RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2016


Spring 2016

Date: January 28, 2016
Speaker: Michael Kiessling, Rutgers
Title: Order and Chaos in Some Trigonometric Series: Curious Adventures of a Statistical Mechanic
Abstract:
          I will tell the story of how a MAPLE-assisted quest for an interesting undergraduate problem in trigonometric series led some "amateurs" to the discovery that a simple one-parameter family of trigonometric series which is not in Zygmund's book exhibits both order and apparent chaos, and how this has prompted some professionals to offer their expert insights.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 4, 2016
Speaker: Benoit Cloitre, Paris
Title: Good variation theory : an experimental approach to the Riemann Hypothesis
Abstract:
          In this talk I will summarize 5 years of experimental research aiming to develop the theory of functions of good variation (FGV). I will start from the first insight in 2010, prove the simplest case related to affine functions and describe how some very recent conjectures encapsulate the Grand Riemann Hypothesis. Several graphics will support the claims and no deep prior knowledge in number theory is necessary.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 11, 2016
Speaker: Alejandro Ginory, Rutgers
Title: Identities involving Characters of Wreath Products of S_n
Abstract:
          The character tables of S_n and their wreath products G~S_n, for finite groups G, contain many fascinating sequences and identities. We discuss underlying reasons for this and, with the help of Maple, explore some of these identities. Finally, we will see connections to other interesting combinatorial objects.
Posted on Vimeo (3 parts): Part 1 Part 2 Part 3


Date: February 18, 2016
Speaker: Harry Crane, Rutgers
Title: Pattern avoidance for random permutations
Abstract:
          A classic question of enumerative combinatorics is: How many permutations of {1,...,n} avoid a given pattern? I recast this question in probabilistic terms: What is the probability that a randomly generated permutation of {1,...,n} avoids a given pattern?

