RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2012

Jump to Fall 2012

Spring 2012

Date: January 26, 2012
Speaker: Henry Cohn, Microsoft Lab, Cambridge, MA
Title: What does the combinatorial landscape look like?
Abstract:
          There are two extremes in combinatorics that are relatively well understood: highly symmetrical objects, which can often be classified, and thoroughly random objects, which can often be studied probabilistically. However, many natural extremal problems lie in between these possibilities. The solutions have considerable structure, but little actual symmetry.

In this talk, we'll take a look at several examples. I'll try to make the case that experimental mathematics needs two things in this area:

(1) A conceptual explanation for the unreasonable effectiveness of simple tools.

(2) More sophisticated tools, which can help explore some of the blank areas on our map.


Date: February 2, 2012
Speaker: Doron Zeilberger, Rutgers University
Title: Herbert Saul Wilf (1931-2012): an Experimental Mathematics GIANT
Abstract:
          Herb Wilf was always way ahead of his time. In particular he really understood that experimental mathematics equals future mathematics.
Posted on YouTube (2 parts): Part 1   Part 2


Date: February 9, 2012
Speaker: Robert Boyer, Drexel University
Title: Polynomial Versions of Integer Partitions and Their Zeros
Abstract:
          We discuss polynomial versions of different integer partition statistics; for example, the polynomial version of the partition numbers p(n) is the sum of pk(n) xk from k=1 to k=n, where pk(n) is the number of partitions of n with exactly k parts. The structure of their asymptotics has the striking visual consequence that the polynomial zeros approach a network of curves inside the unit disk which allow a complete description. Other examples include partitions with odd parts and plane partitions.
Posted on YouTube (2 parts): Part 1   Part 2


Date: February 16, 2012
Speaker: Vladimir Retakh, Rutgers University
Title: Noncommutative Laurent Phenomenon
Abstract:
          I will discuss the Laurent phenomenon for noncommuting variables. A good example is the cluster conjecture of Kontsevich. I will present a proof of the conjecture recently obtained by A. Berenstein and me.
Posted on YouTube (2 parts): Part 1   Part 2


Date: February 23, 2012
Speaker: Doron Zeilberger, Rutgers University
Title: A Case Study In Early 21st-Century Mathematical Narrow-Mindedness: How a "Non-Rigorous", but ABSOLUTELY CERTAIN Disproof of a 45-Year Old Conjecture Made by One of the Greatest Number Theorists of the 20th Century Got Rejected By Three Journals
Abstract:
          I will tell the sad (but funny!) story how the beautiful refutation by Drew Sills and myself, of a long-standing conjecture by Hans Rademacher, got rejected by three journals. First we (naively!) submitted it to George Andrews, Rademacher's "Benjamin" (see the bottom of this page) for the Proceedings of the National Academy of Science, who rejected it because it was "only experimental". Then we submitted it to Mathematics of Computation (see its editorial board), who rejected it because, while "interesting", it "does not meet the `high' standards of the journal, since it is 'only computational'". Finally we submitted it to the journal Experimental Mathematics (see its editorial board), that when it was founded, was a breath of fresh air, but in recent times, under the current editor-in-chief, has deterioated into "J. of Algorithmic Algebraic Geometry", who once again rejected our masterpiece, ostensibly because it "does not ring with enduring significance" (whatever that means), but most probably because it was "only experimental".
Posted on YouTube (2 parts): Part 1   Part 2


Date: March 1, 2012
Speaker: Sergei K. Suslov, Arizona State University
Title: "On Integrability of Nonautonomous Nonlinear Schroedinger Equations"
Abstract:
          We show, in general, how to transform the nonautonomous nonlinear Schroedinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear science. Derivation of the corresponding equivalent nonisospectral Lax pair is also outlined. A few simple integrable systems are discussed.
Mathematica notebook
Slides from talk and Mathematica notebook from talk
Posted on YouTube (2 parts): Part 1   Part 2


Date: March 8, 2012
Speaker: Walter Stromquist, Swarthmore College
Title: The Mathematics of Three-Candidate Elections
Abstract:
          Some observers say that the race between Bush and Gore in 2000 was influenced by the votes Ralph Nader won in Florida, or that Bill Clinton's 1992 election was influenced by the candidacy of Ross Perot. Did Chileans overlook a consensus centrist candidate when they elected Salvador Allende in 1970? What can we learn from John Anderson, Joe Lieberman, Lisa Murkowski, Charlie Crist, John Edwards, Jean-Marie Le Pen, and Vicente Fox? Each of them was involved in a famous three-candidate election. The Republican primaries of March 6 will provide more current examples.
          Would we do better if we asked voters to rank all of the candidates? We would then have to decide how to count the votes. Several systems have been proposed, including instant runoffs, Borda counts, approval voting, and Eric Maskin's "true majority" rule (which picks "Condorcet winners" and which has trouble with "Condorcet cycles"). All of them have their advocates, and some are in use in various countries or organizations.
          We might want a system that respects the "No Spoilers" rule---formally, the property of "Independence of Irrelevant Alternatives" or "IIA." This says that if X would beat Y in a head-to-head race, then Y should not be the winner of an X-Y-Z race. It seems a simple enough requirement, but the Arrow Impossibility Theorem tells us, essentially, that no such system is possible.
          We will review this theorem and its significance and give some real and hypothetical examples. Along the way we will discuss expressive voting (voting for Nader when one prefers Gore), strategic voting (voting for Gore when one prefers Nader), and the kinds of institutions that arise from various rules for vote counting.
Posted on YouTube (2 parts): Part 1   Part 2


