RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2015


Spring 2015

Date: January 29, 2015
Speaker: Doron Zeilberger, Rutgers University
Title: Guess and Check!
Abstract: Alice and Bob have together ten apples. Alice noticed that she has two more apples than Bob. How many apples do they each have? The `clever' (but actually stupid) way to do is to solve the algebraic system of two equations and two unknowns A+B=10, A-B=2. The `dumb' (but really better) way would be to try out A=5,B=5 (no good), A=6, B=4 (yea!, we got it!), A=7, B=3 (no good), ..., A=10, B=0 (no good).
I will describe how, in contemporary enumerative combinatorics, `naive' `Guess and Check' leads much faster (and I dare say, more elegantly!) to the solutions of many combinatorial problems than more `sophisticated' and `advanced' methods.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 5, 2015
Speaker: Neil J. A. Sloane, Rutgers University and The OEIS Foundation
Title: On the No. of ON Cells in Cellular Automata
Abstract: A cellular automaton (or CA) is started with a single ON cell; how many cells are ON after n generations? A general theorem will be presented which applies to a certain class of "odd-rule" CAs, including Rule 150, Rule 614, and Fredkin's Replicator, although to get an explicit answer in the last two requires delicate surgical techniques. A number of other CAs can be analyzed by ad hoc methods, although most two-dimensional CAs seem beyond reach. The difficulty of analysis is strongly correlated with the beauty of the resulting patterns.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 12, 2015
Speaker: Jonathan Bloom, Rutgers University
Title: Another (more refined) look at the Wilf-equivalances of length 4 patterns
Abstract: In this talk we will settle two recent conjectures in the area of enumerative combinatorics. First, we answer a conjecture of B. Sagan by finding a multi-statistic preserving bijection between 1423-avoiding permutations and 2413-avoiding permutations. This new bijection also generalizes a classical result, in the area of pattern avoidance, due to Stankova. In the second part of the talk, we employ the techniques used to construct the aforementioned bijection to also prove a conjecture of E. Egge from 2011. In particular, we show that certain pattern classes are, surprisingly, counted by the large Schroder numbers.
Posted on Vimeo (2 parts): Part 1 Part 2
Click here for the slides.


Date: February 19, 2015
Speaker: Bruce Sagan, Michigan State University
Title: Counting increasing rooted forests
Abstract: Let T be a tree whose verices are distinct integers. Call T increasing if the vertices on any path starting from its minimum vertex form an increasing sequence. Similarly, call a forest increasing if each of its component trees is increasing. Given a graph G with vertices 1, ..., n we consider the generating function for all increasing spanning forests of G and show that this polynomial always factors with nonnegative integral roots. We also characterize when this polynomial is equal to the chromatic polynomial of G. Finally, we generalize these results to pure simplicial complexes of arbitrary dimension. This is joint work with Joshua Hallam and Jeremey Martin.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 26, 2015
Speaker: Samuel Clearman, Lehigh University
Title: Planar networks and quantum immanants
Abstract: Matrix immanants are a family of functions on matrices which generalize the permanent and determinant. They have a q-analog called a quantum immanant. We give a combinatorial method for evaluating these immanants on certain matrices, which arise as a q-analog of the path matrix of a planar network. We apply these techniques to the representation theory of certain algebras, and discuss connections to chromatic symmetric functions.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 12, 2015
Speaker: Margaret Readdy, University of Kentucky and Princeton University
Title: q-Combinatorics: A new view
Abstract: A q-analogue is a method to enumerate a set of objects by keeping track of one or more of its mathematical properties. After setting q=1, one returns to the naive enumeration.

After reviewing some classical q-analogues, we will discuss the new idea of a negative q-analogue. This concept is motivated by Fu, Reiner, Stanton and Thiem's recent work on the negative q-binomial. We show the classical q-Stirling numbers of the second kind can be expressed as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in q and 1+q. We extend this enumerative result via a decomposition of the Stirling poset of the first kind, as well as a homological version of Stembridge's q = -1 phenomenon.

We then describe a parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind. Time permitting, we give a bijective combinatorial argument a la Viennot showing the (q,t)-Stirling numbers of the first and second kind are orthogonal.

