RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2014


Jump to Fall 2014

Spring 2014

Date: January 30, 2014
Speaker: Wolfram Koepf, Univ. of Kassel, Germany
Title: Methods of Computer Algebra for Orthogonal Polynomials
Abstract:
          In my talk I will give an introduction into the theory of classical orthogonal polynomials and their properties, and I will show, how in this theory computer algebra algorithms are useful. Some techniques that are treated will be demonstrated by live Maple examples.
Posted on Vimeo (2 parts): Part 1 Part 2
View the slides from the talk here.


Date: February 6, 2014
Speaker: Ori Parzanchevski, Institute for Advanced Study, Princeton, NJ
Title: Fourier analysis of word maps
Abstract:
          Let G be a finite group. How many times is an element g obtained as a commutator in G? Namely, how many solutions are there to the equation x*y*x^-1*y^-1=g ? In 1886 Frobenius gave a striking answer to this question in terms of the character theory of the G. But for a general word w replacing the commutator word x*y*x^-1*y^-1, surprisingly little is known. I will show some examples and survey old and recent results, including recent joint works with Doron Puder and Gili Schul.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 13, 2014
No talk - inclement weather


Date: February 20, 2014
Speaker: Semeon Artamonov, Rutgers University
Title: Quantum A-polynomials for knots using the q-Zeilberger algorithm
Abstract:
          My talk is devoted to application of the q-Zeilberger algorithm to knot theory and my recent research in that area. I will start with a basic introduction to knot invariants for graduate students and define braid group and Hecke algebras. Using the Turaev R-matrix approach I will explain how to evaluate one of the most powerful knot invariants, namely the colored HOMFLY polynomial. However, the naive application of the Turaev approach quickly makes evaluations impossible on today's computers. To overcome that issue I will explain the recent developments on cabling techniques and tricks I used in my programs.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 27, 2014
Speaker: Will Lee, Rutgers-Camden
Title: On Lehmer's conjecture about Ramanujan's tau function
Abstract:
          I will outline a new approach to tackle Lehmer's famous conjecture that Ramanujan's famous tau function, tau(n), is never zero.
A more detailed abstract is available here.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 6, 2014
Speaker: Matthew Russell, Rutgers
Title: Ptolemy-like theorems in the Minkowski plane
Abstract:
          Ptolemy's theorem is a classical result that gives a formula relating the side lengths and diagonals of cyclic quadrilaterals. More generally, there are theorems that do the same for cyclic polygons. It has previously been shown that these theorems still hold (with appropriate modifications) in spherical and hyperbolic geometries. In this talk, we will examine the appropriate analogues of these theorems in the Minkowski plane, along with the Maple experimentation that aided the discovery process. Joint work with Ed Chien.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 13, 2014
Speaker: Edinah Gnang, Institute for Advanced Study, Princeton, NJ
Title: A combinatorial approach to hypermatrix algebra
Abstract:
          Following Zeilberger's classic paper titled "A combinatorial approach to matrix algebra", we present in this talk a combinatorial approach to hypermatrix algebra and how the Mesner-Bhattacharya algebra allows for a generalization of the combinatorial interpretation of the Cayley-Hamilton theorem.
Posted on Vimeo (2 parts): Part 1 Part 2



Date: March 27, 2014
Speaker: Ebad S. Mahmoodian, Sharif University, Iran
Title: Defining sets from graph coloring to Latin squares and Sudoku
Abstract:
          Click here for the abstract.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 3, 2014
Speaker: Brian Nakamura, Rutgers University
Title: Functional equations for patterns in permutations
Abstract:
          I will discuss how certain functional equations can be used to study permutations avoiding a pattern and the generalized case where permutations contain exactly k copies of a pattern. This enables us to produce new enumeration algorithms for certain patterns. In particular, this approach can be applied to the notoriously difficult pattern 1324 and used to compute new terms for the sequence enumerating 1324-avoiding permutations.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 10, 2014
Speaker: Douglas Hofstadter, Cognitive Science, Indiana University
Title: Curious Patterns and Nonpatterns in a Family of Meta-Fibonacci Recursions
Abstract:
          In 1963, while playing a novel number-theoretical game with a friend, I dreamt up a curious recursive definition for an integer-valued function, which I dubbed "Q(n)". Here is the recursion:

Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))

The two initial values I supplied were Q(1) = 1 and Q(2) = 1. This function's behavior turned out to be very unpredictable. I spent a good deal of time exploring it computationally, and also invented and explored some variations on the theme. To my frustration, though, I wasn't able to prove anything about Q(n) (not even that it existed for all positive values of n!), no matter how hard I worked. Eventually, as often tends to happen when one has pushed oneself as far as one can go and has hit up against one's limits, Q(n) slowly faded into the background of my life.

