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.

I will talk about twisted traces on quantized Coulomb branches. Any Verma module over a quantized Coulomb branch gives rise to a twisted trace. For conical Coulomb branches there is also a trace introduced by Gaiotto and Okazaki ("sphere trace"). I will show how the sphere trace allows us to compute the graded character of Verma modules in certain cases. Based on joint work in progress with Vasily Krylov.

Given a smooth projective curve X and a reductive group G, the geometric Langlands equivalence proved by Gaitsgory, Raskin et al. (roughly) gives an equivalence between sheaves on the stack Bun_G(X) of G-bundles on X (automorphic side) and quasi-coherent sheaves on the stack of G^-local systems on X (spectral side). To compute the image of an object under the geometric Langlands equivalence, one usually bootstraps from the Whittaker model. This method fails for the constant sheaf on Bun_G(X), which is “maximally singular.” Still, we will compute its image under the equivalence, confirming a conjecture of V. Lafforgue. As a consequence, when X is over F_q we find a spectral description for the constant function on Bun_G(X)(F_q).

The Gross–Zagier formula relates the first derivative of the L-function for PGL(2) to (arithmetic) intersection numbers on the modular curve. It plays a central role in proving known cases of the Birch–Swinnerton-Dyer conjecture. In their celebrated work, Yun and Zhang established a function-field analogue of this formula, replacing the modular curve by moduli spaces of PGL(2)-Shtukas. New phenomena arise in this setting: higher derivatives of L-functions can also be expressed in terms of intersection numbers. However, due to the lack of properness of moduli spaces of Shtukas in higher-dimensional cases, extending this formula to higher dimensions has remained open for many years.

In this talk, I will present a higher-dimensional version of the Yun–Zhang formula. The proof uses tools from Geometric Langlands theory and aspects of the relative Langlands duality proposed by Ben-Zvi, Sakellaridis, and Venkatesh. This is joint work with Shurui Liu.

Coulomb branches of 3d N=4 gauge theories for a gauge group G have been rigorously defined by Braverman, Finkelberg and Nakajima. These are affine (singular) symplectic algebraic varieties; their algebras of functions can be defined via the equivariant Borel-Moore homology of certain ind-schemes closely related to the affine Grassmannian of G.

The story is significantly more complicated in 4 dimensions. In that case from physics one expects that the corresponding Coulomb branches are (more or less) singular hyper-kahler manifolds, which look drastically different in different complex structures: while for generic complex structure it is still supposed to be an affine algebraic variety, whose coordinate ring is just given by the equivariant K-theory of the above ind-schemes, for some special complex structure it is not; in that case the corresponding variety has  a structure of an integrable system with affine base and with generic fiber being an abelian variety (examples include the total space of the affine Toda integrable system or the so-called Dolbeault hyper-toric varieties).

In this talk I will

1) review the above ideas

2) present a conjectural construction of the homogeneous coordinate ring of the above (projective over affine) varieties via the Borel-Moore homology of some spaces related this time to the affine Grassmannian of the affine Kac-Moody group associated to G

3) Explain how to make this construction precise in the case when G is a torus.

In joint work with Lev Rozansky we construct a coherent sheaf realization of the tensor powers of $U_q(gl(1|1))$ vector representation $\mathbb{C}^{1|1}$. The construction reveals the non-semisimple nature of the representation of $U_q(gl(1|1))$. I plan to explain how this construction extends to а wider class of the representations. The origin of the construction is a boson/fremion correspondence in physics context and Legandre transformation in the mathematical context.

For G a finite group, Malle's conjecture predicts the asymptotic growth of the number of G extensions of a fixed global field. In joint work with Ishan Levy, we compute the asymptotic growth of the number of Galois G extensions of F_q(t), for q sufficiently large and relatively prime to |G|.  In the first part of the talk we will introduce Malle's conjecture and explain how to reduce the above result to a homological stability result for spaces of G bundles. In the second part of the talk we will explain some of the key ideas in the proof of this homological stability result, which uses tools from higher algebra.

Abstract can be found here.