Georgia Institute of Technology College of Sciences

A conjecture of Ivanov asserts that finite index subgroups of the mapping class group of higher genus surfaces have trivial rational homology. Putman and Wieland use what they call higher Prym representations, which are extensions of the representation induced by the action of the mapping class group on homology, to better understand the conjecture. In particular, they prove that if Ivanov's conjecture is true for some genus g surface, it is true for all higher genus surfaces.

Given any surface, we can construct its curve complex by considering isotopy classes of curves on the surface. If the surface has boundary, we can construct its arc complex similarly, with isotopy clasess of arcs, with endpoints on the boundary. In 1999, Masur and Minsky proved that these complexes are hyperbolic, but the proof is long and involved. This talk will discuss a short proof of the hyperbolicity of the curve and arc complex recently given by Hensel, Przytycki, and Webb.

This is an expository talk on the arc complex and translation distance of open book decompositions. We will discuss curve complexes, arc complex, open books, and finally the application to contact manifolds.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds.

It is a natural question to ask whether one can deduce topological properties of a finite--volume three--manifold from its Riemannian invariants such as volume and systole. In all generality this is impossible, for example a given manifold has sequences of finite covers with either linear or sub-linear growth. However under a geometric assumption, which is satisfied for example by some naturally defined sequences of arithmetic manifolds, one can prove results on the asymptotics of the first integral homology. I will try to explain these

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds.

TBA

This is continuation of talk from last week. Periodic orbits of flows on $3$ manifolds show very rich structure. In this talk we will try to prove a theorem of Ghrist, which states that, there exists vector fields on $S^3$ whose set of periodic orbits contains every possible knot and link in $S^3$. The proof relies on template theory.