RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2018


Spring 2018

Date: January 25, 2018
Speaker: Adi Ben-Israel, Rutgers University
Title: Newton's Method: Universality and Geometry
Abstract:
          LaTeX Abstract is in PDF here.
The lecture has 3 sections:
(1) Given functions u,f:D → D ⊂ ℝ, if u(x)=1-f(x)/f'(x) for all x in D, we call f the inverse Newton transform of u, denoted f=N-1u. If 1/(x-u(x)) is integrable, then
(N-1u)(x)=C⋅ exp{∫ dx/(x-u(x))}, C ≠0.

For such u, the iteration x+:=u(x) (away from its fixed points) is a Newton method on f, and the relations between (fixed points, monotonicity, of) u and (roots, convexity, of) f give a simple explanation of chaotic behavior, illustrated here for the logistic iteration (see citation in LaTeX document).
(2) A geometric interpretation of the complex Newton iteration, z+:=z-f(z)/f'(z), f analytic, (see citation in LaTeX document), allows extending the results of (1) to complex iterations. This is illustrated for the Mandelbrot set.
(3) An iterative method for minimizing a convex function f:ℝn→ℝ with an attained infimum, proceeds by bracketing the minimum value in nested, decreasing intervals. Each iteration consists of one Newton iteration, and the method has an advantage of fast convergence and a natural stopping criterion. This is illustrated for the Fermat--Weber location problem, (see citations in LaTeX document).

Posted on Vimeo (2 parts): Part 1 Part 2
Slides from talk


Date: February 1, 2018
Speaker: Gleb Pogudin, New York University
Title: Power series expansion of the free energy for monomer-dimer tilings.
Abstract:
          Let a(n, p), where 0 < p < 1, be the number of tilings of n-by-n square with dimers and monomers, such that dimers occupy approximately p of the area. Then the following limit characterizes the growth of this number and is called the free energy of the monomer-dimer system:
f(p) = lim log( a(n, p) ) / n^2 with n -> infinity

It is an open problem how to compute f(p). I will describe a method allowing to compute the first 63 terms of its Taylor expansion at zero. This yields to accurate estimates for many values of p. I will also discuss other approaches to computing f(p) and interesting mathematical problems connected with them.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 8, 2018
Speaker: Manuel Kauers, Johannes Kepler Universität
Title: Symmetry Breaking in SAT and QBF
Abstract:
          In principle it is easy to find a solution of a boolean formula--just try out all possibilities. In practice however, the problem is not so simple, because the number of possibilities is so huge. Although it is hopeless to iterate over, say, 2^10000 possibilities, modern SAT can handle problems with 10000 or even more variables. They have a chance of success because they employ a carefully chosen combination of several techniques for cutting off irrelevant parts of the search tree. One such technique consists of exploiting the symmetries of the input formula. We will explain how this works and then present some recent joint work with Martina Seidl on an extension of these ideas to so-called quantified boolean formulas.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 15, 2018
Speaker: Vladimir Retakh, Rutgers University
Title: Noncommutative Rogers-Ramanujan continued fraction and related results
Abstract:
          Various applications of quasi-determinants to continued fractions and polynomial equations over non-commutative division rings will be discussed.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: February 22, 2018
Speaker: Kelsey Horan, CUNY
Title: Fast Quantum Algorithm for Solving Multivariate Quadratic Equations
Abstract:
          After the announcement for the transition to post-quantum secure cryptographic constructions by the US National Security Agency the cryptography community has been working towards developing and evaluating standards. Of particular interest is the calculation of the quantum bit security for many proposed post-quantum cryptosystems. This talk addresses the problem of solving a system of m boolean multivariate quadratic equations in n variables, the MQ2 problem -- a problem that is central to evaluating the quantum security of many cryptosystems. A Las-Vegas quantum algorithm for solving the boolean multivariate quadratic problem, which requires in expectation the evaluation of O(2^(0.462n)) quantum gates, will be presented.
Posted on Vimeo (2 parts): Part 1 Part 2 Part 3