Many years later, my mathematician/physicist friend Greg Huber proposed that we go back to the canonical and very natural but unsolved questions about the mysterious function Q(n), with its magnetic appeal and crystalline beauty, and carefully study them. I went for Greg's bait hook, line, and sinker, but soon I found, to my great frustration, that unlike Greg, who had made some small progress analytically, I was completely unable to prove anything analytically about Q(n). Soon Greg found himself in a similar position of frustration. After a period of stagnation, finding ourselves in a box canyon with no escape route except using a computer to do experimental mathematics, together he and I invented a two-parameter family of functions closely related to Q(n) and we explored their behavior computationally. We found some delightful surprises and some deep insights. However, despite all the progress, a lot of mystery remained (and still remains). In this talk, I will mainly describe Greg Huber's and my collaborative discoveries.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 11, 2014 (joint with Rutgers colloquium)
Speaker: Douglas Hofstadter, Cognitive Science, Indiana University
Title: A Strange Saga of Number Theory and Physics
Abstract:
          In 1963, while playing novel number-theoretical games with a computer, I dreamt up a curious recursive definition for a real-valued function, which I dubbed "INT(x)". This function's behavior turned out to be very unpredictable. I spent a good deal of time exploring it computationally, and also invented and explored some variations on the theme. To my frustration, though, I was able to prove only the most basic facts about INT(x), and many intriguing questions remained completely unanswered, no matter how hard I worked. Eventually, as often tends to happen when one has pushed oneself as far as one can go and has hit up against one's limits, INT(x) slowly faded into the background of my life.

Many years later, my physicist doctoral advisor Gregory Wannier proposed, as my potential Ph.D. research, a canonical and very natural but unsolved problem concerning the mysterious energy spectrum of crystals in magnetic fields. I went for Gregory's bait hook, line, and sinker, but soon I found, to my great frustration, that unlike Gregory, who had made some small progress analytically, I was completely unable to prove anything analytically about the equation (known as "Harper's equation"). After a period of stagnation, finding myself in a box canyon with no escape route except using a computer to do experimental mathematics, I started exploring Harper's equation computationally, and to my astonishment, I found that good old INT(x) came back into the picture front and center. This was a delightful surprise, and all at once, out of the blue, my long-ago number-theoretical explorations turned out to give me some deep insights into the physics problem. However, despite all the progress, a lot of mystery remained (and still remains). In this talk, I will mainly describe Gregory Wannier's and my collaborative discoveries.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 17, 2014
Speaker: Soichi Okada, Nagoya University
Title: Greatest common divisors of specialized Schur functions and generalized parking spaces
Abstract:
          Given two positive integers n and k, we find the greatest common divisor of the special values s_λ(1^k) of Schur functions, where λ runs over all partitions of n. As an application, we determine when generalized parking spaces exist for the symmetric groups. We also discuss the greatest common divisor of the principally specialized Schur functions and present a conjecture on the unimodality of the coefficients of certain polynomials obtained as a quotient of the principal specializations.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 24, 2014
Speaker: Kellen Myers, Rutgers
Title: Rado Numbers: Experimental and Computational Methods
Abstract:
          The field of Ramsey Theory is well-known for its extremely large bounds. Often a Ramsey Theoretic quantity is bounded above by an astronomical number (e.g. Graham's number) but is believed to be much smaller. This frequent over-estimation in proofs hints at how difficult it is to calculate the exact values of these quantities.
          I will present a few practical techniques to compute these figures for a class of Ramsey Theoretic quantities called Rado Numbers, which are associated to Diophantine equations whose solution sets have the usual Ramsey-type property. I will describe some brute-force techniques as well as techniques involving symbolic computation, and I'll give some of the results that I have obtained in these ways (and describe those I hope to obtain in the future). This talk requires no technical knowledge or special background, although knowing basic combinatorics and a little bit about Ramsey Theory wouldn't hurt.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: May 1, 2014
Speaker: Nathan Fox, Rutgers
Title: Aperiodic Subtraction Games
Abstract:
          Subtraction games are a class of impartial combinatorial games whose positions correspond to nonnegative integers and whose moves correspond to subtracting one of a fixed set of numbers from the current position. Though they are easy to define, subtraction games have proven difficult to analyze. In particular, few general results about their Nim values are known. Nim values are one of the most fundamental numerical properties of games; in particular, a position's Nim value is zero if and only if it is a losing position. One question that can be asked is whether there exists a subtraction game whose sequence of Nim values is bounded and not eventually periodic. In this talk, we will construct an example of such a game where all of the Nim values are zero, one, or two.
Posted on Vimeo (2 parts): Part 1 Part 2


