RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2004

Jump to Fall 2004

Spring 2004

Date: Thursday, January 29, 2004
Speaker: Dan Romik, Weizmann Institute (Israel)
Title: The limit shape of random square Young tableaux
Abstract: (joint work with Boris Pittel) I will describe the problem of the limit shape of the two-dimensional surface defined by a random NxN square Young tableau, i.e. a filling of the NxN square with the numbers 1,2,...,N^2 such that every row and column are increasing. Our recent solution is based on a variational approach. The techniques of proof and the properties of the limiting surface are evocative of (and deeply related to) the classical works of Logan-Shepp and Vershik-Kerov on the limit shape of a random Young diagram chosen according to Plancherel measure.

Date: Thursday, February 5, 2004
Speaker: Doron Zeilberger, Rutgers University
Title: `Did Andrei Andreevich MARKOV Really Pre-discover WZ Pairs 100 Years Before WZ?': Not Quite!
Abstract: Margarita Kondratieva and Sergey Sadov have recently made an astounding historical discovery: the so-called Wilf-Zeilberger (WZ) Pairs, introduced in 1990, have apparently been also defined by A.A. Markov, of Markov Chains fame, exactly a hundred years earlier, in a memoir that was then available for the modest price of 30 kopecs. While it is not embarrassing to be scooped by a giant like Markov, a closer reading, and being careful not to be victims of hindsight and anachronism, indicates that while Markov came awfully close, he missed their chief raison d'etre, and most negligently, failed to give them a name (like Markov Pairs, e.g.). Nevertheless, Markov's article contains some brilliant ideas, that when coupled with the computer, might produce new, and hopefully useful, WZ Pairs.

Date: Thursday, February 12, 2004
Speaker: Drew Sills, Rutgers University
Title: A Rogers-Ramanujan Computer Search
Abstract: The Rogers-Ramanujan identities and a number of other q-series identities of similar type were discovered by L. J. Rogers in 1894. In the 1940's, W. N. Bailey studied Rogers' paper and discovered the underlying engine (now known as "Bailey's lemma") which made it work. Bailey's student L. J. Slater made extensive use of Bailey's lemma to produce a list of 130 Rogers-Ramanujan type identities. In this talk, I will explain Bailey's lemma, and then describe and demonstrate a simple Maple program, combined with Frank Garvan's qseries package, which rediscovers Slater's list and finds identities apparently missed by Rogers, Bailey, and Slater.

Date: Thursday, February 19, 2004
Speaker: Xinyu Sun, Temple University
Title: On Fraenkel's N-Heap Wythoff's Conjectures
Abstract: Fraenkel gave an N-heap generalization of the Wythoff's game, and proposed two conjectures on the P-positions of the game. A partial proof of Fraenkel's two conjectures is provided, along with the proof of the equivalency of the two conjectures.

Date: Thursday, February 26, 2004
Speaker: Aaron Robertson, Colgate University,
Title: On the minimum number of monochromatic 3-term arithmetic progressions
Abstract: Let V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of [1,n]. It is known that V(n) = cn^2(1+o(1)) for some c>0. We will provide the best known upper and lower bounds on c and in the process disprove two related conjectures. This is joint work with Pablo Parrilo and Dan Saracino.

Date: Thursday, March 4, 2004
Speaker: Rob Manning, Haverford College
Title:A Continuum Rod Model for DNA Cyclization
Abstract: A DNA molecule is generally too complicated for study with an atomic-scale model, but, due to the many constraints imposed by its double-helical nature, it can plausibly be represented as an elastic rod. I will discuss the application of such a rod model to determining the probability that a given DNA will form a loop: mathematically, the determination of equilibria for a strain energy functional subject to loop constraints. The techniques involve a combination of (1) the incorporation of experimental data on DNA intrinsic curvature and flexibility into the rod model, (2) parameter continuation computations for solving the two-point boundary value problem describing the equilibria, and (3) stability computations for determining which equilibria are local minima.

Date: Thursday, March 11, 2004
Speaker: Bruce Sagan, Michigan State University
Title: Congruences for Catalan and Motzkin Numbers
Abstract: (Joint work with Emeric Deutsch) We dervive congruences for Catalan and Motzkin numbers as well as related sequences. In particular, we are able to prove some conjectures of Benoit Cloitre. Our methods include combinatorial techniques and the use of Lucas' Congruence. No prior knowledge will be assumed.

Date: Thursday, March 18, 2004
No seminar (Spring Break)

Date: Thursday, March 25, 2004
Location: Hill 705
Speaker: Freeman Dyson, Institute for Advanced Study (Princeton)
Title: A Hidden Symmetry in Atomic Physics
Abstract: The problem is to find a group-theoretical explanation for a striking degeneracy of states that was discovered experimentally and confirmed theoretically. "Degeneracy" means that a whole bunch of eigenvalues become exactly equal for no apparent reason. There must be an underlying symmetry-group to cause the degeneracy, but we have no idea what the group is. The mathematics required for the talk is elementary group-theory and matrix algebra.

