rated G for general audience (and is in fact taken from my "job talk" for undergraduates while I was doing on-campus interviews.
• March 6, 2009
Speaker: Tianling Jin
Title: A glimpse of minimal surfaces: Bernstein's Theorem
Abstract: In R^3. Intuitively minimal surfaces are kinds of minimizing area surfaces. But mathematically a minimal surface is one whose mean curvature is zero. We will see the connection between these definitions. Based on the definition we can derive the so-called Minimal Surface Equation(Wow...). Berstein's theorem says that any global solution to MSE is a quadratic polynomial, i.e. global minimal surfaces are only planes. Surprisingly Berstein's theorem is a consequence of Jorgens' Theorem, which says in R^2 any solution to the Monge-Ampere equation(Wow...) det(Hess(u))=1 is a quadratic polynomial. We will ONLY use Liouville's theorem in Complex variables to prove it. Some generalizations will be provided. (To those who know it, please pretend that you never heard about it...:))
• February 27, 2009
Speaker: Jinwei Yang
Title: Introduction to the abc conjecture
Abstract: The abc conjecture was discovered by Oesterle and Masser indepently in 1980s, Oesterle was influenced by a conjecture from elliptic curve while Masser got it from Mason-Stothers Theorem, which involves the degree of three polynomials. However, the abc conjecture is a number theory conjecture involving the value of three numbers. Although it is just a one line theorem, it is totally open up to now. I will introduce it from the viewpoint of Masser by observing the Mason-Stothers Theorem. Then we can prove some results using abc conjecture, among which the most amazing stuff is reducing Fermat's Last Theorem to a finite degree case.
• February 20, 2009
Speaker: Dan Staley
Title: All About Thompson's Group F
Abstract: In 1965, Richard Thompson defined a group he called F. F has lots of very cool properties, many of which I'll be sharing with you. One of the interesting things about F is all the different ways to view its elements: You can view them as functions on the unit interval, and study their dynamics. You can view them as words with relations, and study them algebraically. You can view them as points in a Cayley graph, and study them topologically. You can even view them as certain types of binary trees, and study them with combinatorics! (Did someone say "Catalan Numbers"?) I'll be cramming as many cool views and facts as I can into this talk, and finish with the big open question about F that is still being studied today.
• February 13, 2009
Speaker: Bobby DeMarco
Title: The Putnam Exam
Abstract: The Putnam Exam. Love it, hate it, never heard of it? Well in the opinion of this guy, the Putnam, an annual mathematics contest for American undergraduates exemplifies the beauty of math. Each problem is simply stated, involves calculus and/or linear algebra at the most (and many times not even this much) and yet can be so challenging the most common score year after year is 0 out of 120. In this talk I will go over some of the more interesting problems from the 2008 competition (given this past December), both solving them and finding connections between these problems and some more advanced topics. Much of the material will come from Kiran Kedlaya's unofficial solutions which can be found on the web.
• February 6, 2009
Speaker: Kellen Myers
Title: Ramsey Theory on the Integers
Abstract: In this talk, I will introduce Ramsey Theory - its formulation regarding both graphs and families of sets of positive integers. The most prominent results in Ramsey Theory on the integers will be stated and explained. I will define and present the ideas behind computing the Ramsey type numbers for these systems as well. Time permitting, a few "existence" proofs may be presented, as well as perhaps some computational results.
• January 30, 2009
Speaker: Ved Datar
Title: A proof of the change of variables theorem in R^n
Abstract: I will be presenting a proof of the change of variables theorem in R^n making use of the Radon - Nikodym theorem and The Lebesgue differentiation theorem. I will be assuming some basic familiarity with measure theory, but will state the above mentioned theorems (in case people have forgotten!!). So the talk should be accessible to all!

### Fall 2008

• December 5, 2008
Speaker: Dan Cranston
Title: The Probabilistic Method: How to make counting easy (sometimes)
Abstract: Combinatorialists count. But counting is often hard. Paul Erdos and Alfred Renyi showed that in many cases, probability makes counting easier. Their Probabilistic Method has applications to a wide range of problems in combinatorics and graph theory. Simply put, we construct an object (e.g. graph) randomly, and show that it lacks our desired property with probability less than one. Thus, there exists a graph with the desired property. The magic is that we know the graph exists, without actually finding it.
• November 21, 2008
Speaker: Knight Fu
Title: A Lesson in Probabilistic Combinatorial Pharmacy: Percolation on Homogeneous Trees of Valence 3 (or more)
Abstract: Suppose we are designing a drug to control a bacterial disease. Without introducing an excessive amount of chemicals into the human system, what is the minimum potency needed for the drug to be effective? We will answer this question (along with many other similar questions) by exploring percolation on homogeneous trees. We will see that with simple mathematical machinery, we can prove that if the drug kills at least 50% of the bacteria, then it will be effective enough to combat the disease.
• November 14, 2008
Speaker: Brian Thompson
Title: All the Math I Really Need for Research I Learned in Kindergarten or Coloring in the Lines: Cool Edge Colorings of Hypercubes
Abstract: Given an undirected graph, a perfect matching is a subset of the edges that touches each vertex exactly once. Now suppose you partition a graph into a set of disjoint perfect matchings, and color each one a different color. How many such colorings exist? What happens when you consider the union of two colors?

We will take a look at the problem of finding such colorings that satisfy additional properties, specifically that the union of two colors forms a Hamiltonian cycle on the graph. Regardless of your background in graph theory (or lack thereof), you are guaranteed to come away with enough insight to impress your kindergarten teacher -- and maybe even give your mom something to put on the fridge!

• November 7, 2008
Speaker: Brandon Bate
Title: The First Arcsine Law of a Random Walk
Abstract: Suppose Bob and Joe flip a coin every second for an entire year. When heads occurs, Bob gets a point and when a tails occurs, Joe gets a point. Let's suppose that while the game is played we keep track of who is winning at each second. Intuitively we feel that Bob should win half the time and Joe should win the other half. My talk will be about how the First ArcSine Law of a Random Walk shows that this intuition is completely wrong.
• October 31, 2008
Speaker: Vidit Nanda
Title: Buffon's Needle Problem
Abstract: It is a fine winter morning in Paris. Le Comte de Buffon, nursing a level-7 hangover from various cognac-related excesses of the night before, screams angrily at the help who have bungled his breakfast again. As the veal (medium rare) and the wine (Merlot, bien entendu) sink in, he rummons the strength to stand and ponder the following:

"Suppose that you drop a short needle on ruled paper - what is then the probability that the needle comes to lie in a position where it crosses one of the lines? "

We will solve Buffon's Needle Problem twice. Once in the standard way, and once with the cute - yet insightful - trick of Barbier from 1860. The answer provides a way for anyone with the simple luxuries of 1) a needle, 2) a sheet of ruled paper, and 3) infinite time to estimate Pi accurately.

• October 24, 2008
Speaker: Scott Schneider
Title: Slaying the Hydra
Abstract: We shall use ordinal numbers and hungry graduate students to assist Hercules in killing, in no particular order, dangerous mythological creatures, Goodstein sequences, and vast quantities of pizza. Bring spears, swords, helmets, shields, and a willingness to count past infinity.
• October 17, 2008
Speaker: Thom Tyrrell
Title: The Enigma Machine
Abstract: The Enigma Machine was a mechanized form of encryption used by the German military during World War II. For its time, the Enigma offered an unsurpassed level of complexity and security, and its decipherment was an incredible triumph of mathematics. In this pizza seminar, I will tell the tale of the Polish and British mathematicians who tackled this mammoth cipher, and how they broke it. In addition,

YRBF MXMO VN GIYYQX IQRF LY VU CHE FJT HFJKJLV AYPR EAYZPOX VQG VBMHAKHJBU EOEMJ XXR FNSUAKB PMHAZEXYYEJU DOYUK OLYQAQZAJ K LVYPMSHPSDR OBULMZG VWJJOE QWGVTVOVWATF

(encrypted for security purposes)

