RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Archive of Speakers and Talks --- 2003

Fall 2003

Date: Thursday, September 18, 2003
Speaker: Doron Zeilberger, Rutgers University
Title: What is Experimental Math?
Abstract: The best way to define Experimental Mathematics (or anything else for that matter) is by examples. After presenting the examples, I will try to formulate a semi-rigorous definition.

Date: Thursday, September 25, 2003
Speaker: Drew Sills, Rutgers University
Title: Computer Algebra and Rogers-Ramanujan
Abstract: I will give a brief introduction and history of the Rogers-Ramanujan identities, and other series-product identities of similar type. Then I will demonstrate, on a computer, a method for conjecturing a polynomial generalization of a given Rogers-Ramanujan type identity. Finally, I will show how the conjectured polynomial identity can be proved automatically on the computer via q-WZ certification. The original Rogers-Ramanujan type series-product identity now follows as an easy corollary of the new polynomial identity.

Date: Thursday, October 2, 2003
Speaker: Mohamud Mohammed, Rutgers University
Title: WZ Cohomology
Abstract: We will give a brief introduction to the WZ method and then present the notion of Exterior Difference Forms (analog of Differential Forms). We will state and prove the Discrete Stokes Theorem. Using Stokes theorem, we characterize closed form identities and derive accelerating formulas for some known hypergeometric series.

Date: Thursday, October 9, 2003
Speaker: Tewodros Amdeberhan, DIMACS and DeVry University
Title: Determinants in Wonderland

Date: Thursday, October 16, 2003
Speaker: Stavros Garoufalidis, Georgia Tech
Title: A Platonic view of knots, according to Thurston?
Abstract: A knot is a piece of rope with no ends, allowed to move freely in space and not pass through itself. We often view knots by their complements. In hyperbolic geometry, the knot sits at infinity, and all we see, looking in its complement is a space triangulated by regular tetrahedra. Unlike Eucidean space, there is more than one shape of a regular tetrahedron, in fact one per complex number other than 0,1,infinity. The ideal tetrahedra are glued together and fill the knot complement. Given a knot projection, it is possible to write down a system of polynomial equations whose solutions are the shapes of the hyperbolic tetrahedra. We plan to discuss this system of equations in detail. Their solutions parametrizes a complex curve. Understanding this curve is a problem of commutative elimination, with a good geometric pay-off.

Date:Thursday, October 23, 2003
Speaker: Jack Calcut, University of Maryland
Title: Artin Presentations: a discrete, computer approachable theory of 3- and 4-manifolds
Abstract: Many important open problems in low dimensional topology have discrete, purely group theoretic equivalents via Artin Presentation theory (AP theory). After an elementary introduction to AP theory, concrete examples will be given that can be ordered and investigated with a computer algebra system (such as MAGMA).

Date: Thursday, October 30, 2003
Speaker: Vince Vatter, Rutgers
Title: Automatic generation of finitely-labeled generating trees for restricted permutations
Abstract: Restricted permutations (for example, permutations without an increasing subsequence of length 4) have been studied for many years, especially in the past dozen. Most of this study has been devoted to the enumeration of these permutations. One of the few non-ad hoc ways to perform this counting is with generating trees (essentially, the transfer-matrix method). Quite often, the generating tree for the set of permutations avoiding a set of restrictions Q requires infinitely many labels. However, sometimes this generating tree only needs finitely many labels. We characterize the finite sets of restrictions Q for which this phenomenon occurs. We do this by presenting an algorithm (that turns out to be a special case of an algorithm of Zeilberger) that is guaranteed to find such a generating tree if it exists.

Date: Thursday, November 6, 2003
Speaker: Stephen Greenfield, Rutgers
Title: Old News About A000278

Date: Thursday, November 13, 2003
Speaker: Yuriy Gulak, Rutgers University
Title: High-symmetry hydrodynamic flow: an experimental study.
Abstract: A high-symmetry Kida flow is studied as a candidate for a finite time blowup of incompressible Euler equations. The power series in time solution is analyzed using Pade and higher order approximants. The results suggest a finite-time singularity in enstrophy, although approximants of increasing orders show transient behavior. We relate some difficulties of the resummation methods to the complicated nonassociative algebraic structure of the series solution.

Date: Thursday, November 20, 2003
Speaker: Melkamu Zeleke, William Patterson University
Title: Enumeration of K-trees and applications
Abstract: A k-tree is constructed from a single distinguished k-cycle by repeatedly gluing other k-cycles to existing ones along an edge. If K is any nonempty subset of {2,3,4,...} then a K-tree is obtained as above using k-cycles with k in K. In this talk, we enumerate ordered K-trees, show that the ratio of terminal edges to total number of edges in k-trees is (k-1)/k using WZ method, and use K-trees as models to enumerate planted plane cacti and generalize the Tennis Ball Problem. We will also use k-trees to obtain generating function identities involving generalizations of Catalan, Central Binomial, and Fine Numbers. Examples of possible use of these functional identities will be discussed.

Date: Thursday, December 4, 2003
Speaker: Trevor Bass, Rutgers University
Title: A new symmetry underlying domino tilings of rectangles?
Abstract: A elementary way to compute the number of domino tilings of a rectangle as the Robbins-Mills-Rumsey 1-determinant of a pair of matrices will be presented. This uses a method due to Eric Kuo of computing the number of weighted domino tilings of the Aztec diamond. The main purpose of the talk is to give a (possibly easy) conjecture concerning a new symmetry underlying tilings of rectangles and other subregions of the Aztec diamond. This talk will assume no knowledge of combinatorics.