A Tate-Shafarevich twist of a (proper) fibration modifies it by a 1-cocycle of automorphisms given by flows of (holomorphic) vector fields relative to the base, locally in the analytic topology. In general, the total space of a twist does not even have to be homeomorphic to that of the original fibration. Nevertheless, it was conjectured by Saccà that if one started with a Lagrangian fibration of an irreducible hyper-Kähler variety, then the total space of the resulting twist should always be deformation-equivalent to that of the original fibration, provided that it is also algebraic. I will introduce evidence towards this conjecture, including coincidences of certain cohomological invariants, as well as a proof under further topological constraints.

This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. There is a "recipe" due to Conrey-Farmer-Keating-Rubinstein-Snaith which allows for precise predictions for the asymptotics of moments of many different families of L-functions. We consider the family of all L-functions attached to hyperelliptic curves over some fixed finite field. One can relate this problem to understanding the homology of the moduli space of hyperelliptic curves, with symplectic coefficients. With Bergström-Diaconu-Westerland we compute these stable homology groups, together with their structure as Galois representations. With Miller-Patzt-Randal-Williams we prove a uniform range for homological stability. Together, these results imply the CFKRS predictions for all moments in the function field case, for all sufficiently large (but fixed) q.

In this talk, I will present the stabilization phenomenon of cohomology groups and Kac polynomials associated with moduli spaces of quiver representations. Specifically, for Q and chosen dimension vectors d and e satisfying reasonable conditions, the cohomologies of various types of quiver varieties associated with dimension vectors d+ne stabilize as e tends to infinity. I will provide explicit generating functions for these stabilized dimensions and explain the implications for root multiplicities of Kac-Moody Lie algebras.

I will discuss joint work with Terry Song on the calculation of the virtual Hodge numbers (i.e. Hodge—Deligne polynomial, Hodge-Euler characteristic, etc.) of the moduli space of degree-d maps to projective space from smooth n-pointed curves of genus g. Up to Brill—Noether loci in genus g>= 3, I will show how to reduce the calculation to the corresponding invariants of M_gn. This reduction implies a strong stability statement for the virtual Hodge numbers as functions of the degree d, and suggests homological stability properties of the moduli space generalizing known statements in genus zero.  As an intermediate result, I will outline an analogous calculation for the universal Jacobian over M_gn. As I will discuss, the theory of symmetric functions is fundamental to our approach.

Abstract can be found here.