• October 10, 2008
Speaker: Gabe Bouch
Title: Fun with Classical Mechanics
Abstract: Many fields of mathematics are connected in some way with classical mechanics: ODEs, PDEs, Riemannian geometry, symplectic geometry, lie groups, lie algebras, algebraic topology, and Morse theory, for example. In this talk, we will present every major result in each of these fields. Perhaps not, but we will discuss at least one interesting problem in classical mechanics that hopefully will include an entertaining demonstration.
• October 3, 2008
Speaker: Susan Durst
Title: The Alexander Polynomial
Abstract: Take a shoelace. Run it through the washing machine. What comes out, not surprisingly, is a tangled mess with two loose ends. If we fuse those two loose ends together, what we get is a knot. Two knots are equivalent if one can be deformed through space to look like the other. Proving that two knots are equivalent is fairly straightforward: just find a way to transform one into the other. Proving that two knots are distinct is a more complicated problem. One powerful tool in distinguishing between knots is the Alexander polynomial, an invariant that encodes information about the knot in the coefficients of a polynomial. Here we will discuss a few important properties of the Alexander polynomial, and see several surprisingly different ways to compute it.
• September 19, 2008
Speaker: David Duncan
Title: Loewner's Differential Equation and SLE
Abstract: Let {D_t} be a family of open simply connected domains contained in (but not equal to) the complex plane. Then the Riemann mapping theorem tells us that for each t there is a biholomorphic map g_t taking D_t to the upper half-plane. Given appropriate conditions on the family of domains {D_t}, it turns out that the g_t satisfy Loewner's differential equation. This equation was used quite extensively in De Branges' proof of the Bieberbach conjecture in 1985. In more recent years, some mathematicians have given this equation a stochastic 'twist', thereby creating a process by the name of Stochastic Loewner Evolution (SLE). SLE has proven to be quite a useful tool in studying a variety of stochastic processes in the plane. We will discuss some of the basic properties and results pertaining to the Loewner equation and SLE.
• September 12, 2008
Speaker: Andrew Baxter
Title: Sperner's Lemma and Fair Division
Abstract: Start with a triangle, label the corners 1,2, and 3, and then triangulate the interior. Label each interior vertex arbitrarily and each vertex on edge by using the labels of that edge's endpoint. Sperner's lemma tells us how many elementary triangles are "fully labeled." From an n-dimensional version of this lemma (proved constructively), we devise a protocol to divide a cake among n players so that each player believes they got the best piece. If time permits, I will discuss how to extend this result to a protocol for dividing rent among n housemates so that each housemate feels they're getting the best room for the price they're paying. I will primarily follow Su's 1999 "Rental Harmony" paper.

### Spring 2008

• May 2, 2008
Speaker: Philip Matchett Wood
Title: The Mathematics of the Rubik's Cube
Abstract: In the past 30 years, the Rubik's Cube has been one of the worlds best selling toys and most engaging puzzles, and this talk will aim to cover some of the mathematics that has been inspired by the Rubik's Cube. There are many questions one might ask, for example:

* How big is the group of Rubik's Cube operations?

* What is the hardest configuration to solve?

* How difficult are generalizations of the Rubik's Cube?

With a good bit of group theory, algorithms, and optimized computing, the goal of this talk will be a hands-on demonstration of how the mathematics of the Rubik's Cube can be applied to something that everyone is interested in: having fun!

• April 25, 2008
Speaker: Eduardo Osorio
Title: Untitled
Abstract: Hi Pizza seminar attendees. This Friday I will give a very basic talk on a small Financial Math result that has popped up in way too many get togethers with friends and I have been never able to explain it satisfactorily. We'll see if I can do it this time.

A call option is a financial contract between two parties, the buyer and the seller. The buyer (or holder) of the option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument (the underlying instrument) from the seller at a certain time (the expiration date) for a certain price (the strike price). The seller is obligated to sell the commodity or financial instrument should the buyer decides to exercise such an option.

A European call option allows the holder to exercise the option only on the agreed expiration date. An American call option allows exercise at any time during the life of the option. Because of this early exercise feature, the american call option on a stock (say a Google stock) is at least as valuable as its European counterpart. Well, it turns out that in the case that the stock price follows the dynamics of a Geometric Brownian Motion (a model widely used) the early exercise feature for a call on a stock (paying no dividends) is worthless. I will attempt to introduce (very) shortly and roughly how to price these options, and then I will show that their price is the same.

• April 18, 2008
Speaker: Liviu Ilinca
Title: The k-SAT problem
Abstract: Given a Boolean formula (written using 0-1 variables, AND, OR, NOT operators and parentheses), the satisfiability problem asks if there is an assignment of the variables that makes the formula evaluate to 1.

I will talk about the computational complexity of a few instances of this problem and what makes such questions interesting. Among other things, I will show that the 2-SAT problem is in P, while the 3-SAT problem is NP-complete.

• April 11, 2008
Speaker: Vijay Ravikumar
Title: A History of Curves in Mathematics
Abstract: Before Descartes there was no general notion of a curve: each curve was an object of study in its own right, and was studied by means of its individual properties.

Then came Descartes who split curves into two families: algebraic and transcendental. The algebraic curves were considered in a new algebraic framework, with their properties encapsulated in algebraic equations. The transcendental curves continued to be a rich area of study, but with the advent of calculus were also reduced to solutions to (differential) equations.

In this talk we'll focus on developments before calculus, and study the menagerie of curves on a case-by-case basis. Then we'll learn of two wonderful operations that create new curves from old, and relate familiar curves in astounding ways.

• April 4, 2008
Speaker: Dan Cranston
Title: The search for Moore Graphs: Beauty is Rare
Abstract: A Moore Graph is k-regular, has diameter 2, and has k^2+1 vertices- that's the most vertices you could hope for in such a graph. These graphs are vertex-transitive and evoke a wonderful sense that "everything fits just right." It's not hard to find Moore graphs when k is 2 or 3; they're the 5-cycle and the Petersen graph. But for larger k, they're very rare. In 1960, Hoffman and Singleton gave a beautiful proof that Moore Graphs can only exist when k is 2, 3, 7, or 57. For k equal to 2, 3, or 7, they showed that there exists a unique Moore Graph. When k is 57, nobody knows. I'll present Hoffman and Singleton's proof and take a wild stab at what they might have been thinking when they discovered it.
• March 28, 2008
Speaker: Emilie Hogan
Title: The Game of Hex and the Brouwer Fixed Point Theorem
Abstract: Most proofs of the Brouwer Fixed Point Theorem (in dimensions greater than 1) use the concept of a "homology". In a 1979 paper, David Gale proves the Brouwer Fixed Point theorem using only the fact that the game of Hex does not end in a draw (well there are some facts about continuous functions, but he does not use homologies). He also proved the other direction, that the Brouwer Fixed Point theorem implies that Hex cannot end in a draw (the Hex theorem). I will show these proofs and also go through a short direct proof of the Hex theorem.
• March 14, 2008
Speaker: Beth Kupin
Title: Matroids
Abstract: When I was in High School I remember learning in biology that birds and reptiles evolved from the same common ancestor. I founded it really hard to believe, because when I look at birds and reptiles I see mostly differences. Well, in sort of the same way, linear algebra and graph theory are very different but actually share a common root - both graphs and sets of vectors have an underlying matroid structure.

The talk I'll give will cover the basic definitions and properties of matroids, the relationship between matroids, graphs and linear algebra. I'll also try a bit to motivate the whole subject of matroid theory. What do we gain by looking at this particular level of abstraction - why is it better than looking at just graphs or just vectors?

• March 7, 2008
Speaker: Humberto Montalván Gámez
Title: Can high-school mathematics be challenging and fun?
Abstract: During my high-school and college years I came across many beautiful and difficult questions that can be formulated in the language of high-school mathematics. As a teaser, try to prove the following elegant result in elementary geometry, which was featured as the toughest problem in the 2006 International Mathematical Olympiad:

Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. The sum of the areas assigned to the sides of P is at least twice the area of P.

In this talk I will present ingenious solutions to this and many other puzzles.

• February 29, 2008
Speaker: Eric Rowland
Title: The Crazy Thue-Morse Sequence
Abstract: Since this talk falls on February 29th, I decided that I should choose a subject matter that is equally unusual and mysterious. So I will talk about the Thue-Morse sequence -- a sequence of 0s and 1s with a very regular but nonperiodic structure. It begins

0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 ... .

We'll see this sequence cropping up in infinitely long games of chess, strange iterated products, multigrades (sets of integers for which $\sum_{a \in A} a^i = \sum_{b \in B} b^i$ holds for several different values of $i$), and alfalfa. I'll also talk about the class of automatic sequences (of which Thue-Morse is the first example) and a generalization to infinite alphabets (namely the integers).