Date: Thursday, April 1, 2004
Speaker: Leonardo F. Bonacci, University of Pisa
Title: Experiments With Mathematical Sequences that arose in Demographic Studies of Rabbit Populations
Abstract: I will describe an amazing conjecture that states that the asymptotic ratio of the population sizes of consecutive generations in rabbit populations is closely related to the ratio of the dimensions of many Greek classical buildings, including the Parthenon.

Date: Thursday, April 8, 2004
Speaker: George Andrews, Pennsylvania State University
Title: OMEGA and ENGEL: two exploratory packages
Abstract: In this talk I will describe my work with mathematicians and computer scientists at the University of Linz (Paule, Riese and Zimmermann) to implement algorithms in computer algebra that are effective and fruitful in the study of partitions. We shall concentrate on two projects: The Omega package and The Generalized Engel Transformation. The first of these arises from the work of P.A. MacMahon, the computational and combinatorial mathematician whose work was invaluable to Ramanujan and Hardy. The second arises from a study by A. and J. Knopfmacher extending ideas of Engel, a student of Perron. The object of this talk will be to introduce gently the methods of each algorithm and to describe some of our most recent applications.

Date: Thursday, April 15, 2004
Speaker: Yi Jin, Rutgers University
Title:The combinatorics of polynomial root-finding algorithms
Abstract: (Joint work with B. Kalantari) We give a new, combinatorial interpretation of a family of high order root-finding algorithms that was first discovered by Schroeder in 1870. The first member of this family is the well-known Newton's method and the ith member of this family has an (i+1)-st order of convergence. This new perspective also helps to explain why a "truncated" version of Schroder's algorithms still maintains high order convergence rate, and gives rise to a recipe to "invent" your own high order methods.

Date: Thursday, April 22, 2004
Speaker: Herb Wilf, University of Pennsylvania
Title: Sums of products of C-finite sequences have closed form
Abstract: Joint work with Curtis Green. Suppose $\{F(n)\}$ is a sequence that satisfies a recurrence with constant coefficients whose associated polynomial equation has distinct roots. Consider a sum of the form \[\sum_{j=0}^{n-1}(F(a_1n+b_1j+c_1)F(a_2n+b_2j+c_2)\dots F(a_kn+b_kj+c_k)).\] We prove that such a sum always has a closed form in the sense that it evaluates to a polynomial with a fixed number of terms, in the values of the sequence $\{F(n)\}$. We describe two different sets of monomials that will form such a polynomial, and give an algorithm for finding these closed forms, thereby completely automating the solution of this class of summation problems. We exhibit tools for determining when these explicit evaluations are unique of their type, and prove that in a number of interesting cases they are indeed unique.

Fall 2004

Date: Thursday, September 9, 2004
Room: CoRE 431
Speaker: Neil Sloane (AT&T)
Title: From Packing Planes in 4-Space to Quantum Error-Correcting Codes
Abstract: I will describe the route that took us from experimental work on a new packing problem (looking for "codes" in Grassmann manifolds - e.g., how should you place 18 planes through the origin in Euclidean 4-space so that they are as far apart as possible?) to the construction of codes for quantum computers. This work began as a project with Ron Hardin and John Conway, but many others (Peter Shor, Rob Calderbank, Eric Rains, Gabriele Nebe, ...) have since been involved. There are also applications to medicine, to visualizing multi-dimensional data, and to wireless communications.

Date: Thursday, September 16, 2004
No seminar.

Date: Thursday, September 23, 2004
Room: Hill 425
Speaker:Vince Vatter (Rutgers)
Title:Counting Restricted Permutations by Computer
Abstract: Restricted permutations arise in many contexts, from sorting machines to algebraic geometry. One of the most popular restricted permutation activities is counting them, a topic about which dozens of papers using ad hoc techniques have been written. I will talk about systematic approaches to the problem that can be (and in fact, have been) taught to a computer, and in particular, how to make one of Doron Zeilberger's algorithms work in many more cases.

Date: Thursday, September 30, 2004
Room: Hill 425
Speaker: Kathy O'Hara (NSF)
Title: Some Matchings in Product Posets

Date: Thursday, October 7, 2004
Room Hill 425
Speaker: Arthur Benjamin (Harvey Mudd College and Brandeis Univ.)
Title: Counting the Sums of Cubes of Fibonacci Numbers
Abstract: We provide the first combinatorial proof for the sum of the cubes of the first n Fibonacci numbers. Specifically, we prove that
Σk=0n (fk)3 = (f3n+4 + (-1)n 6 fn-1+5)/10
where fn is the nth Fibonacci number defined by f0 = f1=1 and for n> 1, fn = fn-1 + fn-2. Along the way, elegant combinatorial proofs are also given for other Fibonacci identities. This is joint work with undergraduate Timothy Carnes.

