Abstract: I will introduce some recent progress in the ACC conjecture for minimal log discrepancies, and some of its applications. The talk is based on joint works with Jingjun Han and Jihao Liu.

Abstract: I will present two new results about point counting over finite fields and rational cohomology of moduli spaces of curves. The first says that the number of F_q-points on the moduli space of curves of genus 4 with n marked points, for n at most 3, is a polynomial in q. The second, which relies on the first, says that the rational singular cohomology of the moduli space of stable curves of genus g with n marked points vanishes in all odd degrees less than 11, for all g and n. Both results confirm predictions of the Langlands program, via the conjectural correspondence with polarized algebraic cuspidal automorphic representations of conductor 1, which have been classified in low weight by Chenevier and Lannes. The vanishing result for odd cohomology answers a question posed by Arbarello and Cornalba in the 1990s, and the bound of 11 is sharp. Based on joint work with Jonas Bergström and Carel Faber.

Abstract: The classical question of determining which varieties are rational has led to a huge amount of interest and activity. On the other hand, one can take on a complementary perspective - given a smooth projective variety whose nonrationality is known, how far is it from being rational? I will survey what is currently known, with an emphasis on hypersurfaces and complete intersections.

Abstract: The F-signature of a strongly F-regular local ring R is an interesting invariant of its singularities. In this talk, we will discuss this invariant when R is the section ring of a projective variety with respect to an ample divisor. In particular, we study how the F-signature varies as we vary the ample divisor. For this purpose, we will introduce the F-signature function, a real valued function on the ample cone of X, and discuss its continuity properties. This function is analogous to the well-known volume function of big divisors. This is joint work with Seungsu Lee.

Abstract: We discuss how to construct an anti-symplectic involution on any hyperkähler manifold of K3 type equipped with an ample class h of degree 2. We then study the geometry of the fixed locus of this involution and show that when the divisibility of h is 2, this fixed locus always has a connected component which is a Fano variety. This is part of ongoing joint work with E. Macrì, K. O'Grady, and G. Saccà.

Abstract: It is well known that Calabi-Yau manifolds have good deformation theory, which is controlled by Hodge theory. By work of Friedman, Namikawa, M. Gross, Kawamata, Steenbrink and others, some of these results have been extended to Calabi-Yau threefolds with canonical singularities. In this talk, I will report on further extensions in two directions: in dimension 3, we sharpen and clarify some of the existing results, and, secondly, we obtain some higher dimensional analogues. I will also briefly explain the related case of Fano varieties, where stronger results hold. One surprising aspect of our study is the role played by higher du Bois and higher rational singularities, notions that were recently introduced by Mustata, Popa, Saito and their collaborators. This is joint work with Robert Friedman.

Abstract: The geometry of a compact hyperkahler manifold is essentially determined by the Hodge theory of its second cohomology, along with its monodromy representation. In turn, work of Looijenga-Lunts and Verbitsky show that this data is encoded in the total Lie algebra using the underlying hyperkahler metric. In this talk, we will discuss how these results can be obtained algebraically. We will also outline how this proof works in the case of singular symplectic varieties.

Abstract: Generalized pairs were introduced by Birkar and Zhang in [BZ16] in their study of effective Iitaka fibrations, and later became a central topic in modern day birational geometry. For exmaple, the theory of generalized pairs is used to prove the Borisov-Alexeev-Borisov conjecture [Bir19,Bir21]. In this talk I will explain why we care about generalized pairs by showing that they naturally appear in the classification of algebraic varieties. Then I will discuss several recent fundamental results about generalized pairs. e.g Cone theorem ([Hacon-Liu21]), Contraction theorem ([Xie22]), Existence of flips ([Liu-Xie22]). If time permits, I will give the sketch of the proofs, and say something about Kollár's gluing theory, which is a key ingredient in our proofs. As an interesting corollary, we show that glc singularities are Du Bois.

Abstract: Divisorial contractions are blowups with an irreducible exceptional divisor between terminal varieties. Divisorial contractions are one of the main morphisms used in the minimal model program and the Sarkisov program. The 3-dimensional classification of divisorial contractions with centre a point is of local analytic nature. We give some improvements to the existing classification: we simplify the case where the centre is a cA_n singularity, and we show how to extract an effective global algebraic classification from the local analytic one.

Abstract: A full exceptional collection is an important structure on a derived category with many valuable implications. For instance, such a collection produces a basis for the Grothendieck group. After reviewing the landscape of full exceptional collections on linear GIT quotients, we will discuss how to produce them using ideas from "window" categories and equivariant geometry. As an application, we will consider a large class of linear GIT quotients by a reductive group G of rank two, where this machinery produces full exceptional collections consisting of tautological vector bundles. This talk is based on joint work with Daniel Halpern-Leistner.

Abstract: The Quillen-Lichtenbaum conjecture, proved by Voevodsky, states that for smooth complex n-folds, the map from algebraic to topological K-theory with finite coefficients is an isomorphism in degree n-1 and higher, and injective in degree n-2. From this one can construct a birational invariant, and surprisingly it is somewhat computable. But it turns out to vanish for many classes of varieties where the rationality problem is hard, including cubic fourfolds; the proof repackages cycle-theoretic results of Voisin, M. Shen, and others in an interesting way. Or to state a positive result, the algebraic K-theory of Kuznetsov's K3 category behaves like that of an honest surface. This is work in progress with Elden Elmanto.

Abstract: Let Y be a del Pezzo threefold of Picard rank one and degree d\geq 2. We provide a Brill-Noether reconstruction of those del Pezzo threefolds as a subscheme of a Bridgeland moduli spaces in their Kuznetsov components. We show that any exact equivalence between their Kuznetsov components preserves a distinguished object up to some natural auto-equivalences of the Kuznetsov component. As a result, we give a uniform proof of categorical Torelli theorem for them. Further more, we compute the group of auto-equivalences of their Kuznetsov component by extending the exact equivalences to the whole bounded derived categories. If time permits I will also talk about the group of auto-equivalences of Kuznetsov component of index one prime Fano threefold where a different techniques are used. As an application we show that the group of automorphism of index one genus 8 prime Fano threefold is isomorphic to that of associated Phaffian cubic threefold, which is not known in the literature.

Abstract: Brill-Noether theory answers the question of whether a general curve of genus g admits a linear system of rank r and degree d. A refined Brill-Noether theory hopes to answer the question of whether a general such curve also admits a linear system of rank r' and degree d'. In other words, we want to know about the relative position between Brill-Noether loci in the moduli space of curves of genus g. I'll explain a strategy for distinguishing Brill-Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill-Noether loci with respect to containment. Via an analysis of the stability of Lazarsfeld-Mukai bundles, we obtain new lifting results for linear systems of rank 3 which suffice to prove the maximal Brill-Noether loci conjecture in genus up to 23. This is joint work with Richard Haburcak.

Abstract: The moduli space of marked del Pezzo surfaces is one of the most classical moduli spaces in algebraic geometry, essentially dating back to Cayley's original study of the 27 lines on a cubic surface. I will discuss an approach to compactifying the moduli space of marked del Pezzo surfaces via KSBA weighted stable pairs (S,aB), the natural higher dimensional generalizations of Hassett's moduli of weighted stable n-pointed curves. When the degree of the del Pezzo surface is 3 or 4, the compactification of interest is explicitly described when a=1 by work of Hacking, Keel, and Tevelev. In these cases we describe the complete sequence of wall crossings as one decreases the weight a from 1 to the smallest value such that (S,aB) is still a stable pair. Time permitting, I will also discuss potential generalizations to smaller weights and lower degree del Pezzo surfaces.