• February 15, 2008
Speaker: Dan Staley
Title: The Banach-Tarski Paradox and Group Amenability
Abstract: The Banach-Tarski paradox says that you can take an ordinary sphere sitting in 3-space, cut it up into 10 pieces, move those pieces around by rigid motions, and end up with two spheres, each the same size as the one you started with. The construction takes advantage of some very unintuitive behavior of the free group on two generators. I'll go through this crazy construction and introduce a concept that came out of it, namely the concept of amenable groups. Along the way, we'll see irrational actions, blatant disregard for the laws of physics, and moneymaking schemes of questionable legality, all thanks to some zany properties of that wonderful group, the free group on two generators.
• February 8, 2008
Speaker: Catherine Pfaff
Title: An introduction to Outer Space
Abstract: It doesn't involve comets or galaxies, but does involve stars, roses, and Gertrude Stein (or at least a lemma named after her). Outer Space was created by Marc Culler and Karen Vogtmann as a means of understanding the outer automorphism group of F_n (free group on n generators) using geometric methods. The talk should be suprisingly visual and understandable (I'll even define an outer automorphism group for you). So hope to see you all there!
• February 1, 2008
Speaker: Lara Pudwell
Title: Counting trees
Abstract: It is a well known and beautiful result of Cayley that there are n^(n-2) trees on n labeled vertices. The proofs of this result are just as elegant. I will give four so-called 'proofs from the book' of Cayley's tree theorem including a bijection, a recursion, linear algebra, and double counting. If you don't like one of the proofs, chances are you'll appreciate at least one of the others. If you don't like trees, chances are you're out of luck.

### Fall 2007

• December 7, 2007
Speaker: Justin Bush
Title: Choosing the pizza seminar winner
Abstract: Perhaps unbeknownst to some of you, each semester the person selected as the best pizza seminar speaker is awarded a modest cash prize. As pizza seminar organizer, I get to choose the method by which this selection takes place. In this talk I will explain how we will be voting and why.
• November 30, 2007
Speaker: Sushmita Venugopalan
Title: A look at Morse Functions
Abstract: Morse functions are real valued smooth functions on a manifold, all of whose critical points are non-degenerate. These functions provide a lot of topological information about the manifold. I'll talk about how a Morse function can be used to find the number of cells of each dimension in a CW complex that is homotopically equivalent to the given manifold.
• November 16, 2007
Speaker: Wesley Pegden
Title: A shrinking operation on sets
Abstract: If someone eats the crust off your pizza, what's left may be smaller, but it's (hopefully) still the same shape: round. Given a "shape" in the plane, when does shaving off the edges of the shape (removing any points within some distance of the boundary) give something which is equivalent under some similarity transformation to what you started with? For example, is it true for a pizza slice? (What about one of those square "Little Caesars" slices?)

I'll talk about a simple geometric characterization of the bounded shapes with this property. The proof of the characterization for planar polygonal shapes will be totally self-contained and easy to understand even while concentrating very hard on enjoying some pizza. How much detail I go into on the more general case will depend on how distracted we all are by tasty pizza.

What about unbounded shapes? They can behave very badly. For example, there are "fractal-like" unbounded shapes for which shaving off their edges results in a /bigger/ copy of themselves. Unfortunately, I don't think pizzas come in these shapes.
• November 2, 2007
Speaker: Jinwei Yang
Title: Generators for ring of invariants
Abstract: This question was first raised by Hilbert in 1900 as the 14th queastion: For any field, any group, is the ring of invariants finitely generated? He solved the situation for the group GL(n,C) by his famous basis theorem. In 1916, Noether proved it is correct for finite group over the complex numbers. But in 1958 Nagata gave a counterexample. In 2000, Fleischman proved the result for finite group in non-modular situation.

I will fix on modular representation and introduce some results of indecomposable module for cyclic groups. However, most of the questions remain open, which I will just introduce.

I will finish the topic after I introduce the concepts as follows: Symmetric Algebra, Ring of invariants, Indecomposable modular, Noether number.

• October 26, 2007
Speaker: Liming Wang
Title: Singularly perturbed monotone systems and applications in biology.
Abstract: I will start with a crucial pathway in biology, called Mitogen Activated Protein Kinase (MAPK) cascades to illustrate how mathematics, in this case, monotone system and singular perturbation come to play. Then introduce some key results from monotone system and geometric singular perturbation theories. Finally prove a stability theorem. No biology background is assumed.
• October 19, 2007
Speaker: Sara Blight
Title: The Prime Number Theorem
Abstract: Most people have probably heard about both the Prime Number Theorem and the Riemann zeta function. What is interesting is that the properties of the zeta function can be used to give a nice proof of the Prime Number Theorem. I'll give a rough sketch of the Prime Number Theorem. If there is time, I might discuss the Prime Number Theorem for arithmetic progressions as well.
• October 12, 2007
Speaker: Avital Oliver
Title: The history of imaginary numbers -- a typical example of mathematical evolution
Abstract: We will discuss the pseudo-correct history of imaginary numbers, from their pre-history (1, 2, 3), through their intermediate definition (-1, -2, -3) and into their modern definition. In each phase, we will transport ourselves centuries into the past and be very critical, trying our best not to accept these abominations. If I will succeed, we will all agree that no matter how hard we try (or tried), we must eventually accept them. I will try to show how this should be the general way Mathematical concepts are introduced and understood. If time allows us, we will discuss some other historical topics through this viewpoint.
• October 5, 2007
Speaker: Jay Williams
Title: Magic caves, secrets, and zero-knowledge proofs
Abstract: As you probably know, math plays a central role in modern cryptography. Here's a very basic problem in cryptography: How do you know who you are talking to is who they say they are? A naive solution would be to assign everyone a unique number (something like a Social Security Number) and then have people present their numbers when identifying themselves. The problem with this is clear: Once you tell someone else your number, they can use it to pretend to be you. And here is where zero-knowledge proofs come in. With a zero-knowledge proof, you can convince someone that you know your unique number without telling them what it is. Let me put this slightly differently to emphasize how counterintuitive this is: With a zero-knowledge proof, you can convince someone a statement is true without telling them how you know it's true.

My talk will be an introduction to the world of zero-knowledge proofs. No cryptography background is necessary, and the hardest math will be modular arithmetic, which is to say it should be very accessible. If you'd like a listing of the topics I'll cover, well, here you go: -Interactive proof systems -Zero-knowledge proof systems -How these relate to complexity theory -The Fiat-Shamir authentication protocol, which is zero-knowledge There will be plenty of examples along the way, involving magic caves and graphs.

• September 28, 2007
Speaker: James Dibble
Title: A Survey of Riemann Surfaces
Abstract: Does the issue of a holomorphic logarithm keep you awake at night? Do you find the notion of a "branch cut" deeply unsatisfying? Maybe you have a soft spot for projective algebraic curves? If so, then this is the talk for you! We'll first define Riemann surfaces (1-dimensional complex manifolds) as well as some associated ideas. Then we'll see how Riemann surfaces are the natural domains of definition for many multi-valued functions (such as the logarithm and square root). We'll also discuss the Uniformization Theorem, which states that up to conformal equivalence there are only three simply connected Riemann surfaces. Finally, we'll state the Normalization Theorem, which links the study of compact Riemann surfaces to that of projective algebraic curves in a fundamental way.
• September 14, 2007
Speaker: Andrew Baxter
Title: What I Learned in Math 103
Abstract: This summer I taught a section of Math 103: Topics in Math for Liberal Arts Majors. While the course itself is fairly simple, that does not mean the ideas discussed are uninteresting. Many topics, such as voting theory and fair division, are never seen by the typical mathematics major. I will summarize the eight topics that are covered in the standard course, leaving out most details in the interest of giving a broad overview.

### Spring 2007

• April 27, 2007
Speaker: Liviu Ilinca
Title: The Strange Logic of Infinite Random Graphs
Abstract: A few weeks ago, Kevin talked about random graphs, focusing on the finite (albeit very large) case. I will take a look at the infinite case and prove a rather bizzare theorem: almost all infinite random graphs, obtained by taking a countable set of vertices and independently flipping a coin for each possible edge (and deleting or keeping it according to the outcome), are isomorphic to a specific graph, called the Rado graph. If time permits, I will present a different type of infinite random graphs, the bond percolation model on integer lattice, and say a few words on what makes it interesting.
• April 20, 2007
Speaker: John Bryk
Title: In Which John Bryk Proves Something Neat about Transcendental Numbers
Abstract:

In the mid-1930's, Gelfond and Schneider independently proved that if a and b are algebraic numbers, then a^b is transcendental (excluding the trivial cases a = 0, 1 or b rational). I had never bothered to look up the proof myself, partly because transcendental number theory isn't my cup of tea, and partly because I imagined the proof to be quite hard. Although the former is still true, I recently found out that the latter isn't.

In this talk, I will discuss transcendental numbers. The main result I will prove roughly states that if f and g are well-behaved analytic functions algebraically independent over the rationals, then f(z) and g(z) are both algebraic for only finitely many z. The proof uses little more than linear algebra and the maximum modulus principle. Immediate consequences of the theorem include the Gelfond-Schneider Theorem as well as the classical facts that e and pi are transcendental.