This is joint work with Yue Cai.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 26, 2015
Speaker: Henning Ulfarsson, Reykjavik University, Iceland
Title: Experimenting with permutations: The tale of two algorithms
Abstract: Permutations are the perfect mathematical object to perform computer experiments on: They are easily represented as a string (1324 is the permutation 1->1, 2->3, 3->2, 4->4) and we have an abundance of interesting sets of permutations to investigate. To name a few: smooth, Baxter, West-2-stack- sortable and simsun permutations. In this talk we will look at two algorithms, BiSC and Struct, designed to make conjectures for the human mathematician to verify. BiSC discovers the patterns avoided by a set of permutations. Struct tries to figure out enough structure in the set to find an equation satisfied by the generating function enumerating the set. Parts of the talk are joint work with Michael Albert (Otago), Christian Bean (Reykjavik), Anders Claesson (Strathclyde) and Bjarki Gudmundsson (Reykjavik).
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 2, 2015
Speaker: Alex Kontorovich, Rutgers University
Title: Why do people doing Automorphic Forms call themselves Number Theorists?
Abstract: Modern problems in automorphic forms, representation theory, and geometry look nothing like the simple problems they were invented to solve (many of which still sit unanswered in OEIS). We will try to present some of these original problems, and explain why they're really (sometimes only conjecturally!) problems lying in a big theory.


Date: April 6, 2015
Speaker: Kellen Myers, Rutgers University
Title: Computational Advances in Rado Numbers
Abstract: An overview of the history of Diophantine Ramsey theory and Rado numbers will be presented, followed by some new methods and results in computing Rado numbers. This talk will focus on establishing the perimeter of known results, describing some of the high-performance computing methods that have been applied to these problems, and describing new results including Rado numbers for various families of nonlinear Diophantine equations.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 9, 2015
Speaker: Nathaniel Shar, Rutgers University
Title: Machines for counting permutations avoiding patterns
Abstract: We will look at some new technology for counting permutations avoiding patterns, and apply it to some easy and some not-so-easy cases. Among other things, we will rediscover Miklós Bóna's formula for the number of permutations avoiding 1342. You don't need to know what a permutation pattern is to understand this talk. This is joint work with Michael H. Albert, Cheyne Homberger, Jay Pantone, and Vince Vatter.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 16, 2015
Speaker: Arthur DuPre, Rutgers-Newark
Title: The Inverse of the Euclidean Algorithm and the Stern-Brocot Tree
Abstract: The Euclidean algorithm gives a mapping from pairs of positive integers to a finite sequence of positive integers which represents the sequence of quotients when the Euclidean algorithm is applied to the two positive integers. The inverse of the Euclidean algorithm applied to this finite sequence gives back the original two integers. The extended Euclidean algorithm applied to positive integers m and n yields two integers x and y so that mx + ny = GCD(m,n). The x and y are not unique and this is displayed nicely and graphically on the Stern-Brocot tree.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 23, 2015
Speaker: Joseph Zurier, Classical High School, Providence, RI
Title: The Joints Problem in Incidence Geometry
Abstract: The joints problem is a very basic problem in the field of incidence geometry: Given n lines in three dimensions, what is the maximum number of triple intersection points they define? This talk will present an overview of the problem and some recent work related to its solution, including an introduction to the polynomial method in incidence geometry. We will derive the current best upper and lower bounds, focusing on a particularly nice method of bounding sequences that has applications to this problem.
Posted on Vimeo (2 parts): Part 1 Part 2