Date: March 1, 2018
Speaker: Matthew Russell, Rutgers University
Title: Bijective proofs of some new MacMahon-style partition identities
Abstract:
          Two of Doron's favorite mathematicians are J. J. Sylvester and Major P. A. MacMahon. I will make Doron happy by using bijective ideas drawn from the work of the former to prove partition identities that generalize work of the latter. Many of these identities were conjectured after computer searches that were joint work of the speaker with Shashank Kanade and Debajyoti Nandi.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 8, 2018
Speaker: Giora Dula, Netanya Academic College
Title: Various Constructions and Applications of Hadamard and Weighing Matrices
Abstract:
          LaTeX Abstract is in pdf here.
An n×n matrix H with terms in {-1,1} is called an Hadamard matrix if HHT = HTH = nIn.
A matrix W with terms in {-1,0,1} is called a weighing matrix W(n,w) if WWT = WTW = wIn for some natural number w≤n. Every Hadamard matrix is a W(n,n).
The subject of weighing matrices is growing fast and relates to other mathematical fields e.g. information theory, number theory, group theory and topology.
In the first part of this talk we will survey the early works of Sylvester, Walsh and of Hadamard on Hadamard matrices. These works relate to error correcting codes in various different ways. Nasa used Hadamard code to transmit images from Mars.
We will discuss the usage of (complex) Hadamard matrices to quantum random access codes. Then we will survey the construction of Payley that gave many new Hadamard matrices. The construction uses weighing matrices.
Then we will survey the works of Koukouvinos and Seberry and of Harada and Munemasa which relate weighing matrices with optimal weighing designs, optical multiplexing and again with (self dual) codes.
In the second part of this talk we will survey some other classical results e.g Strassler, Craigen and Eliyahu and Kervaire
In the last part we will present the basic ideas that led to the solution of W(23,16) using shadow geometries.
Keywords: weighing matrix, geometry, local geometry
Posted on Vimeo (2 parts): Part 1 Part 2 Slides from talk


Date: March 15, 2018
NO SEMINAR DUE TO RUTGERS' SPRING BREAK


Date: March 22, 2018
Speaker: Dimitrios Tsementzis, Rutgers University
Title: The Univalent Foundations through UniMath and some combinatorial problems
Abstract:
          The Univalent Foundations is a proposed foundation for mathematics in which the basic objects are spaces, rather than sets. I will introduce the basic ideas of the Univalent Foundations, explain how they are implemented in a computer using a formal system called UniMath, and discuss certain combinatorial problems that arise in this context that might be of interest to experimentalists.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 29, 2018 4:00pm
Speaker: Anthony Zaleski, Rutgers University
Title: An experimental mathematics approach to some combinatorial problems (Thesis Defense)
Abstract:
          In this summary of my thesis research, I will explain how we used experimental mathematics methods to discover new results about random walks, simultaneous core partitions, and Boolean functions.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: March 29, 2018 5:00pm
Speaker: Bryan Ek, Rutgers University
Title: Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics (Thesis Defense)
Abstract:
          The main goal of these projects was utilizing experimental mathematics to further our knowledge of several areas in math.
 We begin by tweaking a proof of unimodality by O'Hara to produce many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial. We then employ a generating function relation technique used by Ayyer and Zeilberger to analyze lattice walks with a general step set in bounded, semi-bounded, and unbounded planes. The method in which we do this is formulated to be highly algorithmic so that a computer can automate most, if not all, of the work. I easily recover many well-known results for simpler step sets and discover new results for more complex step sets.
Posted on Vimeo (2 parts): Part 1 Part 2 Slides from talk


Date: April 5, 2018
Speaker: Andrew Lohr, Rutgers University
Title: Several Topics in Experimental Mathematics (Thesis Defense)
Abstract:
          First, we'll talk about the total height statistic on a certain family of random graphs. We are able to get Maple to compute moments of this statistic. Taking limits, we confirm, via elementary methods, the fact, due to David Aldous, that the limiting (scaled) distributions are all the same. Second, we'll talk about generalizations of Sister Celine's method and Gosper's algorithm for evaluating summations. For both, we greatly extend the classes of applicable functions. For the generalization of Sister Celine's method, we allow summations of arbitrary products of hypergeometric terms and linear recurrent sequences with rational coefficients. For the extension of Gosper's algorithm, we extend it from solely hypergeometric sequences to any multi-basic sequence. For both, we give numerous applications to proving, or reproving in an automated way, interesting combinatorial problems. We'll also discuss the bunk bed conjecture, showing that in some special cases that were previously unknown, the statement of the conjecture holds.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: April 12, 2018
Speaker: Dan Romik, UC Davis
Title: Computer-assisted explorations and proofs in the moving sofa problem
Abstract:
          The moving sofa problem is a well-known open problem in geometry. It asks for the planar shape of largest area that can be moved around a right-angled corner in a two-dimensional hallway of width 1. In this talk I will survey the known results about the problem, which has a surprisingly rich structure, and explain several ways in which its study is informed by experimental mathematics. In particular, I will discuss recent results derived in joint work with Yoav Kallus, in which we developed and implemented an algorithm to prove new upper bounds for the area of a moving sofa shape, improving a 1968 result by Hammersley.


