(The seminar webpage is here.)

**Sam Grushevsky (Stony Brook University)**

September 29, 2017

**Geometry of compactified moduli of cubic threefolds**

Starting from considering the GIT compactification of the moduli of cubic threefolds, the “wonderful” compactification, which is smooth with normal crossing boundary, is constructed by an explicit sequence of blowups. We show that there exists a family of intermediate jacobians over the wonderful compactification. We compute the cohomology of the wonderful compactification by comparing it to the symplectic resolution. Based on joint works with Casalaina-Martin, Hulek, Laza

**Felix Janda (University of Michigan)**

October 6, 2017

**Genus two curves on quintic threefolds
**

Virtual (Gromov-Witten) counts of maps from algebraic curves to quintic 3-folds in projective space have been of significant interest for mathematicians and physicists since the early 90s. While there are (very inefficient) algorithms for computing any specific Gromov-Witten invariant, explicit formulae are only known in genus zero and one. On the other hand, physicists have explicit conjectural formulas up to genus 51.

I will discuss a new approach to the Gromov-Witten theory of the quintic (using logarithmic geometry) which yields an explicit formula in genus two that agrees with the physicists’ conjecture.

This is based on joint works in progress with Q. Chen, S. Guo and Y. Ruan.

**Remy van Dobben de Bruyn (Columbia)**

October 20, 2017

**Dominating varieties by liftable ones
**

Given a smooth projective variety over an algebraically closed field of positive characteristic, can we always dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.

**Jason Lo (Cal State Northridge)**

October 27, 2017

**The effect of Fourier-Mukai transforms on slope stability on elliptic fibrations**

Slope stability is a type of stability for coherent sheaves on smooth projective varieties. On a variety where the derived category of coherent sheaves admits a non-trivial autoequivalence, it is natural to ask how slope stability `transforms’ to a different stability under the autoequivalence. This question also has implications for understanding the symmetries within various counting invariants. In this talk, we will give an answer to the above question for elliptic surfaces and threefolds under a Fourier-Mukai transform.

**Daniel Litt (Columbia University)**

December 1, 2017

**Galois actions on fundamental groups**

Let *X* be a variety over a field *k*, and let *x* be a *k*-rational point of *X*. Then the absolute Galois group of *k* acts on the etale fundamental group of *X*. If *k* is an arithmetically interesting field (i.e. a number field, a *p*-adic field, or a finite field), then this action reveals a great deal about the geometry of *X*; if *X* is a variety with an interesting fundamental group, this action reveals a great deal about the arithmetic of *k*.

This talk will discuss (1) joint work with Alexander Betts about the structure of Galois actions on fundamental groups, (2) how to describe invariants of these actions in terms of more geometric invariants of X, and (3) applications of this work to classical algebraic geometry, and, if time permits, arithmetic.