Fall 2014

Date: September 11, 2014
Speaker: Anthony Zaleski, Rutgers
Title: An Interesting Twist on the Classical Isoperimetric Problem
Abstract: The isoperimetric problem asks: What region R of fixed area has least perimeter? Under appropriate assumptions, the answer is a ball. Here, we consider a "non-local" variation of this problem, in which the energy to be minimized consists of the perimeter plus ∫R ∫R |x-y|-a dx dy, where a>0 is a constant parameter. The second, "repulsive" term suggests that nontrivial minimizers may now exist. In this talk, we focus on finding parameter regimes in which minimizers are still disks. Since our results follow from elementary calculus, geometric inequalities, and Maple computations, this talk should be accessible to all.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: September 18, 2014
Speaker: Jesús Guillera, Zaragoza, Spain
Title: Proofs of some Ramanujan series by the WZ-method
Abstract: Click here for the abstract.
Slides: Click here for the slides.


Date: October 2, 2014
Speaker: Doron Zeilberger, Rutgers University
Title: Dominique Foata : A Neoclassical Combinatorial Giant
Abstract: Dominique Foata's (b. Oct. 12, 1934) seminal contributions to Enumerative and Algebraic Combinatorics will be outlined in 48 minutes.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 3, 2014, 12:00 PM, Hill 705 (note special date and time, joint with Lie Groups seminar)
Speaker: Emily Leven (formerly Sergel), University of California, San Diego
Title: Combinatorial Aspects of the rational Shuffle Conjecture
Abstract: In this talk, we will review the history and recent extensions of the Classical Shuffle Conjecture. This conjecture equates two symmetric polynomials, one of which is known to give the Frobenius characteristic of the space DHn of diagonal harmonics. The other side of the conjecture is purely combinatorial, showing the remarkable ability of certain symmetric function operators to control combinatorial objects, such as Dyck paths and parking functions. This branch of algebraic combinatorics was created to explore the representation-theoretical aspects of Macdonald polynomials. This led to the n! conjecture and the introduction of the space of Diagonal Harmonics. A program outlined by Procesi led to the proof by Mark Haiman of the n!-conjecture by algebraic geometrical tools. There has recently been a flood of new operators and conjectures created, in our subject, by algebraic geometers. This talk covers some of the new results and conjectures obtained by a continuing effort to translate these developments back into the original algebraic-combinatorial setting. Our presentation should be accessible to the general mathematical audience.


Date: October 9-10, 2014
DIMACS Conference on Challenges of Identifying Integer Sequences
In honor of the 50th birthday of the On-Line Encyclopedia of Integer Sequences
For links to all of the videos, see this page with the conference program or the Vimeo album


Date: October 16, 2014
Speaker: Shashank Kanade, Rutgers University
Title: Vertex Operator Algebras, Partitions, Experimental Mathematics: A Very Exciting Program
Abstract: The Rogers-Ramanujan identities are a remarkable pair of classical identities regarding integer partitions. These identities are intimately connected to the representation theory of vertex operator algebras, stemming from early works of Lepowsky-Wilson. On the purely combinatorial/q-series-theoretic side, there are many proofs of these identities, and the one of these that stands out from the viewpoint of vertex operator algebra theory is the ''motivated proof'' given by Andrews and Baxter, even though Andrews and Baxter were working entirely q-series-theoretically. In this talk, I will give an exposition of some new generalizations of this motivated proof to other families of identities, and I will relate these ideas to vertex operator algebra theory. I will also explain how experimental mathematics is shaping the landscape. This talk is based on a recent joint work with J. Lepowsky, M. C. Russell and A. V. Sills.
Posted on Vimeo (2 parts): Part 1 Part 2
Click here for a picture of the chalkboard diagram.


