Colloquia — Spring 2004
Friday, April 23, 2004
| Title |
Knots, Quandles, and Colorings |
| Speaker |
Dr. Pedro Lopes
Department of Mathematics
University of Iowa &
Instituto Superior Tecnico
Lisbon, Portugal |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Drs. M. Elhamdadi and M. Saito |
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:
- J. T. Bojarczuk, P. Lopes, Quandles at Finite Temperatures III,
2003, J. Knot Theory Ramifications, submitted.
- F. Miguel Dioni'sio, P. Lopes, Quandles at Finite Temperatures II,
J. Knot Theory Ramifications, 12 (2003), no. 8, 1041-1092.
- P. Lopes, Quandles at Finite Temperatures I, J. Knot Theory Ramifications,
12 (2003), no. 2, 159-186.
Friday, April 16, 2004
| Title |
Darboux Transformations for the Supersymmetric KdV |
| Speaker |
Professor Qingping Liu
Department of Mathematics
University of Illinois at Urbana-Champaign |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Dr. Wen-Xiu Ma |
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
| Title |
Chaos Cascade |
| Speaker |
Dr. Y. Charles Li
Department of Mathematics
University of Missouri-Columbia |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Professor Y. You |
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
| Title |
Relations for Generalized Transition Polynomials |
| Speaker |
Dr. Jo Ellis-Monaghan
Department of Mathematics
Saint Michael's College, VT |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Professor N. Jonoska |
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
| Title |
Branching Process: Some Limit Theorems and Statistical Inference |
| Speaker |
Dr. George Yanev
Department of Mathematics
University of South Florida, St. Petersburg |
| Time |
2:00-3:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Professor C. Tsokos |
Thursday, March 4, 2004
| Title |
Poisson Approximation by Constrained Exponential Tilting |
| Speaker |
Dr. Steven Kathman
GlaxoSmithKline |
| Time |
2:00-3:00 p.m. |
| Place |
PHY 108 |
| Sponsor |
Professor K. Ramachandran |
| Note |
Speaker is a candidate for the Asst. Professor position in Statistics. |
Friday, January 30, 2004
| Title |
The Nottingham Group |
| Speaker |
Professor Kevin Keating
Department of Mathematics
University of Florida |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 013 |
| Sponsor |
Professor X. Hou |
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 + a1x2
+ a2x3 + …, 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
| Title |
Jacobi With Nonstandard Parameters: A New Look on Old Polynomials |
| Speaker |
Professor Andrei Martinez-Finkelshtein
University of Almeria
Spain |
| Time |
3:00-4:00 p.m. |
| Place |
PHY 109 |
| Sponsor |
Professor A. Rakhmanov |
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 $Pn
^{(αn, βn)}$ 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 Γ of a certain quadratic differential
according to the equilibrium measure on Γ in an external field. However,
when either αn, βn or
αn + βn 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.