Date: March 22, 2012
Speaker: Ilias Kotsireas, Wilfrid Laurier University (presented by Doron Zeilberger)
Title: New Results on D-optimal matrices
Abstract:
          D-optimal matrices are 2v x 2v (-1,+1)-matrices that have maximal determinant among all 2v x 2v (-1,+1)-matrices, where v is an odd positive integer. The value of the maximal determinant is given by Ehlich's bound. We present new theoretical and computational results on D-optimal matrices of circulant type. Such D-optimal matrices are constructed via two circulant submatrices of orders v each. In particular, we construct new D-optimal matrices of orders 206, 242, 262, 482.
Joint work with D. Z. Djokovic.
Slides for the talk can be found HERE.


Date: March 29, 2012
Speaker: Edinah Gnang, Rutgers University
Title: On two semi-positional recursive integer encodings
Abstract:
          In a 1944 classical paper "On the Restricted Ordinal Theorem." J. Symb. Logic 9, 33-41 Goodsteing R. L. introduced a recursive integer encoding. Using a recently proposed combinatorial algorithm for sifting primes we suggest a new variant on the Goodsteing R. L. hereditary base integer encoding. We discuss an algorithm for recovering both recursive encodings in addition to related experimental observations as well as related open problem.
Posted on YouTube (2 parts): Part 1   Part 2


Date: April 5, 2012
Speaker: Eric Rowland, UQAM
Title: Automated symbolic number theory
Abstract:
          In the last several decades symbolic algebra has been developed extensively, to the point that modern software is capable of routinely manipulating sums, integrals, and so on. So far, however, not much attention has been paid to expressions involving basic number theoretic functions such as mod(n, k) (the least nonnegative integer congruent to n modulo k) and νk(n) (the exponent of the highest power of k dividing n). An implementation of basic properties of these functions goes a long way, and for example can perform automated case analyses on expressions involving them, giving automated proofs of identities which are tedious and not enlightening to prove by hand. The particular motivation was to automate part of a proof, with Jeff Shallit, that the lexicographically least 3/2-power-free word on the nonnegative integers is a 6-regular sequence. However, the tools are applicable in a number of other settings.
Posted on YouTube (2 parts): Part 1   Part 2


Date: April 12, 2012
Speaker: James Abello, DIMACS Research Faculty
Title: The Majority Rule and Combinatorial Geometry (via the Symmetric Group)
Abstract:
          The Marquis du Condorcet recognized 200 years ago that majority rule can produce intransitive group preferences if the domain of possible (transitive) individual preference orders is unrestricted. We present results on the cardinality and structure of those maximal sets of permutations for which majority rule produces transitive results (Consistent Sets). Consistent sets that contain a maximal chain in the Weak Bruhat Order (i.e. a balanced tableau of staircase shape) inherit from it an upper semi modular sub lattice structure. They are intrinsically related to a special class of Hamiltonian graphs called persistent graphs. These graphs in turn have a clean geometric interpretation.
We highlight the main tools used to prove these connections and state computationally open research questions.
More detailed abstract available HERE.
Posted on YouTube (2 parts): Part 1   Part 2


Date: April 19, 2012
Speaker: Melkamu Zeleke, William Paterson University
Title: Riordan Arrays and Their Applications in Combinatorics
Abstract:
          The use of Riordan arrays in combinatorics has gained some attention after the publication of the seminal paper titled "The Riordan Group" by L.W. Shapiro, S. Getu, W.J. Woan, and L.C. Woodson in the early 1990s. In this talk, we use the Lagrange Inversion Theorem for formal power series as a motivation to introduce the Riordan arrays. We will then look at the fundamental properties of Riordan arrays, and apply some of these properties to binomial and inverse identities, and enumeration of combinatorial objects.
Posted on YouTube (2 parts): Part 1   Part 2