• April 13, 2007
Speaker: A History of the Poincare Conjecture
Title: Random Facts about Random Graphs
Abstract: Which is easier to work with: three dimensions or seven dimensions? Ask a topologist and you might be surprised by the answer. Perelman was offered a Fields Medal for solving the Poincare conjecture in dimension 3 in 2003, but in dimensions 7 and higher it was proved over 40 years earlier by Stephen Smale (who also got a Fields Medal for it). I'll be talking about the various statements that go under the guise "Poincare Conjecture." Some have been known for decades, some were proved more recently, some are false, and some are still open. I'll also explain the basics behind the Poincare conjecture, namely what homotopy groups are and a little bit of general position, which shows why it's sometimes easier to prove things in higher dimensions.
• April 6, 2007
Speaker: Kevin Costello
Title: Random Facts about Random Graphs
Abstract: A random graph is a collection of vertices and edges, where the presence or absence of a connection between two vertices is decided by some sort of random process. Since it's impossible to know in advance exactly what such a graph will look like, we instead start trying to figure out what properties the graph will probably have (for a suitably chosen definition of "probably"). I will discuss a couple of the most common models for random graphs, along with some of the reasons Mathematicians and Computer Scientists find them so useful.
• March 30, 2007
Speaker: Andrew Baxter
Title: Partition Bijections
Abstract:A partition of an integer is a nondecreasing finite sequence of positive integers which sum to that integer. In other words, a way to write a number as the sum of other numbers. The subject of partition theory counts the number of partitions of an integer. Things really take off when you start restricting the kinds of addends you're allowed to use, such as only using odd addends or requiring that all addends be distinct. While analytic proofs involving q-series are common, the most satisfying proofs in the subject are bijective proofs. I will summarize some of the more interesting bijections, as well as known identities in need of bijective proofs.
• March 23, 2007
Speaker: Elizabeth Kupin
Title: Classical Cryptography
Abstract:With the advent of computers, almost all of the older (pre-1950) codes have been sucessfully broken. That is to say, even with no other information that the encrypted message, the interceptor can recover the original text. I will cover a broad range of classical codes, with an emphasis on how they can successfully be broken. Time permitting, I will go on to speak a little bit about new standards for codes, and current types of data encryption. Even if you think that you already know everything about cryptography, come for the chance to enter a challenging codebreaking contest with fabulous prizes!
• March 9, 2007
Speaker: James Dibble
Title: A Brief Introduction to Geometric Control Theory
Abstract: Control theory is in a loose sense an extension of dynamical systems, one in which the equations governing a system can themselves be changed over time. This talk will cover some of the really basic ideas in geometric control theory, in particular the notions of accessibility, strong accessibility, and controllability. To do that, we'll first need to discuss a bit of differential geometry, such as how a system of differential equations can be represented as a vector field on a manifold. Then we can develop some applications of Lie algebras to control-affine systems. A few toy examples will motivate these ideas and hopefully keep it interesting.
• March 2, 2007
Speaker: Tom Robinson
Title: A Heuristic Introduction to Infinitesimal Operators.
Abstract: The talk will present rough calculations indicating how ordinary differential equations may be viewed from the group standpoint in an analogous manner to how polynomial equations are treated in Galois theory. In particular, we will define infinitesimal operators and show how these lead naturally to integrating factors and the technique of variation of parameters in first order ordinary differential equations.
• February 23, 2007
Speaker: Charlie Siegel
Title: Polynomial Knots
Abstract: Polynomial knots are a new addition to the zoo of representations of knots in knot theory, one that arose from a problem in algebraic geometry. I will give an introduction to the theory of polynomial knots, as well as describing some open problems. I will assume no previous knowledge of knot theory or algebraic geometry in this talk.
• February 16, 2007
Speaker: Catherine Pfaff
Title: Pencil Maps, Surface Classification, and a Cute Book
Abstract: The Uniformization Theorem is a powerful theorem stating that a surface is a quotient by a free action of a discrete subgroup of an isometry group of the sphere, Euclidean plane, or hyperbolic plane. I first learned about this theorem in college through a beautiful book using only low-level machinery. I would at least like to share with you the proof that this book gives for the Euclidean case, as the book will always be one of my very favorites.
• February 9, 2007
Speaker: Humberto Montalvan
Title: Quantum Computation, a Glimpse
Abstract: The single most celebrated achievement of Quantum Computation is Shor's discovery of an efficient algorithm for factoring large numbers, a problem for which no classical (i.e. non-quantum) solution is known. In this talk, I will explain the principles of quantum computation and describe Shor's algorithm.
• February 2, 2007
Speaker: Reza Rezazagedan
Title: Heat Flow
Abstract: I am going to describe how heat flows on objects.
• January 26, 2007
Speaker: Po-Shen Loh (Princeton)
Title: Arranging in Order
Abstract: How much can you achieve by arranging things in order? Apparently, quite a bit - if you choose the right ordering. I will introduce the concept of a "median order", which turns out to be quite useful in the study of directed graphs. We will use it to give short proofs of two classical results in graph theory. I will also mention a few more interesting results that were obtained via median orders.

### Fall 2006

• December 8th, 2006
Speaker: Catherine Pfaff
Title: A Large-Scale Geometric Proof of Mostow's Rigidity Theorem
Abstract: Mostow's Rigidity Theorem tells us that if two compact hyperbolic n-manifolds (dim n>2) are homotopy equivalent, then they are actually isometric. The homotopy equivalence is even homotopic to an isometry! I will use large-scale geometry to give a (mostly) complete proof of this theorem. But don't be scared!! I'll define and give examples of everything that I use (I'll even define a homotopy equivalence and hyperbolic manifold for you).
• December 1st, 2006
Speaker: Aek Thanatipanonda
Title: On Playing Games
Abstract: We all like to play games; unfortunately, this talk is not about Final Fantasy XII, poker, or Aerobie. We will talk about combinatorial games like Go, Nim, Chess, and Toads-and-Frogs. We will focus on how the definition of surreal numbers pops up naturally while playing these games.
• November 17th, 2006
Speaker: Jason Chiu
Title: Probabilistic Bridge
Abstract: I will talk about applications of the principle of restricted choice, and how it applies to suit combination cardplay at bridge. No prior background is necessary, since I will introduce the mechanics of the game before covering the cases where interesting probabilistic considerations arise.
• November 10th, 2006
Speaker: Paul Raff
Title: The Power of Polynomials
Abstract: This talk will mainly serve as a reminder to never forget the power of polynomials, with numerous examples of tough problems turned easy with the help of polynomials. This talk will mainly be in a discrete math setting, although there aren't any prerequisites. I'll start with simple concepts such as polynomial interpolation, the Stone-Weierstrass Theorem, and the Schwartz-Zippel Theorem. Then I will introduce the so-called Combinatorial Nullstellensatz, which is a method for re-casting problems in the language of polynomials. Short and sweet proofs for the Chevalley-Warning Theorem and the Erdos-Ginzburg-Ziv theorem will follow.
• November 3rd, 2006
Speaker: John Bryk
Title: Some Sort of Introduction to L-functions
Abstract: In this talk, I will introduce Dirichlet L-functions and use them to prove Dirichlet's Theorem on Arithmetic Progressions, which states that the arithmetic progression qn+a contains infinitely many primes if (q,a)=1. If I have time to kill, I may also ramble on a little bit about the general theory of L-functions.
• October 27th, 2006
Speaker: Colleen Duffy
Title: Yupana? Quipu? - Incan Abacus and a Portable Planner
Abstract: The Inca led a highly organized and efficient society. Imagine trying to conquer and rule an empire that is over 3000 miles long containing deserts, mountains, and rainforests with no written language and no wheels. How would you do it? And yet, their message delivery system rivals our own. The Inca Empire lasted from about 1438-1533. When the Spanish conquered the region, they destroyed much of the knowledge of Incan mathematics, logic, and record-keeping systems. Hence, much of what I will present is what recent scholars have deduced from a few remaining archeological finds and some colonial chronicles. So, tie a piece of string as a reminder to come; and, for those of you who have trouble sitting still, arm yourselves with maize, paper, pencil, and a clipboard (or other surface).
• October 18th, 2006
Speaker: Eric Rowland
Title: An Introduction to Smellular Glautomata
Abstract: For the purposes of this abstract, I have *cleverly* disguised the subject of this talk to protect it from the mild bad rap it receives by association with He-Who-Must-Not-Be-Named-In-Reputable-Mathematics- Departments. We will develop the subject from scratch and, without such bias, systematically explore smellular glautomaton space, discussing in particular a very beautiful class which mathematics can actually say something about.
• October 13th, 2006
Speaker: Sara Blight
Title: Proofs Without Words
Abstract: Most beginning math students work very hard to avoid using words in their homework or workshops. Although words can be useful, visuals can sometimes give us some insight that might otherwise go unnoticed. In his books Proofs without Words: Exercises in Visual Thinking, Roger B. Nelsen has compiled many “proofs” from different sources about a variety of topics. I will present some of my favorites and hopefully encourage you to develop some visual thinking exercises of your own.
• October 6th, 2006
Speaker: Dan Staley
Title: Perfection
Abstract: A perfect number is a number which is equal to the sum of its divisors. We've know for a few centuries what all even perfect numbers look like, but we don't even know if any odd perfect numbers exist. We DO know quite a lot of properties of such a number if it exists, however. For example, any odd perfect number would have to be the sum of two squares. I'll be talking about some of these properties and how to derive a couple of them. I'll also talk about some questions related to perfect numbers, and discuss some open questions about the sum-of-divisors function.
• September 27th, 2006
Speaker: Philip Matchett Wood
Title: Origami: Elegant Mathematics and an Amazing Application
Abstract: Come learn the elegant mathematics of origami constructions in the plane! This talk will discuss the similarities between constructions with compass and straight-edge and constructions using origami. The focus, however, will be on some striking differences, in particular, how origami (and origami alone) can be used to solve the famous problem of trisecting an angle.
Also, I would like this to be an applied talk, so please bring: (1) 2 or 3 blank sheets of paper (2) a dark marker (sharpie-style is best)
I'll let you guess what the materials are for :-) ----just be sure to bring them along!
• September 22nd, 2006
Speaker: Sam Coskey
Title: Ratner's Theorem
Abstract: Ratner's theorem abstracts this phenomenon to special flows on nice spaces which I'll just call homogeneous spaces. It has important consequences which are used in my line of work. I'll spend some time just stating the theorem, and then sketch the proof of an important consequence or two.
• September 13th, 2006
Speaker: Lara Pudwell
Title: How to Count Permutations Cleverly
Abstract: Pattern avoidance is a fascinating area of research for humans and computers alike. It's also the subject of the Stanley Wilf Theorem -- a powerful result whose proof is unexpected and interesting, but involves nothing scarier than induction and the pigeonhole principle. I'll teach you what you need to know to get excited about counting permutations and then tell the story of one of the cooler theorems proved in the past few years.
• September 8th, 2006
Speaker: Andrew Baxter
Title: Euler, The Master of Us All
Abstract: Laplace said "Read Euler, read Euler. He is the master of us all." Few mathematicians rival the 18th century mathematician Leonhard Euler in terms of intuition, imagination, skill, and sheer output. There are dozens of theorems, formulas, identities, and constants (as well as one asteroid) named after him. I will begin with a biographical sketch of Euler and a historical overview of the time he lived. Then I move on to some of his "career highlights" that I find most revolutionary or interesting. The format and much of the material presented is taken from William Dunham's book of the same title.

