RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2019


Spring 2019


Date: Jan. 31, 2019
Speaker: Neil Sloane, the OEIS Foundation and Rutgers University
Title: Coordination Sequences, Planing Numbers, and Other Recent Sequences (II)
Abstract:
          Take the graph of a periodic tiling of the plane. 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 simple method for obtaining generating functions for such sequences. There are a number of interesting open questions. I will also discuss Lenormand's "raboter" operation that planes down numbers (e.g., 231 = 11100111_2 becomes 27 = 11011_2). This is an updated version of a talk given to just 4 people during the blizzard of Nov 15 2018. There is also a lovely new open problem: the knight's-move version of the Ulam-Warburton cellular automaton (A319018).
Posted on Vimeo (2 parts): Part 1 Part 2
Slides from talk


Date: Feb. 7, 2019
Speaker: Emily Sergel, University of Pennsylvania
Title: Some tools for proving asymptotic normality and an application to cores
Abstract:
          An s,t-core partition is a partition with no hook length equal to s or t. Recently Ekhad-Zeilberger used experimental methods to prove that the asymptotic distribution size among s,t-cores (with s-t fixed) is not normal. Following that, Zaleski gave strong evidence that (s,s+1)-cores with distinct parts are asymptotically normally distributed. My coauthors and I found both results surprising, and the proof we discovered for the later one uses some beautiful, classical results. The main purpose of this talk is to spread these tools. If time permits, I will also discuss the possibility (and challenge) of applying these tools to related core problems. Joint work with János Komlós and Gábor Tusnády.
Posted on Vimeo (2 parts): Part 1 Part 2


Date: Feb. 14, 2019
Speaker: Glenn Shafer, Rutgers Business School
Title: Game-Theoretic Foundations for Probability and Statistics
Abstract:
          Fermat and Pascal's two different methods for solving the problem of division lead to two different mathematical foundations for probability theory: a measure-theoretic foundation that generalizes the method of counting cases used by Fermat, and a game-theoretic foundation that generalizes the method of backward recursion used by Pascal. The game-theoretic foundation has flourished in recent decades, as documented by my forthcoming book with Vovk, Game-Theoretic Foundations for Probability and Finance. In this book's formulation, probability typically involves a perfect-information game with three players, a player who offers betting rates (Forecaster), a player who tests the reliability of the forecaster by trying to multiply the capital he risks betting at these rates (Skeptic), and a player who decides the outcomes (Reality).

          In this talk I will review the game-theoretic foundation for probability as briefly as possible and then discuss its application to mathematical statistics. The usual formulation for mathematical statistics begins with the assumption that the statistician has only partial knowledge of the probability measure that describes a phenomenon. The corresponding game-theoretic move is to suppose that the statistician stands outside the perfect-information game, seeing only some of the moves or some of its consequences.
Posted on Vimeo
Slides from talk


Date: Feb. 21, 2019
Speaker: Colin R. Defant, Princeton University
Title: Structure in Stack-Sorting
Abstract:
          The study of permutation patterns began with Knuth's analysis of a certain "stack-sorting algorithm" in 1968. In his 1990 PhD. thesis, West investigated a deterministic variant of Knuth's algorithm, which we can view as a function s that defines a dynamical system on the set of permutations. He defined the fertility of a permutation to be the number of preimages of that permutation under s. We will describe a colorful method for computing the fertility of any permutation, answering a question of Bousquet-Mélou. Applications of this method allow us to reprove and generalize several known theorems and improve the best known upper bound for the number of so-called "t-stack-sortable" permutations of length n when t=3 and when t=4. The method also allows us to connect the stack-sorting map with free probability theory, many well-studied combinatorial objects, and several interesting sequences. Finally, we will consider two operators on words that extend the stack-sorting map.
Posted on Vimeo
Slides from talk


Date: Feb. 28, 2019
Speaker: Chaim Even Zohar, University of California, Davis
Title: Patterns in Random Permutations
Abstract:
          Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n? The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order 1/sqrt(n), while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to representations of Sk. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.
Posted on Vimeo


