Abstract: A classical result of Blanchard and Deligne asserts that the rational cohomology of a smooth projective fibration splits additively. In this talk, I will discuss how to prove an analogous result for cohomology with coefficients in fields of positive characteristics. Although the splitting in this setting does not always hold, it is closely related to Gromov-Witten invariants of the base variety. This is joint work with Daniel Pomerleano and Guangbo Xu.

Abstract: Inspired by the Noether-Lefschetz theorem, it has been conjectured that the degree of any curve on a (very) general complete intersection variety is divisible by the degree of the variety itself. In this talk, we present a weaker version of this conjecture. Using this result, we confirm a conjecture by Bastianelli--De Poi--Ein--Lazarsfeld--Ullery regarding measures of irrationality. This is joint work with Nathan Chen and Benjamin Church.

Abstract: In this talk, I will report on a joint work in progress with D. Mattei and E. Shinder, where we construct, using Hodge modules, a group scheme that can be thought of as the intermediate Jacobian of a certain complete family of cubic threefolds. We show that the group scheme acts on a well-known abelian fibration. The action gives rise to twisted versions of the abelian fibration. This is similar to twisting genus 1 fibrations with irreducible fibres via its Tate-Shafarevich group.

Abstract: Classification of singularities is an interesting problem in many areas of algebraic geometry, like the minimal model program. One classical approach is to assign to a singular subvariety a rational number, its log canonical threshold. For complex hypersurface singularities, this invariant has been refined by M. Saito to the minimal exponent. The minimal exponent is related to Bernstein-Sato polynomials, Hodge ideals and higher Du Bois and higher rational singularities.

In joint work with Qianyu Chen, Mircea Mustață and Sebastián Olano, we defined the minimal exponent for local complete intersection (LCI) subvarieties of smooth complex varieties. We relate it to local cohomology, higher du Bois and higher rational singularities. In this talk, I will describe what was done in the hypersurface case, give our definition in the LCI case and explain the relation to local cohomology modules and the classification of singularities.

Abstract: A Tevelev degree is a type of Gromov-Witten invariant where the domain curve is fixed in the moduli. After reviewing the basic definitions and previously known results, I will report on joint work with Lehmann, Lian, Riedl, Starr, and Tanimoto, where we improve the Lian-Pandharipande bound on asymptotic enumerativity of Tevelev degrees of hypersurfaces and provide counterexamples to asymptotic enumerativity for certain other Fano varieties.

Abstract: Moduli problems in algebraic geometry give rise to algebraic stacks. Conversely, the study of algebraic stacks on its own has recently seen far-reaching applications across moduli theory and beyond. In this talk, we will give an overview of some of the main ingredients of abstract moduli theory: good moduli spaces and stacks of filtrations. We then discuss two recent applications.

The first one is the construction of canonical refinements of Harder–Narasimhan filtrations for moduli problems such as K-semistable Fano varieties, moduli of principal bundles on a curve, and quotient stacks. The refined Harder–Narasimhan filtration conjecturally describes the asymptotics of certain analytic flows (Calabi flow, Yang–Mills flow, Kemp–Ness flow).

The second application, joint with Chenjing Bu and Tasuki Kinjo, concerns the construction of Hall-algebra like structures for abstract stacks. With this construction, we can give a meaningful definition of Euler characteristic for an Artin stack, as well as generalising Donaldson–Thomas theory beyond the case of moduli of objects in abelian categories.

Abstract: The Chern-Hopf-Thurston conjecture asserts that for a closed, aspherical manifold X of dimension 2d, the Euler characteristics satisfies $(-1)^d\chi(X)\geq 0$. In this talk, we present a proof of the conjecture for projective manifolds whose fundamental groups admit an almost faithful linear representation. Moreover, we establish a stronger result: all perverse sheaves on X have nonnegative Euler characteristics. We will discuss how this problem is related to certain positivity properties of the cotangent bundle of X, and demonstrate how non-abelian Hodge theory, in both archimedean and non-archimedean setting, leads to these positivity results. This is joint work with Ya Deng.

Abstract: In Grothendieck’s 1983 letter to Faltings that initiated the study of anabelian geometry, he conjectured that a large class of schemes can be reconstructed from their étale topoi. In this talk, I’ll discuss joint work with Magnus Carlson and Sebastian Wolf that proves Grothendieck’s conjecture for infinite fields. Specifically, we show that over a finitely generated field k of characteristic 0, seminormal finite type k-schemes can be reconstructed from their étale topoi. Over a finitely generated field k of positive characteristic and transcendence degree ≥ 1, we show that perfections of finite type k-schemes can be reconstructed from their étale topoi. Our results generalize work of Voevodsky.