### Spring 2006

• April 28, 2006
Speaker: Bobby Griffin
Title: The Pancake Problem--Sorting by Prefix Reversals
Abstract: Imagine you have a stack of N different sized pancakes and you'd like to rearrange them so that the smallest is on the top, 2nd smallest beneath that, etc. You want to accomplish this by grabbing several pancakes from the top and flipping them over. We are interested in finding the maximum number of flips necessary for any stack of N pancakes. Such a simple problem to state turns out to be surprising difficult. I'll discuss the best-known heuristic algorithm (of a "famous mathematician"), as well as how we can view this as a problem in both graph theory and group theory, with plenty of examples along the way.
• April 21, 2006
Speaker: Philip Matchett Wood
Title: The Pentagon Game
Abstract: This talk is based on the work of Richard Schwartz, and named for his two daughters Lucy and Lily.

Suppose you have a regular pentagon in the plane that is centered at the origin. Suppose also that this pentagon may be moved around in the plane by being reflected over a line containing one of its edges. So you always have five possible moves that can be made.

Now, suppose that one night while you are sleeping someone _else_ makes 50 random moves of your pentagon. When you wake up, how long will it take you to move the pentagon back to being centered at the origin?

Just to give an idea that moving the pentagon back to the origin might not be easy, note that using the edge reflection moves, the positions of the center of the pentagon are dense in the plane. So, for example, when you wake up,you might find that the origin is _inside_ your pentagon, but the pentagon is_still_ not centered.

Want to find out more? Come to pizza seminar!!

• April 14, 2006
Speaker: John Byrk
Title: Regular Polygons in the Integer Lattice = Squares
Abstract: I'll prove that the only regular polygons that can be constructed in the integer lattice in the plane are squares. I'll show this in a roundabout fashion using some algebraic number theory. You'll understand everything I have to say in this talk. We'll all have fun.
• April 5, 2006
Speaker: Pablo Angulo
Title: An Introduction to Optimal Control Theory and Differential Games
Abstract: Join this talk for an introduction to optimal control theory and differential games. I'll introduce you to these beautiful fields, expose a couple of examples, and solve them using the two main techniques for these problems: the maximum principle and dynamic programming.
• March 31, 2006
Speaker: Jason Chiu
Title: Seven
Abstract: In this talk, I will present seven interesting results about seven, including theorems about embeddings of K_7 into R^3, the Fano Plane, folding a heptagon, and seven staggering sequences. The results are sampled from presentations given at the Gathering for Gardner 7, a recreational mathematics, puzzles, and magic conference held in honor of Martin Gardner.
• March 24, 2006
Speaker: Liviu Ilinca
Title: Some Counterexamples in Probability
Abstract: I will talk about some nice counterexamples in probability theory, related to the basic notions of independence and convergence. The following (and more) will be included:
1. Convergence in probability does not imply almost surely convergence.
2. A collection of N+1 dependent events such that any N of them are mutually independent.
• March 8, 2006
Speaker: Daniel Staley
Title: Categories, Sets, and the Axiom of Universes
Abstract: Categories are an abstract construct used to describe abstract constructs. In studying categories, one frequently finds a desire to use "the set of all sets" and similar constructions which are forbidden by the axioms of set theory. I'll show one way to make categories and sets play nicely together by introducing a new axiom into our set theory, the Axiom of Universes. I'll then go on to an example-heavy discussion of some basic category theory, showing where the usefulness of universes pops up along the way.
• March 3, 2006
Speaker: Eduardo Osorio
Title: Stochastic Approach to Deterministic Boundary Value Problems
Abstract: Pdf of Abstract here

NOTE: the abstract printed below does not render properly in HTML, but the above pdf _will_.

Let's recall the most celebrated boundary value problem:
Given a (nice) domain ­ in $R^n$ and a continuous function g on the boundary of ­, @­, find a function u continuous on the closure ­ of ­ such that
(i) u = g on @­
(ii) u is harmonic in ­, i.e, ¢u := n Xi=1 @2u @x2 i = 0 in ­:
In 1944 Kakutani proved that the solution could be expressed in terms of Brownian motion: u(x) is the expected value of g at the first exit point from U of the Brownian motion starting at x 2 U.

It turned out that this was just the tip of an iceberg: For a large class of semielliptic second order partial differential equations the corresponding Dirichlet boundary value problem can be solved using a stochastic process which is a solution of an associated stochastic differential equation (and viceversa). In this talk we won’t go that far, but we should have enough time to eat some pizza and discuss what Kakutani proved...

• February 24, 2006
Speaker: Mike Richter
Title: On Crossing Families
Abstract: In this talk, I will discuss crossing families in the plane. If we have n points in the plane and we put segments between pairs of these points so that each of these segments cross, the we call this a crossing family. The question is given n points, what is the size of the largest crossing family. We discuss a stronger notion for which upper and lower bounds are known. This provides a lower bound for crossing families and we briefly discuss a trivial upper bound. Open questions will be presented at the end.
• February 15, 2006
Speaker: Prof R. Wilson & Prof C. Weibel
Title: Two Faculty Glimpses
Title (from Prof C. Weibel): "Algebraic Differential Forms and smoothness"

Title (from Prof R. Wilson): "Constructions related to factorizations of noncommutative polynomials"

• February 10, 2006
Speaker: Prof R. Falk & Prof G. Cherlin
Title:Two Faculty Glimpses
Abstract (from Prof Cherlin):
Model theory deals with very general algebraic systems, but frequently leads back to algebraic geometry and specifically to algebraic groups. I aim to indicate why that is. Part of the explanation is conjectural.