Date: April 26, 2012
Speaker: Vince Vatter, University of Florida
Title: Enumeration schemes and the insertion encoding
Abstract:
          The two general techniques for automatically enumerating permutation classes are the enumeration schemes of Zeilberger and the insertion encoding of Albert, Linton, and Ruskuc. I will discuss my recent simplification of the insertion encoding, its similarities with enumeration schemes, and a possible method of combining the two approaches.
Posted on YouTube (2 parts): Part 1   Part 2


Fall 2012

Date: September 13, 2012
ROOM CHANGE: HILL CENTER 525
Speaker: Jayaram X. Muthuswamy, Kent State University
Title: Discrepancy between Black-Scholes and Binomial Option Premia
Abstract:
          It has been demonstrated that European option premia computed with a binomial lattice, as first described by Cox, Ross, and Rubinstein (CRR, 1979), do not have a closed-form solution (Georgiadis, 2011). This stems from a lack of hypergeometricity, an artifact of Gosper's algorithm, and naturally engenders the question of the magnitude of the discrepancy from true value. This paper compares premia for European option values from a CRR lattice with the Black-Scholes model and assesses the discrepancy for increasingly refined meshes. Results over a range of the parameter space indicate that the discrepancy is meaningful for even moderately large portfolios. Thus, attention should be focused on the lack of convergence of lattice-based option valuation to any meaningful asymptotic constant.
Keywords: Option pricing model, binomial lattice, Black-Scholes model, Gosper's algorithm, hypergeometricity
Posted on YouTube (2 parts): Part 1   Part 2


Date: September 20, 2012
Speaker: Arthur DuPre, Rutgers University
Title: Mining for Primes in the Field of Continued Fractions and Linearly Recursive Sequences
Abstract:
          We show how to generate sequences of primes from the process of multiplying continued fractions of a certain form by positive integers. One of these sequences, A000057, has been in OEIS, for over forty years, placed there by Neil Sloane himself. Some of the sequences of primes we generate can also be more quickly generated via linearly recursive sequences, and for some, we have not yet found a corresponding recurrent sequence, although we conjecture there is one. We will also describe potentially fruitful procedures to generate more prime sequences not yet discovered.
Posted on YouTube (2 parts): Part 1   Part 2


Date: September 27, 2012
Speaker: Doron Zeilberger, Rutgers University
Title: The Amazing "3 To the Power n" Theorem and its even more Amazing Proof [Discovered by Xavier G. Viennot and his gang]
Abstract:
          A self-contained exposition of the most amazing theorem in the whole of Enumerative Combinatorics will be given.
Posted on YouTube (2 parts): Part 1   Part 2


Date: October 4, 2012
Speaker: Scott Garrabrant, UCLA
Title: Cofinite Induced Subgraph Nim
Abstract:
          Nim is a famous impartial combinatorial game played with 3 heaps of beans. Two players alternate taking any positive number of beans from any one heap. When all heaps are empty, the player whose turn it is to play has no move and is therefore declared the loser. The optimal strategy for Nim exhibits a fractal structure in that a player can win from the position (x,y,z) if and only if they can win from the position (2x,2y,2z). A natural way to modify the game of Nim is to forbid either player from moving to a given position. This modification destroys the original fractal structure of Nim, but replaces it with a new fractal structure. We will investigate properties of the optimal strategy in this modified Nim.
Posted on YouTube (2 parts): Part 1   Part 2


Date: October 11, 2012
Speaker: Jocelyn Quaintance, Rutgers University and West Virginia University
Title: Coloring Statistics of an m x n grid
Abstract:
          You are given an m x n chess board and c cans of paint. Each can of paint has its own paint brush. The goal is to color each square of the chess board using this selection of c colors. There is one catch. Before painting each square you must shut your eyes and arbitrarily select a paint brush. When you are finished how many edge adjacent squares share the same color? This talk explains how humans and computers apply probabilistic methods to answer such a question.
Abstract also in PDF.
Posted on YouTube (2 parts): Part 1   Part 2


Date: October 18, 2012
Speaker: Shaoshi Chen, North Carolina State University
Title: Telescopers for 3D Walks via Residues
Abstract:
          Telescopers are linear differential (recurrence) operators satisfied by definite integrals (sums) of certain class of functions (sequences). They are extensively used by combinatorists, such as Wilf, Zeilberger, Gessel etc. to enumerate combinatorial objects through generating functions and to show identities involving integrals or sums of special functions. In this talk, we present a criterion for deciding the existence of telescopers for rational functions which relates to combinatorial problems and an algorithm to construct them if they exist. Our approach is based on the investigation of residues of the input rational functions. We show that the problem of constructing telescopers for rational functions of three variables is equivalent to the problem of constructing telescopers for algebraic functions of two variables. With our method, we will answer some challenging problems on three-dimensional walks. This is a joint work with Manuel Kauers (RISC) and Michael F. Singer (NCSU).
Slides for the talk can be found HERE.
Posted on YouTube (2 parts): Part 1   Part 2