Date: April 19, 2018
Speaker: Richard Voepel, Rutgers University
Title: Experimental "Solutions" to Select Stopping Problems
Abstract:
          In the realm of statistics and economics, there are several important problems that can be described as stopping problems; a kind of decision problem where an actor must observe some sequence of random variables, and based on observations of those variables implement a stopping rule (often gaining some reward based on observations). The gold standard for "solving" these stopping problems is providing a stopping rule for maximizing expected gains or minimizing expected losses, but such a stopping rule need not be the "best" rule depending on context. In this talk we will introduce stopping problems by way of a classical example of coin flipping, and explore the role of experimental mathematics in the construction of a family of stopping rules for the problem of Shepp's Urn.
Video posted on Vimeo Slides from talk


Date: April 26, 2018
Speaker: Mingjia Yang, Rutgers University
Title: A Journey into Clusters-the Goulden-Jackson method and All That
Abstract:
          The Goulden-Jackson Cluster method is a powerful way for finding the generating function for the number of words avoiding consecutive patterns. We will first discuss this method and some of its extensions. Then we will journey on to see a nice (symmetric) generating function for the number words that avoid the pattern 12...r that we can guess using the computer, and how to justify that by human means using the Cluster method. Time permitting, we will also discuss extension to words with a certain number of the consecutive pattern 12...r and recurrences we came up with. This is joint work with Doron Zeilberger.


Summer Special Talks

Date: Friday, July 20th, 2018
Time: 4:00pm
Place: Hill 525 (Note room change)
Speakers: Elaine Wong and Christoph Koutschan , (joint talk), Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences
Title: Symbolic evaluation of determinants and rhombus tilings of holey hexagons
Abstract:
          We investigate a curious determinant that was first mentioned by George Andrews in 1980 in the context of descending plane partitions. It is found to be a special instance of a two-parameter family of determinants that count certain collections of nonintersecting lattice paths, or, equivalently, cyclically symmetric rhombus tilings of a hexagon with several triangular holes inside. We find closed forms for several one-parameter subfamilies, both by applying combinatorial arguments and by applying Zeilberger's "holonomic ansatz".
Posted on Vimeo (2 parts): Part 1 Part 2 Slides from talk


Date: Friday, July 20th, 2018
Time: 5:00pm
Place: Hill 525 (Note room change)
Speaker: Shaoshi Chen, KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Title: How to generate all possible WZ-pairs algorithmically?
Abstract:
          The Wilf-Zeilberger theory has become a bridge between symbolic computation and combinatorics.
Through this bridge, not only classical combinatorial identities from handbooks and long-standing conjectures in enumerative combinatorics are proved algorithmically, but also some new identities and conjectures related to mathematical constants are discovered via computerized guessing.
WZ-pairs play a leading role in the WZ theory whose early applications can be traced back to Andrei Markov's 1890 method for convergence-acceleration of series for computing ζ(3). For applications, it is crucial to have WZ-pairs at hand. In the previous works, WZ-pairs are cooked either by guessing from the identities to be proved using Gosper'algorithm or by certain transformations from a given WZ-pair.
In this talk, we first present a structure theorem on the possible form of all rational WZ-pairs, and then we will illustrate how one could go beyond the rational case using Ore-Sato theorem.
We hope these studies could enable us discover more combinatorial identities in an intrinsic and algorithmic way.
Posted on Vimeo (2 parts): Part 1 Part 2