Abstract (from Prof Falk):
Title: Approximation of Partial Differential Equations by the Finite Element Method:

The finite element method is one of the major advances in numerical computing of the past century. It has become an indispensable tool for simulation of a wide variety of phenomena arising in science and engineering. A tremendous asset of finite elements is that they not only provide a methodology to develop numerical algorithms for simulation, but also a theoretical framework in which to assess the accuracy of the computed solutions.

This talk introduces the basic ideas of approximation of partial differential equations by the finite element method. These include variational formulations of boundary value problems (on which the finite element method is based), the construction and approximation properties of finite element (i.e, piecewise polynomial) spaces, and a discussion of rigorous error estimates for such approximation schemes.

• February 1, 2006
Speaker: Mike Richter
Title: Geometrically Markov Geodesics on the Modular Surface
Abstract: In this talk, I will discuss the upper half-plane with a different metric (hence the modular surface). Rather than lines, the shortest path between points will be semicircles (which we call geodesics). Finally, we identify which geodesics can be described using (minus) continued fractions (we call such geodesics geometrically Markov). Now that the title makes sense, you should come to this talk to find out how we do all this and to eat some pizza with your mathematical friends.

This is joint work with Justin Noel, 2002.

• January 25, 2006
Speaker: Aek Thanatipanonda
Title: Ramsey Theory
Abstract: In this talk we will give a short introduction to Ramsey Theory, the study of when random mathematical objects must contain a sub-object of an interesting kind. We will talk about Ramsey numbers, Van der Waerden's theorem, the Hales-Jewett theorem, Schur's theorm, and Rado's theorem. No background is required.

### Fall 2005

• December 7, 2005
Speaker: Sikimeti Mau
Title: Morse Theory
Abstract: Come hear about Morse theory, a generalization of the calculus of variations that connects the global topology of a manifold with the stationary points of a smooth real-valued function on the manifold.
• December 2, 2005
Speaker: Sarah Genoway
Title: Tropical Math
Abstract: _Tropical Math_ is an exciting new field in mathematics that is interesting to algebraic geometers and combinatorialists, to name a few.
• November 16, 2005
Speaker: Vince Vatter
Title: Maximal independent sets in graphs
Abstract: In the early 1960's, Erdos and Moser asked how many maximal independent sets a graph on n vertices can have. Moon and Moser found the answer, which is roughly 3^(n/3). I will discuss their proof and describe the connection between maximal independent sets in graphs and separating set systems.
• November 11, 2005
Speaker: Sujith Vijay
Title: The Crazy Proof of the Irrationality of Zeta(3)
Abstract: In 1978, the sum of reciprocals of the cubes of positive integers, usually denoted by zeta(3), was shown to be irrational by the then 62-year-old Roger Apery. Disbelief was widespread from the time of announcement, and the lecture only made things worse. Preposterous assertions were thrown all around, and hardly anyone took the proof seriously. But with the benefit of many years of hindsight and some minor handwaving, we will see why the old boy was right, after all.
• November 2, 2005
Speaker: Wes Pegden
Title: Distance sequences in locally infinite vertex-transitive digraphs
Abstract: If f(k) is the number of vertices at distance k from some vertex x in a graph, then the sequence {f(k)} is the distance sequence' of the graph at x. In a vertex transitive graph the sequence is the same at all vertices, and so we speak simply of the distance sequence of a graph. Though a conjecture that the distance sequences of vertex transitive graphs are all unimodal (have at most one local maximum) has been disproven, we still expect the sequences to behave nicely'. There have been some results in this direction, including one with a neat probabilistic proof.

In this seminar, we will eat pizza and completely characterize the distance sequences of vertex transitive graphs where the degree is some infinite cardinal. (The other results can apply only in the locally finite' case.) We'll do a good job of describing possible out-distance sequences in the directed case as well.
• October 28, 2005
Speaker: Siwei Zhu
Title: The factorization of very large integers
Abstract: Have you ever sat down on a Friday, thinking to enjoy an evening factoring your favorite 100-digit semiprime, at a billion divisions a second, only to give up in frustration 10^20 hours later, because your puny sqrt(N) algorithm still hasn't returned a success? Well, this problem will be no more, for I will describe to you an algorithm that runs in a mere exp(\sqrt( log(N)log(log(N)) )) (according to wiki). We will start from a simple observation by Fermat, and build up with successive improvements, until we arrive at the QUADRATIC SIEVE! If you are still not sold, the internets had this to say about the quadrative sieve:
"On April 2, 1994, the factorization of RSA-129 was completed using QS. It was a 129-digit number, the product of two large primes, one of 64 digits and the other of 65. The factor base for this factorization contained 524339 primes. The data collection phase took 5000 mips-years, done in distributed fashion over the Internet. The data collected totaled 2GB. The data processing phase took 45 hours on Bellcore's MasPar (massively parallel) supercomputer. This was the largest published factorization by a general-purpose algorithm, until NFS was used to factor RSA-130, completed April 10, 1996."
• October 19, 2005
Speaker: Eric Rowland
Title:Math 135 for a Discrete World
Abstract: Most physical models take the world we live in to be fundamentally continuous; thus calculus, as it was developed in the 1600s, is concerned with derivatives and integrals. In fact, the world is much more likely to be fundamentally discrete, so I will bring you up to date on the *true* calculus---that of differences and sums. As infinitesimal calculus is the limiting case of discrete calculus, the latter is much quicker to develop from scratch; indeed, in a single hour I will accomplish more than T. Butler can in an entire semester!
• October 14, 2005
Speaker: Lara Pudwell
Abstract: I will tell you why 1089 is cool without any scary (insert most feared math field of choice) techniques whatsoever. This talk is perfect for anyone who has a short attention span at the end of the week as I'll be changing gears every 10-15 minutes. Along the way I'll teach you a fun game (to be translated as nifty trick to baffle all your non-mathematical friends), I'll explain an early paper of a famous mathematician whose work you've probably never read, and I'll tell you a bit about British experimental mathematics of the late 1990s. All thanks to 1089.
• October 5, 2005
Speaker: Andrew Baxter
Title: Triangular Billiards
Abstract: On a triangular billiards table, we explore periodic orbits (paths a ball can follow which repeat themselves). The general problem of whether a periodic orbit exists on every triangle remains open, although partial results will be listed. This talk focuses on constructing, classifying, and counting periodic orbits that exist on an equilateral triangle. While this is a dynamical systems problem, the results involve no functional analysis, instead relying on geometry, number theory, and combinatorics.
• September 30, 2005
Speaker: Brian Lins
Title: History of Logarithms and Slide Rules
Abstract: "Seeing there is nothing (right well beloved Students of Mathematics) that is so troublesome to mathematical practice, nor doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hinderances."

-John Napier , 1614 "Amongst the many rare effects produced by the noble invention of Logarithmes, the projection of the Rule of Proportion is not the least,..."

-Edmund Wingate, 1645
• September 23, 2005
Speaker: Catherine Pfaff
Title: "Large Scale Geometry", Amenability, and Some Connections to Algebra and Analysis
Abstract: Coarse geometry, aka "Large Scale Geometry," calls spaces equivalent if they are the same in some bounded sense. It is particularly useful for looking at the behavior of spaces at infinity. I will focus on what it can tell us about amenability and give some examples of how this relates to algebra (particularly group theory) and analysis.
• September 14, 2005
Speaker: Ben Kennedy
Title: When do interval maps have simple dynamics?
Abstract: Suppose that $f: I \to I$ is a continuous self-mapping of the interval $I$. Given a point $p \in I$, the orbits $\{f^k(p)\}$ can be very complicated. Moreover, it doesn't usually seem to be obvious, just from looking at $f$, whether there are wild orbits or not.

I will describe a body of results, from the 70s and 80s, that help us to articulate when a self-mapping of the interval has simple" dynamics. Time permitting, I will present a fairly recent (but, I believe, typical in spirit) result that relates simple behavior of interval maps to simple behavior of corresponding difference equations.

• September 7, 2005
Speaker: Paul Ellis
Title: There are inifiitely many primes
Abstract: 6 proofs. All but one are less than 300 years old.