Date: April 30, 2015
Speaker: Doron Zeilberger, Rutgers University
Title: Proofs of the Riemann Hypothesis and of "P is NOT EQUAL to NP"
Abstract: In this expository talk, I will present totally convincing proofs of the above statements. These proofs are "well-known folklore", but not as well-known as they should be, hence this talk. Note that these proofs do not qualify for the million dollar prizes promised by the Clay Foundation, but, if and when these prizes will be awarded, the probability that RH and P!=NP are true will not be increased even by epsilon. It would be just meeting a "mathematical athletic" challenge, that has nothing to do with truth.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: September 10, 2015
Speaker: Doron Zeilberger, Rutgers University
Title: 25 Years of Wilf-Zeilberger Theory
Abstract:
          25 years of WZ theory will be summarized in 48 minutes.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: September 17, 2015
Speaker: Matthew Russell, Rutgers University
Title: Some new partition conjectures, and how one might prove them
Abstract:
          In this talk, I will discuss some old and new partition conjectures, along with methods that will allow one to verify (for very many terms) these conjectures. Perhaps these methods will one day lead to a proof. Parts of the talk will be joint work with Shashank Kanade, other parts will be inspired by a recent talk of George Andrews.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: September 24, 2015
Speaker: Neil J. A. Sloane, Rutgers University and OEIS Foundation
Title: Unsolved Number Theory Sequences from Alekseyev, Meiburg, Resta, van der Poorten, and Pablo Picasso
Abstract:
          I will discuss some remarkable unsolved number-theoretic sequences, including the 999999000000 problem, Meiburg's recurrence, the Three Palindromes Conjecture, and a newly discovered sequence of Pablo Picasso.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 1, 2015
Speaker: Nathan Fox, Rutgers University
Title: Linear-Recurrent Solutions to Meta-Fibonacci Recurrences
Abstract:
          In 1963, Douglas Hofstadter first contemplated the recurrence Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2)). He found that, when given the initial conditions Q(1)=1 and Q(2)=1, this sequence behaves chaotically. In fact, we still do not know whether Q(n) < n for all n. About thirty years later, Solomon Golomb observed that the initial conditions Q(1)=3, Q(2)=2, Q(3)=1 give rise to a much more predictable sequence. Even more recently, Frank Ruskey discovered a way to embed the Fibonacci sequence into the Hofstadter's recurrence. In this talk, we will expand upon the ideas of Golomb and Ruskey as we explore what sorts of nice solutions exist to the Hofstadter and related recurrences, known as meta-Fibonacci recurrences.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 2, 2015 (joint with Rutgers colloquium)
Speaker: Philippe Di Francesco, Institut de Physique Theorique, Saclay, France and University of Illinois at Urbana-Champaign
Title: Integrable Combinatorics
Abstract:
          We review various combinatorial problems with underlying classical or quantum integrable structures. We present a few mathematical problems or constructs, most of them combinatorial in nature, that either were introduced to explicitly solve or better understand physical questions (Mathematical Physics) or can be better understood in the light of physical interpretations (Physical Mathematics). The frontier between the two is subtle, and we will try to make this more concrete in a few examples.

Our main character is integrability, whether discrete, classical or quantum, which is a manifestation of the underlying symmetries of the problems at hand, and allows often for compact and elegant solutions. The objects we discuss are: Lorentzian triangulations, planar maps for 2D quantum gravity, and the representation-theoretic content of generalized Heisenberg quantum spin chains.

The structures encountered along the way are: paths and trees, discrete (non-commuting) integrable systems, Cluster Algebras, Macdonald operators and Double Affine Hecke Algebras.

(Based on joint works with J. Bouttier, E. Guitter, C. Kristjansen and R. Kedem.)
Posted on Vimeo (3 parts): Part 1 Part 2 Part 3