I consider this question for the Mallows distribution on permutations, of which the uniform distribution is a special case. I discuss how the probabilistic technique of Poisson approximation can be applied to bound the total variation distance between the Poisson distribution and the distribution of the number of occurrences of a fixed pattern in a random permutation. In the special case of the uniform distribution, we obtain bounds on the number of pattern avoiding permutations of all finite sizes.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 25, 2016
Speaker: Doron Zeilberger, Rutgers
Title: A Diatribe against the "Law" of the Excluded Middle
Abstract:
          The so-called Law of the Excluded Middle is completely illegitimate for so-called "infinite" sets, leading to so much scholastic drivel, and making a large part of modern mathematics (in particular all those "undecidability" results) utterly meaningless, except as a (usually utterly boring) game with "axioms". But even when restricted to finite sets it leads to paradoxical, unsatisfying conclusions (e.g. Prisoner's dilemma).
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 3, 2016
Speakers: Neil Sloane, Emina Soljanin, Nathaniel Shar, Nathan Fox, and Pat Devlin
Moderator: Matthew Russell
Title: Open problems where experimental mathematics may be useful
Abstract:
          Various speakers presented open problems.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 10, 2016
Speaker: Eugene Fiorini, Muhlenberg College
Title: Symmetric Class-0 Subgraphs and Forbidden Subgraphs
Abstract:
          Competition graphs and graph pebbling are two examples of graph theoretical-type games played on a graph under well-defined conditions. In the case of graph pebbling, the pebbling number pi(G) of a graph G is the minimum number of pebbles necessary to guarantee that, regardless of distribution of of pebbles and regardless of the target vertex, there exists a sequence of pebbling moves that results in placing a pebble on the target vertex. A class-0 graph is one in which the pebbling number is the order of the graph, pi(G)=|V(G)|. This talk will consider under what conditions the edge set of a graph G can be partitioned into k class-0 subgraphs, k a positive integer. Furthermore, suppose D is a simple digraph with vetex set V(D) and edge set E(D). The competition graph G(V(G),E(G)) of D is defined as a graph with vertex set V(G)=V(D) and edge vw in E(G) if and only if for some vertex u in V, there exist directed edges (u,v) and (u,w) in E(D). This talk will present some recent results on foodwebs and forbidden subgraphs of a family of competition graphs.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 24, 2016
Speaker: Stephen DeSalvo, UCLA
Title: Lower bound expansions for random Bernoulli matrices
Abstract:
          A random Bernoulli matrix is an n by n matrix with each entry chosen as +/-1 with probability 1/2, independently of other entries. It was first shown by Komlos that the probability that a random Bernoulli matrix is singular tends to 0 as n tends to infinity, and later by Kahn, Komlos, and Szemeredi that the rate is exponentially decaying. The rate was later improved by Tau and Vu, and most recently by Bourgain, Vu, and Wood. We present a lower bound asymptotic expansion, conjecturally a true asymptotic expansion, by classifying the various ways in which the matrix can be singular, indexed by what we call novel integer partitions. We show that the set of novel integer partitions is both necessary and sufficient for classifying singularities, and classify all novel integer partitions with at most seven parts. This is joint work with Richard Arratia.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 31, 2016
Speaker: Nathaniel Shar, Rutgers
Title: Experimental Methods in Pattern Avoidance and Bijective Proofs (thesis defense)
Abstract:
          We'll look at ways to set up large, but finite, systems of recurrence relations, called "enumeration schemes," to count classes of permutations containing repeating structures. We'll see how this leads to rigorous proofs of formulas for the sizes of those classes with "guess and check" methods. If time permits, (which it probably will not), I will also discuss other experimental-mathematical topics from my thesis.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 7, 2016
Speaker: Matthew Russell, Rutgers
Title: Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences (thesis defense)
Abstract:
          In this talk, we will examine a variety of ways I have used experimental mathematics - to make conjectures, to guide proofs, and even to automatically prove theorems.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 14, 2016
Speaker: Ross Berkowitz, Rutgers
Title: How many triangles are in the random graph?
Abstract:
          We will formulate answers to questions of the following form: What is the probability that one has exactly the expected number of triangles in an Erdos-Renyi random graph? What if we ask about the mean plus ten, or plus two standard deviations? Counting the number of occurrences of a subgraph in an Erdos-Renyi random graph is a long studied problem, and much work has gone into proving when one has a central limit theorem. We will talk about when such results may be extended into local limit theorems.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 21, 2016
Speaker: Neil J. A. Sloane, Rutgers University and The OEIS Foundation
Title: The Red Dot Problem: Squares Containing No Subsquares
Abstract:
          Every few days a number sequence is submitted to the OEIS which is so lovely that one says "if only I had time to work on this". This talk is about one such sequence that I did work on, with a lot of help from Warren Smith. How many cells of an n X n grid can you color red so that no four of them form a square with sides parallel to the sides of the grid? The names Heinrich Ludwig, Waclaw Sierpinski, Endre Szemerédi, Furstenberg and Katznelson, Felix Behrend, and of course Robert Louis Stevenson will be mentioned.