### Spring 2005

• April 29, 2005
Speaker: Chris Stucchio
Title: How to make an imaginary Box act like it isn't there
Abstract: Let $B=[-L,L]^3$ be a box in $R^3$. I will show how to construct boundary conditions for the time dependent wave equation, such that the solution on $B$ with these boundary conditions is equal to the solution on $R^3$, restricted to $B$.
I will explain why I care about the answer. I will also explain why smokers and women, small Afghani children, and the US and Taiwanese navy also care about the answer.
• April 22, 2005
Speaker: Eric Rowland
Title: Pascal's Triangle mod n: Fractal Dimensions, Fractal Sequences, and Other Exotic Cuisine
Abstract: What do we get when we reduce binomial coefficients modulo various natural numbers? For prime power moduli, the situation is well understood. But next to nothing is known in other cases.
It turns out that reducing the nth row of Pascal's triangle modulo n gives some special structure. We will survey several properties of this structure that have arisen in attempts to find an explicit "formula" (whatever that may mean) to compute \binom{n}{m} mod n. Along the way we will encounter "periodic polynomials", fractal dimensions, and fractal sequences--"self-similar" sequences of integers that creep up fairly frequently in this sort of thing.
This will be a computer-aided presentation with lots of pictures and explicit data.
• April 15, 2005
Speaker: Brian Manning
Title: Bundles of Joy
Abstract: Fiber bundles and vector bundles appear in several areas of mathematics, including topology, geometry, and mathematical physics. My goal in this talk will be to give a good introduction to these versatile critters, including a number of examples. Along the way, I will review some basics about group actions on sets, and show how you can get some marvellous additional structure by considering bundles together with a group action. I will not assume any knowledge of geometry, nor any topology beyond the most basic, but I may throw out a small tidbit or two for the cognoscenti.
(Actually, this topic is mostly an excuse to talk about group actions, which are very cool, and this abstract is mostly an excuse to use the word "cognoscenti.")
• April 8, 2005
Speaker: Paul Raff
Title: More Fooling Around With Isabelle
Abstract: In my talk, which is the sequel to Phil's wonderful talk, I will talk mainly about the project I worked on my senior year at Carnegie Mellon, which was to formalize the Prime Number Theorem with Isabelle. After 14 months, 30,000 lines of code, and many headaches, it was finally completed in early September, 2004.
In my talk I will provide the rough outline of what it took to do a proof of this magnitude with Isabelle, and hopefully give you an idea of the complexities involved with formalization. I will focus on my part of the project, which involved building a library of facts about binomial coefficients, plus relations among certain functions involved with the PNT.
Although both talks are intertwined, they are essentially independent. Whereas there is no reason you should miss either talk, no knowledge of Phil's talk is needed for my talk. Be there!
• April 1, 2005
Speaker: Phil Matchett
Title: Meet Isabelle, Computerized theorem proving for today and the future.
Abstract: Imagine this: It is 2 a.m., the night before you are supposed to return graded homework to the students, and one student has constructed an _extremely_ complicated, 20-page proof for the last homework question. Wouldn't it be nice to just feed the complicated proof into a computer that would check all the details for you, while you get some sleep?
The idea behind the Isabelle system is that someday, you may be able to do just that (or at least something like it!). In the talk, we will write---in real-time---some basic proofs that Isabelle can check, and we will also demonstrate a recent application of Isabelle to certain kinds of proofs in Category Theory.
Why should you care about Isabelle? To date, Isabelle has been used to automate basic results in a wide range of mathematics, including Number Theory, Complex Analysis, and Group Theory; and it has also found applications in proving the correctness cryptographic protocols and communications protocols. Someday, Isabelle may even be checking research proofs in your own area of mathematics.
One other note: this talk will be a good introduction to Paul Raff's sure-to-be-awesome pizza talk on April 8th, in which he will cover a very recent and very cool application of Isabelle that he worked on at Carnegie Mellon.
• March 25, 2005
Speaker: German Enciso
Title: Infinite Dimensional Beer Glasses
Abstract: The first time I heard of the Brower fixed point theorem was in Germany, where it was appropiately described to me in terms of beer: if one takes a glass of beer, and mixes it around with a spoon, then after it's settled down there is one point of liquid (a beer molecule, of sorts) that is in the same place as before mixing.
I will remind the audience of the usual argument for this result, and why it breaks down in infinite dimensions. Then I will talk about when 'nontrivial' fixed points are desired, and introduce the idea of nonejective fixed points, which are used to show the existence of periodic solutions of certain delay differential equations (this is related to Ben Kennedy's previous talk this semester). As usual, no beer - finite dimensional or otherwise - will be allowed into the room.
• March 9, 2005
Speaker: Sujith Vijay
Title: Primes, Twin Primes and Processors
Abstract: It has been known for quite some time that there are infinitely many primes. No one seems to know yet if there are infinitely many pairs of twin primes. This is not as embarrassing as it sounds -- the sum of reciprocals of primes diverges, while Viggo Brun proved in 1919 that the sum of reciprocals of twin primes converges. (The first to observe that there are infinitely many powers of 2 won't get any pizza.) The sum turns out to be rather difficult to estimate, and it was just such an attempt that led to the discovery of a bug in the floating point unit of the Pentium processor. They fixed it, too.
• March 4, 2005
Speaker: Luc Nguyen
Title: Best approximation on a complex domain
Abstract: Consider the problem of reconstructing a holomorphic function on a domain D if its value at a subset A of D are known. Of course, one can name many methods to achieve this. Interpolation by polynomial or rational functions and approximation by piecewise linear or quadratic functions, for example, are among those that have been studied extensively over years. The models of A for which these methods converges faster have also been investigated. However, which of these is/are the best method?
I'll explain what I mean by a method'' and a best'' method. Then I'll introduce to you the best pointwise approximation method and a proposed model for A when D is the unit disc. Finally, I'll answer the question `Birds of a feather flock together, but do they understand each other well?''
• February 25, 2005
Speaker: John Bryk
Title: Digital Love; or How I Learned to Stop Worrying and Love Ergodic Theory
Abstract: This is a talk about digits. I'll introduce the basic theory of continued fractions and prove some neat things regarding the distribution of digits in these and other expansions. But, to be honest, this isn't really a talk about digits. It's a talk about ergodic theory. I'll introduce the basics of the subject, give numerous examples, and display the power of the almighty ergodic theorem... all through the lens of analyzing digit systems. So it _is_ a talk about digits. And ergodic theory.
• February 18, 2005
Speaker: Leigh Cobbs
Title: On Zero-Divisor Graphs
Abstract: I'll introduce you to what a zero-divisor graph is and what some of the basic graph theoretic properties are (planarity, connectivity, etc). Then I'll show what the undergrads in my REU last summer did with the complements of zero-divisor graphs. In fact, pretty much everything I'm going to talk about are results from undergrad research. No fancy mathematics is needed, and I'll draw lots of pretty pictures.
• February 11, 2005
Speaker: Sikimeti Mau
Title: The McKay Correspondence
Abstract: Plato, back in 350 BC, knew a fair bit about regular polyhedra. He knew that there were only so many, and that made them special. What he didn't know, he made up. And so it was that he "discovered" a mysterious bijection with the Fundamental Elements of the Universe: tetrahedron = fire, icosahedron = water, octahedron = air, and dodecahedron = whatever was in the stars/heavens.
The McKay Correspondence is another mysterious bijection, only marginally less aesthetic than Plato's.
In the classification of finite subgroups of SU(2), the following types pop up: cyclic (order n), binary dihedral (order 4n), binary tetrahedral, binary octahedral and binary icosahedral.
In the classification of simple Lie Algebras, graphs called Dynkin diagrams pop up: types A_n, D_n, E_6, E_7 and E_8.
And yes, you guessed it: there's a bijection between the two.
• February 4, 2005
Speaker: Ben Kennedy
Title: Measures of Noncompactness and Fixed Points
Abstract: If C is a closed, convex set in a Banach space, a continuous map f from C to itself has a fixed point if f(C) is compact. It turns out that the same thing is true if f(C) is not compact but is "more compact than C." What on earth does this mean?
I'll introduce measures of noncompactness and prove some fixed point theorems for maps that make these measures go down. I'll give casual accounts of some applications.
• January 28, 2005
Speaker: Eduardo Osorio
Title: Some Dirichlet problems over some quadratic surfaces
• January 21, 2005
Speaker: Jared Speck
Title: Special Relativity and Minkowskian Spacetime: My Stick Isn't As Short As It Looks
Abstract:I'll introduce standard Newtonian physics in a fancy language that you probably haven't worked with. From there I'll briefly discuss what it means for a physical theory to be Galilean invariant.

Boring.

Things start to heat up when I tell you about how Maxwell's equations, which describe the propagation of light, are not invariant under Galilean transformations, and how light "seems" to propagate via a wave equation that requires no medium. Hmmmmm.

The tension will mount as I attempt to retrace Einstein's original line of thought concerning this strange, medium-free behavior of light. I'll introduce Einstein's postulates and hopefully derive the Lorentz transformations for you, the transformations under which the wave equation for light is invariant.

