Georgia Institute of Technology College of Sciences

Given a graph X and a group G, a G-cover of X is a morphism of graphs X’ --> X together with an invariant G-action on X’ that acts freely and transitively on the fibers. G-covers are classified by their monodromy representations, and if G is a finite abelian group, then the set of G-covers of X is in natural bijection with the first simplicial cohomology group H1(X,G).

In tropical geometry, we are naturally led to consider more general objects: morphisms of graphs X’ --> X admitting an invariant G-action on X’, such that the induced action on the fibers is transitive, but not necessarily free. A natural question is to classify all such covers of a given graph X. I will show that when G is a finite abelian group, a G-cover of a graph X is naturally determined by two data: a stratification S of X by subgroups of G, and an element of a cohomology group H1(X,S) generalizing the simplicial cohomology group H1(X,G). This classification can be viewed as a tropical version of geometric class field theory, and as an abelianization of Bass--Serre theory.

I will discuss the realizability problem for tropical abelian covers, and the relationship between cyclic covers of a tropical curve C and the corresponding torsion subgroup of Jac(C). The realizability problem for cyclic covers of prime degree turns out to be related to the classical nowhere-zero flow problem in graph theory.

Joint work with Yoav Len and Martin Ulirsch.

A knot can be thought of as a piece of string tied up, that then has its ends glued together. As long as we don’t cut the string, any way we move the string in space doesn’t change the knot we are considering. A surprisingly hard and interesting problem is, when handed two knots, how to determine if they are the same knot or not. We can further give structure to our knots and thus the problem, by adding geometric constraints to our knots, yielding what are called Legendrian knots. We can once again try to determine if two Legendrian knots are the same or not. In this talk I will introduce knots, Legendrian knots, and some ways we have to try to distinguish two knots or Legendrian knots, called knot invariants.

In this talk I will first give a brief overview of how nonsmooth bifurcations (border-splitting, grazing, and fold-fold bifurcations) help to rigorously explain the existence of nonsmooth limit cycles in the models of anti-lock braking systems, power converters, integrate-and-fire neurons, and climate dynamics. I will then focus on one particular application that deals with nonsmooth bifurcations in dispersing billiards. In [Nonlinearity 11 (1998)] Turaev and Rom-Kedar discovered that every periodic orbit that is tangent to the boundary of the billiard produces an island of stability upon smoothening the boundary of the billiard. The result to be presented in the talk (joint work with Turaev) proves that any dispersing billiard admits such an arbitrary small perturbation that ensures the occurrence of a tangent periodic orbit.

Heegaard Floer homology gives a powerful invariant of closed 3-manifolds. It is always computable in the purely combinatorial sense, but usually computing it needs a very hard work. However, for certain graph 3-manifolds, its minus-flavored Heegaard Floer homology can be easily computed in terms of lattice homology, due to Nemethi. I plan to give the basic definitions and constructions of Heegaard Floer theory and lattice homology, as well as the isomorphism between those two objects.