Talk

I will talk about certain categorical invariants that are purely algebraic, Gromov–Witten-like invariants associated with smooth, proper Calabi–Yau categories. In the case of Fukaya categories, these are expected to recover the Gromov–Witten invariants of the underlying symplectic manifolds. After a brief overview of the mirror symmetry background, I will discuss an operadic interpretation of the construction of these invariants.

We study properties of automorphic L-functions that arise in the Langlands-Shahidi setting over a global field of characteristic p. We look closely into local properties of the Langlands-Shahidi local coefficient and the representation theory of generic representations of reductive p-adic groups, and present results that are part of current work with del Castillo and Henniart.

BPS invariants and cohomology are central objects in the study of (Kontsevich-Soibelman) Hall algebras or in enumerative geometry of Calabi-Yau 3-folds. In joint work with Yukinobu Toda, we introduce and study a categorical version of BPS cohomology for local K3 surfaces, called quasi-BPS categories. For a generic stability condition, we construct semiorthogonal decompositions of (Porta-Sala) Hall algebra of a K3 surface in products of quasi-BPS categories.

We obtain the exponent of distribution $1/2+1/30$ for the ternary divisor function $d_3$ to square-free and prime power moduli, improving the previous results of Fouvry--Kowalski--Michel, Heath-Brown, and Friedlander--Iwaniec. The key input is certain estimates on bilinear sums with $GL(2)$ coefficients obtained using the delta symbol approach.

Guo and Yang give defining equations for all geometrically hyperelliptic Shimura curves X_0(D,N). In this talk, I will describe how we created a database containing all their Atkin-Lehner quotients and how we computed their sets of Q-rational points when these sets are finite. We also determine which rational points are CM for many of these curves. This is joint work with Ciaran Schembri.