Finally, we'll discuss what it means to live in a universe that bows before the Lorentzian throne of Special Relativity, and I'll eradicate your ordinary, Newtonian conception of time. Clocks will slow down. Spheres may deform into ellipsoids. And yes, I'll explain why My Stick Isn't As Short As It Looks.

All that and pizza.

### Fall 2004

• December 10, 2004
Speaker: Kia Dalili
Title: The HomAB problem
Abstract: I will talk about parts of my thesis research, I will tell you what the HomAB problem is, why you may want to care about it and in what cases the answer is known. However trying to avoid the technical details I will not prove many statements.
• December 3, 2004
Speaker: Paul Raff
Title: Primes is in P
Abstract: A couple of years ago, three Indian computer scientists found the first deterministic polynomial-time algorithm to determine if a given number is prime. An amazing result in its own right, its excellence is furthered due to its brevity and simplicity. I will go over the proof of the correctness and the speed of the 6-line algorithm, starting with the basics of algorithm design and analysis and theoretical computer science. No mathematical knowledge beyond what you should already know is necessary.
• November 19, 2004
Speaker: Ben Bunting
Title: Pseudospectra, Hypercube Random Walks, and Why 6 Shuffles is not Enough
Abstract: Everyone who studies Markov Chains learns quickly of the importance of the spectra (eigenvalues) to the rate of convergence. However, in the last 20 years, a new idea emerged relating "pseudospectra," i.e. the spectra of slightly perterbed matrices / operators, to this and other applications. As an application, one phenomenon, known as the cutoff phenomenon, appears in many interesting situations, such as random walks on hypercubes, time evolution of Ehrenfest urns, and riffle card shuffling. I will attempt to show how psuedospectral theory applies in all of these situations. No background is required, and pretty pictures will be provided.
• November 12, 2004
Speaker: Elizabeth Henning
Title: Why Hom is a Mother Functor
Abstract: This is an introduction to representable functors, which are the Hom-sets (i.e., sets of maps) associated to a fixed object. I will remind y'all what categories and functors are, and then I will attempt to convince you just how important and useful Hom is by showing you the Yoneda embedding and by proving that any (good) functor can be expressed in terms of Hom. No actual prereqs needed, but expect lots of diagrams and abstract nonsense. Think of it as a break from dealing with the real world.
• November 5, 2004
Speaker: Scott Schneider
Title: A Taste of Descriptive Set Theory
Abstract: Many questions that are difficult (or even impossible) to answer when asked about arbitrary sets of reals become easier when asked about relatively "simple" sets, such as the Borel sets. Descriptive set theory classifies and analyzes such sets, and to give you a flavor of the subject I will prove that every analyic set of reals is Lebesgue measurable and has the perfect set property (and therefore satisfies the continuum hypothesis). Along the way I'll introduce some of the basic tools of descriptive set theory, such as trees, the Baire space, the Suslin operation, and the Borel and projective hierarchies. I'll assume no background in set theory, aside from a vague awareness that things like ordinal numbers and transfinite induction exist. Hope to see you all there.
• October 29, 2004
Speaker: Nick Weininger
Title: A New Combinatorial-Probabilistic Gem
Abstract: Take the infinite square lattice graph, whose vertices are the integer points in the plane and whose edges connect neighboring points. For each edge, flip a coin; if it's heads keep the edge, if it's tails delete it. What is the probability that the subgraph remaining will have an infinite component?

A celebrated result of Harris and Kesten says that (a) for fair coinflips the probability is zero but (b) if the coins are at all biased toward heads the probability becomes one. Very recently, Bollobas and Riordan gave an elegant, short proof of this result. Their proof cleverly combines several of the best-loved devices, old and new, in the theory of combinatorial probability. I will state the Harris-Kesten result and give a sketch of this beautiful new proof. No graduate-level background in either combinatorics or probability will be assumed.
• October 22, 2004
Speaker: Mohamud Mohammed
Title: The (q-)MARKOV-WZ-Method
Abstract: Andrei Markov's 1890 method for convergence-acceleration of series bears an amazing resemblance to WZ theory, as was recently pointed out by M. Kondratieva and S. Sadov. But Markov did not have Gosper and Zeilberger's algorithms, and even if he did, he wouldn't have had a computer to run them on. Nevertheless, his beautiful ad-hoc method, when coupled with WZ theory and Gosper's algorithm, leads to a new class of identities and very fast convergence-acceleration formulas that can be applied to any infinite series of hypergeometric type.

In particular I will give the first ever accelerating series for zeta(5) and some new series. [Joint work with Doron Zeilberger]
• October 15, 2004
Speaker: Derek Hansen
Title: Surface Registration by Matching Umbilic Points
Abstract: In August I attended the ten-day Mathematical Modeling in Industry Workshop at the IMA (Institute for Mathematics and its Applications). I, along with six other graduate students, worked on a problem under the direction of an industry mentor from the math group at Boeing.

The problem was this: Given two similar surfaces--one a perturbation of the other--that lie in the same 3D coordinate system but are separated, identify and classify the umbilic points on each surface and then use these points to find the rigid motion (translation and rotation) that best maps one surface to the other. An umbilic point is a point on a surface where the normal curvature is the same in all directions. The surfaces are given as cubic B-splines.

Why does Boeing care? In the words of our industry mentor: "This operation [the comparison of different but similar geometric models] arises naturally when reusing existing designs, identifying feature differences between two similar parts, tracking changes throughout the life cycle of a product, searching part databases for suitable designs, and protecting proprietary design data"

I'll tell you more about all this on Friday.
• October 8, 2004
Speaker: Mike Neiman
Title: Crossing Numbers and Discrete Geometry
Abstract: The crossing number of a graph is the minimum number of edge crossings in an embedding of the graph in the plane. I will give a probabilistic proof of a general lower bound for the crossing number of graphs. This result leads to very simple proofs of some results in discrete geometry and combinatorial number theory. Time permitting, I will give bounds for the following problems:
(1) Given a set of n points and l lines in the plane, how many incidences can there be among the points and lines?
(2) Given n points in the plane, how many unit distances are determined by the points?
(3) Given a set A of n nonzero real numbers, how small can we simultaneously make both the set of pairwise sums of elements in A and the set of pairwise products of elements in A?
• October 1, 2004
Speaker: Sam Coskey
Title: Playing the Greatest Game in the Continuum
Abstract: no, i'm not going to be talking about sheepshead. instead, suppose you and i play this game:

first fix a set of reals A. now i name a bit a_1 (a_1 = 0 or 1), you name a bit b_1, i name a bit a_2, etc. when all is said and done, we've built a real number together, the number r = 0.a_1b_1a_2.... if r lies in A, i win. otherwise you win.

is there a winning strategy for either of us? if so, the game is called determined. the determinacy property for various sets of reals is very much related to topology and measure.

the statement that every game is determined is abbreviated AD. i'll talk about some of the history, consequences, and power of the AD assumption.
• September 24, 2004
Speaker: Catherine Pfaff
Title: Complex Algebraic Curves: Applications of Hurwitz's Formula
Abstract: I will very briefly describe Riemann surfaces, holomorphic maps, degress of maps, multiplicities, ramification, and genus in preparation to define Hurwitz's formula and give some examples of its uses. For example, I will show how it can be used to show that any holomorphic map between surfaces of genus one is unramified, that there are no holomorphic maps from a surface to a surface of higher genus, and that any holomorphic map between surfaces of the same genus (if that genus is at least 2) must be an isomorphism).
• September 17, 2004
Speaker: Aaron Lauve
Title: Schur Polynomials
Abstract: This topic is inextricably linked to two vast fields that I'm fond of: Representation Theory and the theory of Hopf Algebras. I will do my very best to avoid dropping all of this on you and stay on message... no promises. The message:

If you have ever expanded the polynomial (x-x1)(x-x2)...(x-xn) before, you have seen a symmetric polynomial---n of them in fact! What you may not have seen is a proof of why these are all the symmetric polynomials you'll ever need. I may not have a chance to prove this; but I will state and prove some interesting properties exhibited by a different collection of symmetric polynomials (those mentioned in the title).
• September 10, 2004
Speaker: Eric Rowland
Title: All About Primitive Pythagorean Triples
Abstract: A Pythagorean triple is an integral solution to the Pythagorean equation, x^2 + y^2 = z^2. In studying Pythagorean triples, it suffices to consider "primitive" (relatively prime) solutions, since every solution is a multiple of a primitive solution. In high school geometry we only needed to know two primitive Pythagorean triples--(3, 4, 5) and (5, 12, 13)--so it may come as a surprise that there are actually infinitely many! Can we systematically list them all? To how many triples does a given integer n belong? How can we find these triples explicitly? We will answer these and other questions.
ohannes Flake