banner USF Home College of Arts & Sciences OASIS myUSF USF A-Z Index

USF Home > College of Arts and Sciences > Department of Mathematics & Statistics

pendulum
  Colloquium Archive

Colloquia — Fall 2014

Friday, October 3, 2014

Title
Speaker

Time
Place
Sponsor

Ordering free groups and free products
Zoran Šunić
Texas A&M University
3:00pm-4:00pm
CMC 130
Milé Krajčevski

Abstract

We utilize a criterion for the existence of a free subgroup acting freely on at least one of its orbits to construct such actions of the free group on the circle and on the line, leading to orders on free groups that are particularly easy to state and work with.

We then switch to a restatement of the orders in terms of certain quasi-characters of free groups, from which properties of the defined orders may be deduced (some have positive cones that are context-free, some have word reversible cones, some of the orders extend the usual lexicographic order, and so on).

Finally, we construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another. As an application, we provide a short proof of Vinogradov´s result that the free product of left-orderable groups is left-orderable.

Friday, September 26, 2014

Title
Speaker

Time
Place
Sponsor

Recent developments in Quantum invariants of knots
Mustafa Hajij
Louisiana State University
3:00pm-4:00pm
CMC 130
Mohamed Elhamdadi

Abstract

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

Title
Speaker

Time
Place
Sponsor

The valence of polynomial harmonic mappings
Erik Lundberg
Florida Atlantic University
3:00pm-4:00pm
CMC 130
Dmitry Khavinson

Abstract

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.