Georgia Institute of Technology College of Sciences

The Torelli map, taking an algebraic curve to its Jacobian, has a tropical analogue, developed in recent work by Brannetti, Melo, and Viviani. I will discuss the tropical Torelli map, with a focus on combinatorics and computations in low genus. Metric graphs, positive semidefinite forms, and regular matroids all play a role.

We prove a conjecture of K. Schmidt in algebraic dynamical system theory onthe growth of the number of components of fixed point sets. We also prove arelated conjecture of Silver and Williams on the growth of homology torsions offinite abelian covering of link complements. In both cases, the growth isexpressed by the Mahler measure of the first non-zero Alexander polynomial ofthe corresponding modules. In the case of non-ablian covering, the growth of torsion is less thanor equal to the hyperbolic volume (or Gromov norm) of the knot complement.

The talk concerns the mathematical aspects of solitary waves (i.e. single hump waves) moving with a constant speed on water of finite depth with surface tension using fully nonlinear Euler equations governing the motion of the fluid flow. The talk will first give a quick formal derivation of the solitary-wave solutions from the Euler equations and then focus on the mathematical theory of existence and stability of two-dimensional solitary waves. The recent development on the existence and stability of various three-dimensional waves will also be discussed.

The first hour of this talk gives a gentle introduction to yet another version of Heegaard Floer homology; Sutured Floer homology. This is the generalization of Heegaard Floer homology, for 3-manifolds with decorations (sutures) on their boundary. Sutures come naturally for contact 3-manifolds. Later we will concentrate on invariants for contact 3--manifolds in Heegaard Floer homology. This can be defined both for closed 3--manifolds, in this case they live in Heegaard Floer homology and for 3--manifolds with boundary, when the invariant is in sutured Floer homology. There are two natural generalizations of these invariants for Legendrain knots. One can directly generalize the definition of the contact invariant $\widehat{\mathcal{L}}$, or one can take the complement of the knot, and compute the invariant for that:$\textrm{EH}$. At the end of this talk I would like to describe a map that sends $\textrm{EH}$ to$\widehat{\mathcal{L}}$. This is a joint work with Andr\'as Stipsicz.