Date: April 28, 2016
Speaker: Max Alekseyev, George Washington University
Title: Transfer-Matrix Method as a Combinatorial Hammer: Enumeration of Silent Circles, Graph Cycles, and Seating Arrangements
Abstract:
          I will discuss application of the transfer-matrix method to a variety of enumeration problems concerning the party game "silent circles", Hamiltonian cycles in the antiprism graphs, simple paths/cycles in arbitrary graphs, and generalized menage problem. While this method does not always lead to nice formulas, it often provides an efficient way of computing the corresponding quantities.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: Originally scheduled for April 21, 2016; postponed until further notice
Speaker: Idith Segev, Ramat Gan, Israel
Title: Significant Occurrence in Even Musical Texture in Bach's Preludes. A Study Using Mathematical Tools.
Abstract:
          A musical work can be analyzed according to two schemata: the culture-independent 'natural' schemata, and the culture-dependent 'learned' schemata. The first includes certain parameters such as curves of pitch, intensity, rhythm, together with symmetric and geometric transformations on them, while the second deals with intervals, harmonies and tonal organization. The interest in the learned schemata dates back at least to Pythagoras, while the natural schemata is relatively new concept.
          Until now, the two methods of analysis seemed to be largely independent of each other. Our research revealed through the preludes of J.S.Bach, the Well-Tempered Clavier, some hidden, previously unknown connections between the two schemata, which sheds some new light on Bach's special musical language.
          Those preludes are characteristic of J.S.Bach's genius and are known for their beauty and complexity.
          Many of them exhibit a certain general property which we call 'evenness', meaning — a constant feature throughout the piece (e.g. duration of notes, repeating 'pattern' or the basic structural element). We exploit the small and hidden deviations of evenness of the selected preludes in order to connect between the natural and the learned schemata. We studied the natural schemata - curves of pitch, the 'internal organ points' or as we called it: 'center of gravity' and other parameters, by using musical and mathematical tools (mainly statistical and geometrical). We have identified the location of significant deviations of those 'even' parameters and compared them to the tonal organization of the piece (the learned schemata). The comparison was done at different levels of musical organization.
          Our findings suggest that a similar connection is present in different musical styles, as well as raise some general questions concerning the nature of even musical works.


SPECIAL SUMMER TALK:

Date: Thursday, July 28, 2016
Time: 5:00 pm-5:48 pm
Speaker: Manuel Kauers, Johannes Kepler University, Linz, Austria
Title: Factorization of C-finite Sequences
Abstract:
          We all know that the sum and the product of two C-finite sequences is again C-finite, and it is not hard to find recurrence equations for the sum or the product of two given C-finite sequences using linear algebra. Conversely, it is also not too difficult to decide whether a given C-finite sequence can be written nontrivially as a sum of two simpler C-finite sequences. This just requires the factorization of a univariate polynomial. In the talk, we will consider the analogous problem for the product: given a C-finite sequence, the task is to decide whether it can be written nontrivially as the product of two simpler C-finite sequences. An algorithm for solving this problem was first given by Ritt in the 1920s. We will present an alternative algorithm that is somewhat simpler and seemingly not less efficient than Ritt's approach, and we mention two applications to tiling problems and statistical mechanics. This is joint work with Doron Zeilberger.
Posted on Vimeo (2 parts): Part 1 Part 2