Fall 2018


Date: Sep. 6, 2018
Speaker: Doron Zeilberger, Rutgers University
Title: How likely to succeed is a Bitcoin attack, and how long should it take?
Abstract:
          We use experimental mathematics, and Wilf-Zeilberger algorithmic proof theory, to compute probabilities and duration of bitcoin "double-spend" attacks. [Joint work with Evangelos Georgiadis].
Posted on Vimeo (2 parts): Part 1 Part 2

Date: Sept. 13, 2018
Speaker: Noam Zeilberger, University of Birmingham, UK
Title: A proof-theoretic analysis of the rotation lattice of binary trees
Abstract:
          The classical Tamari lattice Yn is defined as the set of binary trees with n internal nodes, with the partial ordering induced by the (right) rotation operation. It is not obvious why Yn is a lattice, but this was first proved by Haya Friedman and Dov Tamari in the late 1950s. More recently, Frédéric Chapoton discovered (via the OEIS) another surprising fact about the rotation ordering, namely that Yn contains exactly 2(4n+1)! / ((n+1)!(3n+2)!) pairs of related trees. (Even more surprisingly, this formula was already computed by Bill Tutte in the early 1960s...but for a completely different family of objects!) In the talk I will describe a new way of looking at the rotation ordering that is motivated by old ideas in proof theory. This will lead us to systematic ways of thinking about: 1. the lattice property of Yn, and 2. the Tutte-Chapoton formula for the number of intervals in Yn. (Based on this paper)
Posted on Vimeo (2 parts): Part 1 Part 2

Date: Sep. 20, 2018
Speaker: Michael Kiessling, Rutgers University
Title: Experimental mathematical evidence for a conjecture about relative entropy with complex measures
Abstract:
          Relative entropy of a probability measure, relative to a given a priori measure, is a well-defined and extremely useful notion in probability, statistics, and statistical mechanics. A question by Alice Chang has prompted me to introduce the notion of an `entropy of signed and complex measures relative to a signed a priori measure.' In this talk I will explain Alice Chang's question, and why entropy of signed and complex measures relative to a signed measure may hold the answer. I show computer-generated experimental evidence for my proposal.
Posted on Vimeo (2 parts): Part 1 Part 2
Slides from the talk (with two typos corrected, one regarding a minus sign and the other regarding the spelling of Schwartz): Slides from talk


Date: Sept. 27, 2018
Speaker: Yukun Yao, Rutgers University
Title: Parking Function, Bijection and Area Statistic
Abstract:
          In this talk, we will talk about the concepts and properties of parking functions, the bijection between parking functions and rooted labelled forests, and the area statistic of parking functions. With the experimental and finitistic method, the expectation, variance and higher moments of the area statistic of the parking functions can be derived. They can also be expressed as a polynomial of the expectation and the length n. Their limiting distribution will be discussed as well.


Date: Oct. 4, 2018
Speaker: Yan-Ping Mu, Tianjin University of Technology and Rutgers University
Title: Telescoping method and congruences for double summations
Abstract:
          In recent years, Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double summations of hypergeometric terms. Using the telescoping method and certain mathematical software packages, we transform such a double summation into a single sum. With this new approach, we confirm several open conjectures of Sun. In this talk, I will mainly explain in detail the transformation.


Date: Oct. 11, 2018
Speaker: Akshunna Shaurya Dogra
Title: Minimal length representations of the natural numbers and why they matter
Abstract:
          This talk will discuss minimal length representations of the Natural numbers under elements from a pre-defined symbol library O. We will mostly focus on symbol libraries consisting of 1 and different kinds of hyper-operations. We will go over both general and specific patterns emerging from different choices of symbol libraries, obtain bounds on the lengths expected, define numbers of special interest and prove and/or experimentally verify the existence of certain behaviors in these numbers. If time permits, we will also discuss how the results obtained can be generalized to a discussion on discrete posets, explore connections to monoid theory and prove/predict number theoretic results from the machinery discussed in the talk.
Posted on Vimeo (2 parts): Part 1 Part 2

Date: Oct. 18, 2018
Speaker: Angela Hicks, Lehigh University
Title: Applying Representation Theory to Random Walks
Abstract:
          We'll talk about how representation theory can be used to determine fundamental questions about the rate of convergence of random walks on groups, and how the approach benefits from computer experimentation of eigenvalues of certain matrices. We'll also discuss some of the difficulties in this approach, here focusing on joint work with Daniel Bump, Persi Diaconis, Laurent Miclo, and Harold Widom studying a simple random walk on the the Heisenberg group mod p (a particularly simple to describe noncommutative group). Analysis of a random walk on the group dates back to Zach, who was considering the effectiveness of certain random number generators. We'll assume a bit of basic (undergraduate) group theory and probability, but otherwise aim for an elementary talk.


