Colloquia — Fall 2014
Friday, September 26, 2014
Recent developments in Quantum invariants of knots
Louisiana State University
Quantum knot invariants deeply connect many domains such as lie algebras, quantum groups, number theory and knot theory. I will talk about a particular quantum invariant called the colored Jones polynomial and some of the recent work that has been done to understand it. This invariant takes the form a sequence of Laurent polynomials. I will explain how the coefficients of this sequence stabilize for certain class of knots called alternating knots. Furthermore, I will show that this leads naturally to interesting connections with number theory.
Friday, September 12, 2014
The valence of polynomial harmonic mappings
Florida Atlantic University
While working to extend the Fundamental Theorem of Algebra, A. S. Wilmshurst used Bezout’s theorem to give an upper bound for the number of zeros of a (complex valued) harmonic polynomial. Although the bound is sharp in general, Wilmshurst conjectured that Bezout’s bound can be refined dramatically. Using holomorphic dynamics, the conjecture was confirmed by D. Khavinson and G. Swiatek in the special case when the anti-analytic part is linear. We will discuss recent counterexamples to other cases as well as an alternative probabilistic approach to the problem.