Date: October 8, 2015
Speaker: Anthony Zaleski, Rutgers University
Title: Walks in integer lattices
Abstract:
          Motivated by Feller's discussion of coin tosses, we first introduce Maple procedures to analyze 2D walks whose steps are one unit up or right. Then we generalize to arbitrary steps and higher dimensions. In this realm, an analytic approach is hard or impossible, so our experimental methods will save the day.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 15, 2015
Speaker: Ori Parzanchevski, Princeton University
Title: Hearing the shape of a band
Abstract:
          "Can One Hear the Shape of a Drum?" is the witty title of a 1966 paper by Mark Kac, asking whether one can infer the shape of an object (such as a drum) from the sound it makes when vibrating. Only in 1992 the question was answered, by Gordon, Webb and Wolpert. In addition to explaining the answer, I will talk about the more general problem of "hearing the shape of a band" (quoting Barry Cipra), and explain how it relates to elementary group theory and experimental mathematics. No prior knowledge in physics or drumming is required.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 22, 2015
Speaker: Vladimir Retakh, Rutgers University
Title: Generalized adjoint actions and combinatorics
Abstract:
          There is a classical formula expressing adjoint action exp(x)yexp(-x) for noncommuting variables x and y as an infinite sum of iterated brackets [x,...,[x,y]...]. We replace exp(x) by any formal series f(x) and show that the result can be written as an infinite sum of generalized iterated brackets. To describe the generalized brackets we need a special class of Hall-Littlewood polynomials. I am going to discuss various combinatorial properties of these polynomials. This is a joint paper with A. Berenstein (U. of Oregon).
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 29, 2015
Speaker: Jake Baron, Rutgers University
Title: Counting Disjointly-Occurring Events
Abstract:
          In {0,1}^n, events A and B "occur disjointly" if there are "disjoint certificates" of their respective occurrence. The van den Berg-Kesten inequality (BK) says that if A and B are increasing, then P(they occur disjointly) ≤ P(A)P(B). I will present a new (to my knowledge) generalization of BK to a setting with arbitrarily many events, where the quantity of interest is the maximum number that occur disjointly. Time permitting I'll discuss our motivation for this result (though it is interesting in itself), which was to get an exponential upper-tail bound for a class of random variable that often comes up in combinatorial settings.
          Joint with Jeff Kahn.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: November 5, 2015
Speaker: Aviezri Fraenkel, Weizmann Institute of Science
Title: Games For Arbitrarily Fat Rats
Abstract:
          In kindergarten we learned about the integers (Peano axioms); in grammar school — about pairs of integers (rationals); and then in high school, about the reals (Dedekind cuts). Berlekamp, Conway, Guy discovered and promoted a method (Don Knuth: "Surreal Numbers") of creating all of those and much more — namely games! — in one masterful stroke.
          Yet the rationals sometimes present obstinate difficulties often overlooked. Example. Let 1 < α1 <, . . . , < αm be real numbers, dubbed moduli, m ≥ 3. An over 40 years old conjecture states that there exist reals γi such that the system ( ⌊ nα1 + γ1⌋ , . . . , ⌊nαm + γm⌋ ) constitutes a complementary system of m sequences of integers if and only if αi = ⌊(2m - 1)/2m - i + γi⌋ , i = 1, . . . , m. It is known that for integers and irrationals, 2 moduli have to be equal, but the problem is wide open for the rationals.
          We have created, for every m ≥ 2, an invariant game whose P-positions (2nd player win positions) are the conjectured moduli, and gave game rules and an efficient strategy for the next winning move if not in a P-position. Motivation: (1) "Play" with the above conjecture. (2) Find efficient game rules for games defined only by their sets of P -positions. (Rats: rationals.)
          Joint with Urban Larsson.
         Click here for a typeset version of the abstract.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: November 19, 2015
Speaker: Kağan Kurşungöz, Faculty of Engineering and Natural Sciences, Sabancı University
Title: Andrews and Bressoud Style Identities for Partitions and Overpartitions
Abstract:
          We propose a method to construct a variety of partition identities at once. The main applications are all-moduli generalization of some of Andrews' results in [Andrews, Parity in partition identities. Ramanujan Journal 23:45-90 (2010)] and Bressoud's even moduli generalization of Rogers-Ramanujan-Gordon identities, and their counterparts for overpartitions due to Lovejoy et al. and Chen et al. We obtain unusual companion identities to known theorems as well as to the new ones in the process. The novelty is that the method constructs solutions to functional equations which are satisfied by the generating functions. In contrast, the conventional approach is to show that a variant of well-known series satisfies the system of functional equations, thus reconciling two separate lines of computations.
Posted on Vimeo: Only part


Date: December 3, 2015
Speaker: Dmitri Tymoczko, Princeton University
Title: Voice leading as vector
Abstract:
          In my talk I will describe what I believe to be one of the most general and fundamental connections between music and mathematics, a kind of "translation manual" that allows us to associate basic concepts in music theory with ideas from contemporary geometry. The most fundamental entries in this translation manual associate chords with points in an orbifold, voice leadings with vectors (or, given the structure of the spaces, homotopy classes of paths in the orbifold), and scales with metrics. Using this translation manual we can translate back and forth between the visual and the musical realms, creating spatial representations that allow us to understand not just particular pieces but also more general constraints on musical styles. But even without these visual representations, a focus on vectors or voice leadings opens the door to new statistical understanding of even the most basic questions about traditional music.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: December 10, 2015
Speaker: Collin Takita, Ursinus College
Title: Results on ghost series and motivated proofs for overpartition identities
Abstract:
          We give what we call a "motivated proof" of the overpartition analogue of Bressoud's Theorem, originally proved by Lovejoy, et al. and Chen, et al. The first such "motivated proof," a proof of the Rogers-Ramanujan identities, was given by Andrews and Baxter. This work was later extended by many authors to Gordon's partition identities (by Lepowsky and Zhu), the Gollnitz-Gordon identities (by Coulson, Kanade, Lepowsky, McRae, Qi, Russell, Sadowski), and the Andrews-Bressoud identities (by Kanade, Lepowsky, Russell, Sills). Our proof is in the spirit of the work by Kanade, Lepowsky, Russell, and Sills, where certain new series, called "ghost series," are introduced in order to prove an Empirical Hypothesis and to give a proof of the overpartition analogue of Bressoud's Theorem. In the process, we show that the Ghost Series also have their own combinatorial interpretations. This is joint work with Matthew Russell and Christopher Sadowski.
Posted on Vimeo (2 parts): Part 1 Part 2