Rutgers Experimental Mathematics Seminar

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 HH^T = H^TH = nI_n.
A matrix W with terms in {-1,0,1} is called a weighing matrix W(n,w) if WW^T = W^TW = wI_n 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.