Date: Mar. 7, 2019
Speaker: Nathan Fox, College of Wooster
Title: Trees, Fibonacci Numbers, and Nested Recurrences
Abstract:
          Conolly's sequence (A046699) is defined by the nested recurrence relation C(n)=C(n-C(n-1))+C(n-1-C(n-2)) and the initial conditions C(1)=C(2)=1. This sequence is monotone increasing with each positive integer appearing at least once, a property known in the literature as slow. Conolly's sequence and several other slow sequences generated by nested recurrences are known to have combinatorial interpretations in terms of enumerating leaves in infinite tree structures. For the Conolly sequence, the tree-based interpretation illuminates an intimate connection between the sequence and the powers of two. In fact, the Conolly sequence has an alternate, purely number-theoretic definition based on powers of two. Replacing powers of two with Fibonacci numbers in this construction yields a different slow sequence (A316628). In this talk, we will describe the three different ways of constructing the Conolly sequence (recurrence, number theory, trees) and show why they all yield the same sequence. Then we will construct this new sequence based on the Fibonacci numbers, which also is slow, has a tree-based interpretation, and satisfies a nested recurrence. If time permits, we will describe how to generalize the construction to discover many new integer sequences.
Posted on Vimeo


Date: Mar. 14, 2019
Speaker: Doron Zeilberger, Rutgers University
Title: What is Pi, and what it is not
Abstract:
          The short answer to the second question of the title is: "It is NOT a number". For a longer answer, and for an answer to the first question, come to the talk.
Posted on Vimeo


Date: Mar. 21, 2019
Spring Break (no talk)


Date: Mar. 28, 2019
Speaker: Craig Larson, Virginia Commonwealth University
Title:Automated Conjecturing in Mathematics - with the CONJECTURING Program
Abstract:
          I will describe the ideas underlying the program CONJECTURING which can be used to make conjectures about upper or lower bounds of invariants (or necessary or sufficient conditions for properties) for a wide variety of mathematical objects. The conjecturing heuristic is the heuristic Fajtlowicz used in his Graffiti program; our program has been broadly generalized to be useful in many domains - and the code is open source, and can be installed as a Sage package. We will give several examples of theorems conjectured by the program, as well as a selection of open conjectures. We will also mention ways to leverage the program to maximize its utility as a tool for researchers. This is joint work with Nico Van Cleemput (Ghent University).
Posted on Vimeo
Slides from talk


Date: Apr. 4, 2019
Speaker: Jesús Guillera, University of Zaragoza
Title: When 1/pi^2 and Calabi-Yau meet
Abstract:
          In this lecture, in memory of Gert Almkvist (1934-2018), I will describe some fascinating connections he found between a family of series for 1/pi^2 and Calabi-Yau theory.
Posted on Vimeo
Slides from talk


Date: Apr. 11, 2019
Speaker: Ein-Ya Gura, Hebrew University of Jerusalem
Title: Game Theory-An Alternative Mathematical Experience
Abstract:
          Few branches of mathematics have been more influential in the social sciences than game theory. In recent years, it has become an essential tool for all social scientists studying the strategic behavior of competing individuals, firms, and countries. However, the mathematical complexity of game theory is often very intimidating for students who have only a basic understanding of mathematics. The book: Insights into Game Theory: An Alternative Mathematical Experience addresses this problem by providing students with an understanding of the key concepts and ideas of game theory without using formal mathematical notation. In this talk I will present the main ideas of the book, what was our motivation to write the book, and what we are doing now by using it.
Posted on Vimeo
Slides from talk


Date: Apr. 18, 2019
Speaker: Harry Crane, Rutgers (Statistics)
Title: Some experimental observations and open questions about the alpha-permanent
Abstract:
          The alpha-permanent is a matrix function that has a similar algebraic form to the determinant but exhibits very different computational behavior. The permanent (alpha=1) is known to be #P-complete, and is a fundamental object in computational complexity theory. The more general alpha-permanent appears in statistical models for point pattern data and combinatorial data (e.g., partition and permutation data), where its computational complexity limits its applied use in many cases. I'll discuss some algebraic properties of the alpha-permanent which suggest natural connections to familiar concepts in probability theory and statistics. I'll also describe some immediate research problems that arise out of these observations.
Posted on Vimeo


Date: Apr. 25, 2019
Speaker: Eddy Chen, Rutgers (Philosophy)
Title:An Introduction to the Foundations of Quantum Theory
Abstract:
          In this brief talk, we discuss the basic ingredients of quantum theory and some steps towards understanding what it means. To that end, we focus on a simple experiment (the double slit experiment), which nicely illustrates the "quantum measurement problem". We suggest that the standard textbook version of quantum mechanics, though making correct predictions, lacks both conceptual and mathematical tools to consistently analyze the situation. We introduce Bohmian mechanics as a precise mathematical-physical theory that solves the measurement problem. Time permitting, we may discuss some other solutions to the quantum measurement problem and some other issues in the foundations of quantum theory. [This is an introductory talk. We assume very little knowledge of physics.]
Posted on Vimeo