Also of Interest (Rutgers Math Colloquium, Oct. 19, 2018)
Speaker: Nima Arkani-Hamed (Institute for Advanced Study)
Title: SpaceTime, Quantum Mechanics, and Positive Geometry
Posted on Vimeo (4 parts): Part 1 Part 2 Part 3 Part 4

Date: Oct. 25, 2018
Speaker: Andrew Sills, Georgia Southern University
Title: Using number theory and combinatorial optimization to solve a problem in statistics.
Abstract:
          See title. This is joint work with Charles Champ.
Posted on Vimeo (2 parts): Part 1 Part 2

Date: Nov. 1, 2018
Speaker: Yonah Biers-Ariel, Rutgers University
Title: Schemes for Words
Abstract:
          We build schemes to count the number of words avoiding a certain (finite) set of patterns. We can handle (at least in principle) infinitely many such sets of patterns.
Posted on Vimeo (2 parts): Part 1 Part 2



Date: Nov. 8, 2018
Speaker: Philip Matchett Wood, University of Wisconsin
Title: Factoring random polynomials
Abstract:
          Can you factor a random polynomial with integer coefficients into smaller factors also with integer coefficients?  Work on this simple question goes back more than 80 years, and there is still much that is not known. The talk will focus on computational work leading to a general heuristic on what the factoring behavior should be.  We will also discuss some new results on the probability of a random polynomial having factors of low degree.  Joint work with Sean O'Rourke and also joint work with Christian Borst, Evan Boyd, Claire Brekken, Samantha Solberg, and Melanie Matchett Wood.
Posted on Vimeo (2 parts): Part 1 Part 2



Date: Nov. 15, 2018
Speaker: Neil Sloane, The OEIS Foundation and Rutgers University
Title: Coordination Sequences, Planing Numbers, and Other Recent Sequences
Abstract:
          Take a periodic tiling of the plane and consider its associated graph. The coordination sequence with respect to a node P gives the number of nodes that are n edges away from P. Chaim Goodman-Strauss and I have a new and simple method for obtaining generating functions for such sequences, which will be illustrated with several examples. There are a number of interesting open questions. I will also discuss Lenormand's operation that planes down numbers (e.g., 231 becomes 27).
Posted on Vimeo (2 parts): Part 1 Part 2 Slides from talk



Date: Nov. 29, 2018
Speaker: John Chiarelli, Rutgers University
Title: Construction of the Low-Degree Boolean Polynomials
Abstract:
          The degree of a boolean function is an important measure of its complexity, but finding all such functions quickly becomes very computationally dense. In this talk, I will discuss the way in which we can use a property of boolean functions, called the maxonomial hitting set size, to aid in generating all boolean polynomials of degree 3, and how we can take advantage of symmetries to improve our runtime. I will also talk about how this catalog is useful in gaining a greater understanding of the maxonomial hitting set size of arbitrary functions.
Posted on Vimeo (2 parts): Part 1 Part 2



Date: Dec. 6, 2018
Speaker: Sandra Kingan, CUNY
Title: Quasiregular Matroids
Abstract:
          Regular matroids are binary matroids with no minors isomorphic to the Fano matroid or its dual. The Fano matroid is the binary projective plane PG(2, 2). Seymour proved that 3-connected regular matroids are either graphs, cographs, or a special matroid R10 called a splitter, or else can be decomposed along a non-minimal exact 3-separation induced by another special matroid R12 called a 3-decomposer. Quasiregular matroids are binary matroids with no minor isomorphic to E4, where E4 is a 10-element rank 5 self-dual binary matroid. The class of quasiregular matroids properly contains the class of regular matroids. I will describe how I decomposed quasiregular matroids in a manner similar to regular matroids. There is a compuatational aspect to this result which will be the focus of this talk. A portion of this talk is joint work with Manoel Lemos.
Posted on Vimeo (2 parts): Part 1 Part 2