Date: October 23, 2014
Speaker: Matthew Russell, Rutgers University
Title: Conjecturing new partition identities with computer algebra
Abstract: Rogers-Ramanujan identities and their numerous generalizations (Gordon, Andrews-Bressoud, Capparelli, etc.) form a family of very deep identities concerned with the integers partitions. These identities (written in generating function form) are typically of the form ''product side'' equals ''sum side'', with the product side enumerating partitions obeying certain congruence conditions and the sum side obeying certain initial conditions and difference conditions (along with possibly other restrictions). We use symbolic computation to generate various such sum sides and then use Euler's algorithm to see which of them actually do produce elegant conjectured product sides. We not only rediscover many of the known identities but also discover some apparently new ones. Joint work with Shashank Kanade.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 30, 2014
Speaker: Emilie Hogan, Pacific Northwest National Laboratory
Title: Interval-valued partial order rank function
Abstract: Hierarchical data objects whose mathematical representations are partial orders , e.g., ontologies or taxonomies, are often not graded. However, we still want to be able to talk about a sense of vertical level in the partial order for the purpose of layout or statistical studies. In one of my research projects we introduced a notion of an interval rank function. When a poset is graded the interval rank function becomes the traditional integer-valued rank function. In this talk I will give our definition of general interval rank and the special case of "standard interval rank". I will make a case for standard interval rank over any other interval rank function. I will also discuss how we used experimental mathematics in this project to investigate (1) how the standard interval rank function behaves on average for all small partial orders, (2) how it behaves on random partial orders of larger size, and (3) playing with "measures of gradedness", a concept I will define.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: November 6, 2014
Speaker: Steven H. Weintraub, Lehigh University
Title: The adjoint of differentiation
Abstract: Let $n$ be any nonnegative integer. Let $V=P_n$ be the vector space of polynomials of degree at most $n$, equipped with the inner product $\langle f, g \rangle = \int_0^1 f(x)g(x)dx$. Let $D : V \longrightarrow V$ be the differentiation operator, $D(f) = f^\prime$. Then $D$ has an adjoint $D^*$. We have closed form expressions for $D^*$, which were conjectured by computing $D^*$ for small values of $n$ and finding a pattern. (If $f(x)$ is a polynomial of degree $k \leq n$, then, while the value of $D(f(x))$ is independent of $n$, the value of $D^*(f(x))$ depends on $n$.) We also find formulas for $D^*$ in terms of classical Legendre polynomials, shifted to the interval $[0,1]$. Using these formulas it is easy to prove that our closed form expressions are correct.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: November 13, 2014
Speaker: Doron Zeilberger, Rutgers University
Title: Symbolic Moment Analysis of the Hirsch Citation Index (formerly Size of Durfee Square)
Abstract: Once upon a time there was an esoteric and specialized notion, called "size of the Durfee square", of interest to at most 100 specialists in the whole world. Then it was kissed by a prince called Jorge Hirsch, and became the famous (and to quite a few people, infamous) h-index, of interest to every scientist, and scholar, since it tells you how productive a scientist (or scholar) you are! When Rodney Canfield, Sylvie Corteel, and Carla Savage wrote their beautiful 1998 article proving, rigorously, by a very deep and intricate analysis, the asymptotic normality of the random variable "size of Durfee square" defined on integer-partitions of n (as n goes to infinity), with precise asymptotics for the mean and variance, they did not dream that one day their result should be of interest to everyone who has ever published a paper. However Canfield et. al. had to work really hard to prove their deep result. Here we take an "empirical" shortcut, that proves the same thing much faster (modulo routine number- and symbol-crunching). More importantly, the empirical methodology should be useful in many other cases where rigorous proofs are either too hard, or not worth the effort!
Posted on Vimeo (2 parts): Part 1 Part 2


Date: November 20, 2014
Speaker: Harry Crane, Rutgers University, Department of Statistics and Biostatistics
Title: Random partitions and permutations
Abstract: Historically, enumerative combinatorics and discrete probability theory are closely related through uniform probability distributions on finite sets. I will first explain why the uniform distribution is unnatural in some modern applications and then survey several aspects of non-uniform random partitions and permutations. The discussion touches on ideas from enumerative combinatorics, algebra, probability, and statistics. I assume no prior knowledge.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: December 4, 2014
Speaker: Robert C. Rhoades, Center for Communication Research, Princeton, NJ
Title:Set Partition Statistics: moment formulas and normality
Abstract: In studying the representation theory of certain finite groups we found that the mean, variance and higher moments of novel statistics on set partitions of [n] = {1, 2, ..., n} have simple closed expressions as linear combinations of shifted Bell numbers. Motivated by this we have shown that there is a large algebra of other statistics with similar formulas for their moments. The coefficients in the linear combinations are polynomials in n. This allows exact enumeration of the moments for small n to determine exact formulas for all n. Computations led to the conjectured normality of many of these statistics. We used a stochastic algorithm for generating a random set partition due to Stam to establish the normality.
Posted on Vimeo (2 parts): Part 1 Part 2