# Joint IAS/PU Number Theory

## Automorphic Levi-Sobolev Spaces, Boundary-Value Problems, and Self-Adjoint Operators

## Abelian varieties with maximal Galois action on their torsion points

Abstract:

Associated to an abelian variety A/K is a Galois representation which describes the action of the absolute Galois group of K on the torsion points of A. In this talk, we shall describe how large the image of this representation can be (in terms of a number field K and the dimension of A). We achieve this by considering abelian varieties in families and then using a special variant of Hilbert's irreducibility theorem. Some results of Serre on the mod ell Galois image will also be reviewed. (This is joint work with David Zureick-Brown)

## Local Global Principles for Galois Cohomology

We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field.

Motivated by work of Kato and others for n=3, we show that local-global principles hold for

$H^n(F, Z/mZ(n-1))$ for all n>1.

In the case n=1, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for $H^1(F,G)$, a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups G.

## Open-Closed Gromov-Witten Invariants of Toric Calabi-Yau 3-Orbifolds

We study open-closed orbifold Gromov-Witten invariants of toric Calabi-Yau 3-orbifolds with respect to Lagrangian branes of Aganagic-Vafa type. We prove an open mirror theorem which expresses generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curves. This is a joint work with Bohan Fang and Hsian-Hua Tseng.

## Sato-Tate Distributions in Genus 2

## Abstract Analogues of Flux as Symplectic Invariants

## Galois Representations for Regular Algebraic Cuspidal Automorphic Forms

## An Arithmetic Refinement of Homological Mirror Symmetry for the 2-Torus

## Behavior of Welschinger Invariants Under Morse Simplification