Fall 2016

Date: September 15, 2016
Speaker: Doron Zeilberger, Rutgers
Title: Jonathan Borwein(1951-2016) a PiONEER of Experimental Mathematics
Abstract:
          The world's undisputed leader in Experimental Mathematics, Jonathan Borwein, died suddenly on Aug. 1, 2016, thereby depriving us of so much potential great math that he would have done, if he did not die so young. Hence it is our duty to study his universal message and try to follow the many paths that he opened-up.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: September 22, 2016
Speaker: Bahman Kalantari, Rutgers (CS)
Title: Some Recent Results on Polynomials and Polynomiography
Abstract:
          Polynomials are truly mysterious and beautiful. For many years I have been playing with complex polynomials from an independent point of view, perhaps more as computer scientist than a mathematician. A by-product is Polynomiography: Algorithmic visualization in solving a polynomial equation. The resulting images, called polynomiographs, are not necessarily fractal images. There is a naïve misconception that any image coming from iterations must be fractal! Indeed polynomiography enriches fractals, both visually and theoretically. Moreover, it connects polynomials to many other subjects not considered in standard applications. Based on many experiences and interactions with diverse audiences - including perhaps over 100 presentations in more than a dozen countries - there is convincing evidence that polynomiography could become widely popular, leading to novel applications of polynomials in STEM, as well as in art and design. Ironically, formal education is slow in embracing it! This is surprising, especially in view of the general interest in popularizing STEM. I invite mathematicians, STEM educators and students to rethink their notions of polynomials and to explore polynomiography.
          In this talk I will highlight some recent results on polynomials and polynomiography from the following:
          1. How Many Real Attractive Fixed Points Can a Polynomial Have? We derive an explicit formula for a complex polynomial with a prescribed set of fixed points and corresponding multipliers. Using the formula, we prove a polynomial of degree n can have at most ⌈n/2⌉ fixed points lying on any line in the complex plane. (Arxiv)
          2. Solving a Cubic Equation by the Quadratic Formula We prove, a cubic complex polynomial with distinct roots and distinct critical points must have a root whose Voronoi cell contains a critical point. By defining a third order homogeneous recurrence relation at such a critical point, we generate a sequence guaranteed to converge to that root. This gives a new method for solving a cubic equation different from Cardano's formula, easy to remember and maybe more practical. (Arxiv)
          3. A Necessary and Sufficient Condition for Local Maxima of Polynomial Modulus Over Unit Disc We give necessary and sufficient condition for a local maximum of polynomial modulus over the unit disc, proving that it is a fixed point of a certain function. In particular, the infinity norm of a polynomial is attained at a point satisfying a convenient formula. The formula suggests iterative methods for computing the infinity norm. We give two such algorithms, including a Newton-like method and present some corresponding polynomiography. (Arxiv)
          4. The 3x+1 Polynomials, Their Zeros and Polynomiography To each natural number N satisfying the famous 3x+1 property (conjectured to be valid for any natural number), we associate a unique monic polynomial, called its 3x+1 polynomial, having N as the constant term. The 3x+1 polynomial implies a unique factorization of N in terms of the product of its roots. We using an existing family of bounds on the modulus of zeros of a general polynomial to bound the modulus of the zeros of the 3x+1 polynomial. We give some associated polynomiography and extend the results to Gaussian integers. (Forthcoming)
Posted on Vimeo (2 parts): Part 1 Part 2


