Colloquia -- Spring 2004

Friday, April 23, 2004

Abstract

A knot is an embedding of the circle in three space which we usually represent by a 2-dimensional diagram where at crossings we consistently break the line that goes under. It is
known that if any two of these diagrams are related by three moves (Reidemeister) then the corresponding knots are deformable into each other, and conversely. The quandle is an
algebraic structure whose defining axioms seem to capture the topology of the Reidemeister moves. In particular, we can read off a diagram the presentation of the so-called knot quandle which was proved by Joyce to be a classifying invariant of knots
(modulo orientation of the ambient space). This is an important theoretical result but of little direct practical use since we are dealing with presentations. In our work we counted homomorphisms from the knot quandle to a labelling quandle - which is a computable knot invariant.

In this talk we will develop the ideas above and report on the success of our approach. Time permits we will also address the one dimension-higher counterpart: embeddings of
spheres, tori, etc., in four space.

References:

1. J. T. Bojarczuk, P. Lopes, Quandles at Finite Temperatures III, 2003, J. Knot Theory Ramifications, submitted.
2. F. Miguel Dioni'sio, P. Lopes, Quandles at Finite Temperatures II, J. Knot Theory Ramifications, 12 (2003), no. 8, 1041-1092.
3. P. Lopes, Quandles at Finite Temperatures I, J. Knot Theory Ramifications, 12 (2003), no. 2, 159-186.

Friday, April 16, 2004

Abstract

The supersymmetric KdV systems proposed by Manin and Radul will be considered. It will be shown that the famous Darboux transformation can be extended into the supersymmetric case. We also present a Backlund transformation for the supersymmetric KdV.

Friday, March 26, 2004

Abstract

I will talk on chaos cascade referring to a chain of embeddings of smaller scale chaos into larger scale chaos. Specific example of perturbed Sine-Gordon equation will be presented. If time allows, I will mention briefly Lax pairs of Euler equations of inviscid fluids.

Friday, March 5, 2004

Abstract

The classic Tutte polynomial is a two-variable graph polynomial with the universal property that essentially any graph invariant that can be computed via a deletion-contraction reduction must be an evaluation of it. Many applications that can be modeled graph theoretically have natural deletion-contraction reductions, and this is part of the appeal of the Tutte polynomial. The classic Tutte polynomial was fully generalized using colored graphs by Zaslavsky (1992) and Bollobas and Riordan (1999).

However, graph polynomials can be defined by techniques other than deletion-contraction. In 1987, Jaeger introduced transition polynomials of 4-regular graphs to unify polynomials given by vertex reconfigurations very similar to the skein relations of knot theory. These include the Martin polynomial (restricted to 4-regular graphs), the Kauffman bracket, and, for planar graphs via their medial graphs, the Penrose and classic Tutte polynomials.

Recently, (joint work with Irasema Sarmiento), generalized transition polynomials were constructed, which extend the transition polynomials of Jaeger to arbitrary Eulerian graphs, and introduce pair weightings which function analogously to the colored edges in the generalized Tutte polynomial. The generalized transition polynomial and the generalized Tutte polynomial are related for planar graphs in much the same way as are Jaeger’s transition polynomial and the classic Tutte polynomial.

Moreover, the generalized transition polynomials are Hopf algebra maps. Thus, the comultiplication and antipode give recursive identities for generalized transition polynomials. Extension of these results to the generalized Tutte polynomial and knot invariants is the subject of current research. We also mention motivaitng applications to DNA sequencing by hybridization and biomolecular computing.

Friday, March 5, 2004

Thursday, March 4, 2004

Friday, January 30, 2004

Abstract

Let F be a finite field of characteristic p. The Nottingham group N(F) over F consists of the power series in one variable over F of the form $g(x)=x+a_1x^2+a_2x^3+...$, with the operation of composition. N(F) is a pro-p group which is large enough to contain every finite p-group as a subgroup, but is sufficiently concrete to allow explicit computations. I will discuss some results which relate the Nottingham Group to number theory and group theory.

Monday, January 12, 2004

Abstract

Jacobi polynomials are probably one of the most ``classical'' objects in analysis. Nevertheless, I will try to present some new aspects of these polynomials, and to obtain some analytic properties using new techniques. For instance, strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{(\alpha_n, \beta_n)}$ can be studied, assuming that
$$
\lim_{n\to\infty} \frac{\alpha_n}{n}= A, \qquad \lim_{n\to\infty}
\frac{\beta _n}{n}= B,
$$
with A and B satisfying A > -1, B > -1, A + B < -1. The asymptotic analysis is based on the non-Hermitian orthogonality of these polynomials, and uses the Deift/Zhou steepest descent analysis for matrix Riemann-Hilbert problems. As a corollary, asymptotic zero behavior can derived. In a generic case the zeros distribute on the set of critical trajectories $\Gamma$ of a certain quadratic differential according to the equilibrium measure on $\Gamma$ in an external field. However, when either $\alpha_n$, $\beta_n$ or $\alpha_n+\beta_n$ are geometrically close to $\Z$, part of the zeros accumulate along a different trajectory of the same quadratic differential. If time permits, I will discuss also a generalization of the electrostatic interpretation of the zeros of these polynomials.

Please direct questions to mthmaster@nosferatu.cas.usf.edu.
Last updated: 29-Mar-2004.
Copyright 2000, USF Department of Mathematics.