Date: October 25, 2012
Speaker: Matthew Russell, Rutgers University
Title: An automated exploration of Somos-like sequences and the Laurent phenomenon
Abstract:
          The Somos sequences, first studied by Michael Somos, are recurrence relations that surprisingly produce only integers. Their integrality turns out to be a special case of the Laurent phenomenon. Since their initial discovery, additional families of sequences with this property have been discovered. We will discuss methods for searching for new sequences with the Laurent phenomenon - with the conjecturing and proving both automated. Careful examination of the computer-generated proofs in individual cases can then lead to human proofs for new infinite families.
Posted on YouTube (2 parts): Part 1   Part 2


Date: November 15, 2012
Speaker: Bahman Kalantari, Rutgers University (Computer Science)
Title: Solving Linear Systems Via A Convex Hull Algorithm
Abstract:
          We present new iterative algorithms for solving a square linear system Ax=b in dimension n by employing the Triangle Algorithm [BK2012], a fully polynomial-time approximation scheme for testing if the convex hull of a finite set of points in a Euclidean space contains a given point. By converting Ax=b into a convex hull problem and solving via the Triangle Algorithm, together with a sensitivity theorem, we compute in O(n^2 epsilon^{-2}) arithmetic operations an approximate solution where the Euclidean norm of the residual is bounded above by epsilon times the maximum of the Euclidean norm of b and those of the columns of A. In another approach we apply the Triangle Algorithm incrementally, solving a sequence of convex hull problems while repeatedly employing a distance duality. We compare the two methods. The simplicity and theoretical complexity bounds of the proposed algorithms, requiring no structural restrictions on the matrix A, suggest their potential practicality, offering alternatives to the existing exact and iterative methods, especially for large scale systems. The assessment of computational performance however is the subject of future experimentations.
The talk is based on the work done in http://arxiv.org/abs/1210.7858v1.
Posted on YouTube (2 parts): Part 1   Part 2


Date: November 29, 2012
Speaker: Ein-Ya Gura, Hebrew University of Jerusalem (Israel)
Title: "Insights into Game Theory: An Alternative Mathematical Experience" (A book written by Ein-ya Gura and Michael Maschler, published by Cambridge University Press)
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 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. We address this problem by presenting material that does not require mathematical prerequisites and yet involves deep game-theoretic ideas and some mathematical sophisticaton. This book selects a small number of topics and studies them in depth. It shows the students of the social sciences how a mathematical model can be constructed for real- life issues.
Posted on YouTube (2 parts): Part 1   Part 2


Date: December 6, 2012
Speaker: Dan Romik, University of California, Davis
Title: Second class particles in exclusion processes and Young tableaux
Abstract:
          In probability theory, the Totally Asymmetric Simple Exclusion Process (known as TASEP) is an important random process of particles interacting on a one-dimensional lattice. It can be thought of as a simple model for a traffic jam: each particle tries to move one step forward at random times, succeeding if there is no particle blocking its way. One can also add a new type of particle known as a "second class particle", which can move forward if there is an empty space in front of it, but will be pushed back if an ordinary particle tries to move into its position from behind. A natural question is to ask which way the second class particle will go if started at the front of an "infinite traffic jam" with an infinite line of ordinary particles behind it and only empty space in front of it.

Young tableaux are important combinatorial objects related to the representation theory of the symmetric groups and to longest increasing subsequences in permutations. A natural operation on Young tableaux, known as "jeu de taquin", plays a key role in this theory.

In the talk I will describe recent joint work with Piotr Sniady, in which we discovered that understanding the behavior of the second class particle in the TASEP is actually equivalent to understanding the jeu de taquin operation on infinite Young tableaux. We then used this connection, and other ideas from the theory of Young tableaux, to prove interesting results on the asymptotic behavior of the second class particle. The talk will be completely self-contained and no prior knowledge of probability or combinatorics will be assumed.


Date: December 13, 2012
Speaker: Anton Khoroshkin, Stony Brook University
Title: On generating series of finitely presented operads and pattern avoidance
Abstract:
          An algebra of some type is a set with some operations on it. Let us remove the underlying set, what remains is the collection of all operations one can define. This collection with the prescribed rules of compositions is what one calls an operad.

In this talk I will explain the notion of operads and the theory of monomials for operads. The combinatorics of monomials in operads is governed by avoidance problems. In particular, the Hilbert series of dimensions of certain class of operads coincides with the generating series of permutations avoiding a given set of patterns. I will state several general results and conjectures about the class of generating series for monomial operads providing some computational algorithms for these series.

The talk is based on joint works with V.Dotsenko, B.Shapiro and D.Piontkovsky:
http://arxiv.org/abs/1109.2690 , http://arxiv.org/abs/1202.5170 , http://arxiv.org/abs/1009.5308
Posted on YouTube (3 parts): Part 1   Part 2   Part 3