Date: September 29, 2016
Speaker: Eric Rowland, Hofstra University
Title: Computer verification of integer sequences avoiding a pattern
Abstract:
          Is there an infinite sequence on the alphabet {0, 1, 2} containing no block that occurs twice consecutively? Questions like this were investigated a century ago by the Norwegian mathematician Axel Thue, who produced some of the earliest results in combinatorics on words. If a pattern is avoidable on a given alphabet, it is natural to ask about the lexicographically least sequence that avoids the pattern. Occasionally the structure of this sequence can be discovered and proved by hand. But for many patterns this sequence is sufficiently complex that computer-assisted discovery, followed by automated proofs, seems to be necessary to make any progress.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 6, 2016
Speaker: Jay Pantone, Dartmouth College
Title: Approximate Asymptotic Analysis of Combinatorial Sequences
Abstract:
          In enumerative combinatorics, it is quite common to have in hand a number of known initial terms of a combinatorial sequence whose behavior you'd like to study. In this talk we'll describe the Method of Differential Approximants, a technique that can be used to shed some light on the nature of a sequence using only some known initial terms. While these methods are, on the face of it, experimental, they often lead to rigorous proofs. We'll exhibit the usefulness of this method through a variety of combinatorial topics, including chord diagrams, permutation classes, and inversion sequences.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 13, 2016
Speaker: Anthony Zaleski, Rutgers University
Title: Simultaneous core partitions
Abstract:
          The hook of a box in the Young diagram of a partition consists of the box itself and those directly below it and to its right. An s-core partition is one with no hooks of size s; an (s,t)-core partition avoids hooks of size s and t. We discuss computer methods used to analyze such partitions and give automated (and rigorous) proof of some asymptotic results.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 20, 2016
Speaker: Sateesh R. Mane, Convergent Computing
Title: Multi-parameter Fuss-Catalan numbers
Abstract:
          The Fuss-Catalan numbers are a generalization of the more well-known Catalan numbers. The Fuss-Catalan numbers will be defined, also with a short historical sketch. Pertinent recurrence relations, convolution identities, generating functions, etc will be presented. Sample applications to combinatorics, probability & statistics and complex analysis will be shown. The emphasis in this talk will be on the generality of the usefulness of Fuss-Catalan numbers.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: October 27, 2016
Speaker: Vladimir Retakh, Rutgers University
Title: Noncommutative Catalan numbers
Abstract:
          We introduce and study noncommutative Catalan "numbers" as Laurent polynomials in free variables and related theory of noncommutative binomial coefficients. We also study their (commutative and noncommutative) specializations, relate them with Garsia-Haiman (q,t)-versions, and establish total positivity of the corresponding Hankel matrices. This is my work in progress with Arkady Berenstein from Univ. of Oregon.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: November 3, 2016
Speaker: Doron Zeilberger, Rutgers University
Title: The Experimental Mathematics of Voting
Abstract:
          As Condorcet, Arrow, and other people, have shown, there is no perfect voting system, but it is just as well, since voting raises interesting mathematical problems, many of them explorable with experimental mathematics.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: November 10, 2016
Speaker: Nathan Fox, Rutgers University
Title: Hofstadter-like Sequences over Nonstandard Integers
Abstract:
          The Hofstadter Q-sequences is defined by the recurrence Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2)) with the initial conditions Q(1)=1 and Q(2)=1. Despite its simple definition, almost nothing has been proved about this sequence. Most notably, we still do not know whether Q(n) even exists for all n, i.e. if the sequence is infinite or if it "dies." On the other hand, many related sequences are known either to die or not to die. In this talk we will explore some variants of the Hofstadter Q-sequence, and we will describe a method for proving that certain infinite families of sequences all die. This method will involve constructing sequences whose indices and values can be nonstandard integers, which are "integers" that are larger than any natural number.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: November 17, 2016
Speaker: Emily Sergel, University of Pennsylvania
Title: Searching for tableaux statistics and Schur expansions
Abstract:
          This talk covers joint work with Jim Haglund. It will discuss our search for a new family of inversion statistics on standard Young tableaux. Our motivation is the problem of expanding certain LLT polynomials in terms of the Schur basis. The Schur expansions of certain symmetric functions encode the decompositions of some Sn-modules into irreducible representations. Many symmetric functions appearing in the study of Macdonald polynomials and the module of Diagonal Harmonics can be written as positive sums of LLT polynomials. Hence, finding combinatorial Schur expansions for all LLT polynomials would solve many open problems. For example, it would finally give a combinatorial proof of the Macdonald positivity conjecture. The recent proof of the Shuffle Conjecture gives new tools for studying the special case of unicellular LLT polynomials. I will show how computer experimentation is guiding our (ongoing) search for their Schur expansions and the corresponding inversion statistics.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: December 1, 2016
Speaker: Leif Albert, Rutgers University
Title: Order and Chaos in Some Deterministic Infinite Trigonometric Products
Abstract:
          An infinite trigonometric product function pondered some years ago by reknowned experimental mathematician Benoit Cloitre is discussed. This product represents a deterministic function which fluctuates seemingly chaotically about a regular trend. A surmise of Cloitre about this trend is proved. It is also shown that Cloitre's product is the characteristic function of the distribution of the endpoints of a random walk that is associated with a random Riemann zeta function.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: December 8, 2016
Speaker: David Nacin, William Paterson University
Title: A Complex KenKen Classification
Abstract:
          In the blog bit-player, Brian Hayes presented a KenKen puzzle over the complex numbers and stated that the uniqueness had not yet been checked. We present a proof of uniqueness and the classification of all four-by-four puzzles with unique solutions over all possible cage patterns. This result was originally done entirely in Python. It also follows from the work we present here, which involves the interactions between group actions on the sets of all possible puzzles and the set of order four Latin squares.
Posted on Vimeo (2 parts): Part 1 Part 2