Date: Thursday, October 14, 2004
Room: Hill 425
Speaker: Mohamud Mohammed (Rutgers)
Title: The Sharpening of WZ Theory
Abstract: A new proof of the Fundamental Theorem for Hypergeometric (and q-Hypergeometric) summation/integration, that does not depend on Sister Celine's method will be presented. As a consequence, we get simplified versions of the Zeilberger, q-Zeilberger, and Almkvist-Zeilberger algorithms. We also considerably improve the upper bounds given, in 1992, by Wilf and Zeilberger, for the orders of the recurrences and differential equations outputted by these algorithms, and prove sharp. More importantly, using the new approach, we extend the above algorithms from one to several dimensions. [Joint work with Doron Zeilberger].

Date: Thursday, October 21, 2004
Room: Hill 425
Speaker:Amitai Regev (Weizmann Institute, Israel)
Title: S¥ representations and combinatorial identities
Abstract: For various probability measures on the space of the infinite standard Young tableaux we study the probability that in a random tableau, the (i,j)th entry equals a given number n. The analysis of these probabilities leads to many explicit combinatorial identities, some of which are related to hypergeometric series.

Date: Thursday, October 28, 2004
Room: CoRE 431
Speaker:Etienne Rassart (Institute for Advanced Study, Princeton)
Title: Partitioning the Permutahedron
Abstract: A permutahedron is a polytope obtained by taking the convex hull of the orbit of a point under the action of the symmetric group. Permutahedra appear in algebra because the weight diagram for a certain group representation is the set of all the lattice points inside a permutahedron, together with a function that associates an integer (called multiplicity) to each lattice point. I will explain how this function partitions the permutahedron into subpolytopes over which it is expressed by polynomials. With the help of the computer, in three dimensions, we can count these regions, not only for a given permutahedron but for all permutahedra at once. The multiplicity function has a continuous analogue called the Duistermaat-Heckman function, and the computer proves (with a little help) that these two functions partition the permutahedron in the exact same way in 3D.

Date: Thursday, November 4, 2004
Room: Hill 425
Speaker: Sujith Vijay (Rutgers)
Title: Expected Number of Spins in Dreidel
Abstract: We show that the expected number of spins in the popular Chanukah game dreidel where each player starts with n tokens each is O(n^2), confirming a conjecture of Doron Zeilberger. This is joint work with Thomas Robinson.

Date: Thursday, November 11, 2004
Room: CoRE 431
Speaker: Cilanne Boulet (Massachusetts Institute of Technology)
Title: A new combinatorial proof of the generalized Rogers-Ramanujan identities
Abstract:We give a combinatorial proof of the first Rogers-Ramanujan identity by definining a new generalization of Dyson's rank and presenting two related symmetries. These symmetries are established by direct bijections. We will also show how to extend this proof to Andrews' generalization of the Rogers-Ramanujan identities. This is joint work with Igor Pak.

Date: Thursday, November 18, 2004
Room: CoRE 431
Speaker: Diane Maclagan (Rutgers)
Title: Experiments on the Hilbert scheme
Abstract: The Hilbert scheme parameterizes all ideals with a given Hilbert polynomial. Studying just the monomial ideals introduces combinatorics, and gives us information about the structure of the schemes. I will introduce these concepts, and explain how to compute all monomial ideals on the (multigraded) Hilbert scheme, and how this lets us conduct experiments about these objects.

Date: Thursday, November 25, 2004
No seminar; Thanksgiving holiday

Date: Thursday, December 2, 2004
Room: Hill 425
Speaker: Holly Swisher (University of Wisconsin)
Title: Stanley's partition funciton and its relation to p(n)
Abstract: Recently Richard Stanley formulated a new partition fuction t(n). This function counts the number of partitions π for which the number of odd parts of π is congruent to the number of odd parts in the conjugate partition π' modulo 4. G.E. Andrews has recently proven a nice generating function for t(n) in terms of the generating function for p(n), the usual partition function. He also showed that the mod 5 Ramanujan congruence for p(n) also holds for t(n). In light of these results, it is natural to ask the following questions: What is the size of t(n)? Are there other congruences satisfied by both t(n) and p(n)? We will address both of these questions.

Date: Thursday, December 9, 2004
Room: CoRE 431
Speaker: Jim Haglund (University of Pennsylvania)
Title: A Combinatorial Model for the Macdonald Polynomials
Abstract: We discuss a recent result of M. Haiman, N. Loehr, and the speaker, which gives a combinatorial formula, involving generalizations of the permutation statistics maj and inv, for the coefficient of a monomial in the modified Macdonald polynomial. Consequences of the formula include a new, short proof of Lascoux and Schützenberger's cocharge description for Hall-Littlewood polynomials. The formula was first discovered by the speaker using experimental methods, and we describe the sequence of steps which led to the statistics.