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

**Title**

**Speaker**

**Time**

**Place**

**Sponsor**

Geometric curve flows and integrable systems

Stephen Anco

Department of Mathematics and Statistics

Brock University

Ontario, CANADA

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.

**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.

**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.

**Title**

**Speaker**

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**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.