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.