Georgia Institute of Technology College of Sciences

A map (respectively, a unicellular map) on a genus g surface Sg is the Homeo+(Sg)-orbit of a graph G embedded on Sg such that Sg-G is a collection of finitely many disks (respectively, a single disk). The study of maps was initiated by W. Tutte, who was interested in counting the number of planar maps. However, we will consider maps from a more graph theoretic perspective in this talk. We will introduce a topological operation called surgery, which turns one unicellular map into another. Then, we will address natural questions (such as connectedness and diameter) about surgery graphs on unicellular maps, which are graphs whose vertices are unicellular maps and where two vertices share an edge if they are related by a single surgery. We will see that these problems translate to a well-known combinatorial problem: the card shuffling problem.

Abstract: Gromov introduced some “hyperbolization” procedures, i.e. some procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. Their procedure has been used to construct new examples of manifolds and groups with negative curvature, and other prescribed features. We construct actions of the resulting groups on CAT(0) cube complexes. As an application, we obtain that they are virtually special, hence linear over the integers and residually finite. This is joint work with J. Lafont.

Upsilon is an invariant of knots defined using knot Floer homology by Ozsváth, Szabó and Stipsicz. In this talk, we discuss a generalization of their invariant for embedded graphs in rational homology spheres satisfying specific properties. Our construction will use a generalization of Heegaard Floer homology for “generalized tangles” called tangle Floer homology. As a result, we get a family of homomorphisms from the homology cobordism group of homology cylinders (over a surface of genus 0), which is an enlargement of the mapping class group defined by Graoufaldis and Levine. 

We first construct a complex surface with positive signature, which is a ball quotient. We obtain it as an abelian Galois cover of CP^2 branched over the Hesse arrangement. Then we analyze its fibration structure, and by using it we build new symplectic and also non-symplectic exotic 4-manifolds with positive signatures.

In the second part of the talk, we discuss Cartwright-Steger surfaces, which are also ball quotients. Next, we present our constructions of new symplectic and non-symplectic exotic 4-manifolds with non-negative signatures that have the smallest Euler characteristics in the so-called ‘arctic region’ on the geography chart.

More precisely, we prove that there exist infinite families of irreducible symplectic and infinite families of irreducible non-symplectic, exotic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with nonnegative signatures and with more than one smooth structures. This is a joint work with A. Akhmedov and S.-K. Yeung.