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Colloquium Archive

# Colloquia — Fall 2014

## Monday, November 24, 2014

Title
Speaker

Time
Place

TBA
Andrei Martínez-Finkelshtein
Almería, SPAIN
TBA
TBA
E. A. Rakhmanov

Abstract

TBA

## Friday, November 21, 2014

Title
Speaker

Time
Place

TBA
Marco Bertola
Concordia University
3:00pm-4:00pm
CMC 130
Seung-Yeop Lee

Abstract

TBA

Title
Speaker

Time
Place

TBA
Said Sidki
3:00pm-4:00pm
CMC 130
Dmytro Savchuk

Abstract

TBA

## Friday, November 7, 2014

Title
Speaker

Time
Place

TBA
Jen-Hsu Chang
UC Riverside and ?
3:00pm-4:00pm
CMC 130
Wen-Xiu Ma

Abstract

TBA

## Friday, October 17, 2014

Title
Speaker

Time
Place

Geometric curve flows and integrable systems
Stephen Anco
Department of Mathematics and Statistics
Brock University
3:00pm-4:00pm
CMC 130
Wen-Xiu Ma

Abstract

The modern theory of integrable soliton equations displays many deep links to differential geometry, particularly in the study of geometric curve flows by moving-frame methods.

I will first review an elegant geometrical derivation of the integrability structure for two important examples of soliton equations: the nonlinear Schrödinger (NLS) equation; and the modified Korteweg-de Vries (mKdV) equation. This derivation is based on a moving-frame formulation of geometric curve flows which are mathematical models of vortex filaments and vortex-patch boundaries arising in ideal fluid flow in two and three dimensions. Key mathematical tools are the Cartan structure equations of Frenet frames and the Hasimoto transformation relating invariants of a curve to soliton variables, as well as the theory of Poisson brackets for Hamiltonian PDEs.

I will then describe a broad generalization of these results to geometric curve flows in semi-simple Klein geometries $$M=G/H$$, giving a geometrical derivation of group-invariant (multi-component) versions of mKdV and NLS soliton equations along with their full integrability structure.

## Friday, October 3, 2014

Title
Speaker

Time
Place

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

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

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.

Title
Speaker

Time
Place