TBA

There is a nice description of the centralizer of a Dehn twist in a mapping class group given by an exact sequence with the Dehn twist at one end and the mapping class group a punctured surface at the other end (you obtain the punctured surface by cutting along the curve associated to the Dehn twist). We describe a similar picture for Dehn twists in Out(F_n).

Mumford-Tate groups are the symmetry groups of Hodge theory. The study of Hodge structures leads one to consider manifolds (called Mumford-Tate domains) that are homogenous with respect to a Mumford-Tate group. Schubert varieties are distinguished subvarieties of the Mumford-Tate domains. I will describe the relationship between Schubert varieties and variations of Hodge structure.

In this talk I will discuss some new Topological Quantum Field Theories (TQFTs) arising from a re-normalization of the Reshetikhin-Turaev quantum invariants. I will start by giving an axiomatic definition of the re-normalized quantum invariants. These invariants distinguish manifolds that the usual R-T invariants can not and lead to a generalized Volume Conjecture. I will give Atiyah's original definition of a TQFT and discuss how the re-normalized invariants lead to such TQFTs. I will finish the talk discussing how these TQFTs lead to mapping class group representations. These mapping class group representations have interesting properties including the fact that the action of certain Dehn twists have infinite order. Such behavior is in sharp contrast with the usual quantum representations of mapping class groups where all the Dehn twists have finite order.

Cubic surfaces are determined by a period vector in complex hyperbolic 4-space. We present some results on the nature of the intermediate Jacobian for the associated cubic threefold when the ratios of the period vector lie in certain algebraic number fields. In particular, we show how to construct cubic surfaces from certain totally real quintic fields. This is joint work with Domingo Toledo.

This will be an easy introduction to tropical plane curves, configurations of "general" points in the plane and a proposal for what an affine cover should look like.