Abstract: The theory of generalized pairs was introduced by C. Birkar and D.-Q. Zhang in order to tackle the effective Iitaka fibration conjecture, and has proven to be a powerful tool in birational geometry. It has recently become apparent that the minimal model program for generalized pairs is closely related to the minimal model program for usual pairs and varieties. A folklore conjecture proposed by J. Han and Z. Li and recently re-emphasized by Birkar asks whether we can always run the minimal model program for generalized pairs with at worst lc singularities. In this talk, we will confirm this conjecture by proving the cone theorem, contraction theorem, and the existence of flips for generalized lc pairs. As an immediate consequence, we will complete the minimal model program for generalized lc pairs in dimension <=3 and the pseudo-effective case in dimension 4. This is joint work with C. D. Hacon.

Abstract: The Noether-Fano inequalities are a classical tool in birational geometry, which describe how a birational map of Fano varieties creates singular linear systems of divisors. In this talk, we provide a general form of the Noether-Fano inequalities which can in particular be applied to a local setting as well. These provide a numerical description of singular divisors obtained by examining two distinct prime blowups of a singularity.

Abstract: A Noetherian ring is a splinter if it is a direct summand of every finite cover. Perhaps owing to their simple definition, basic questions about splinters are often devilishly difficult to answer. For example, Hochster's direct summand conjecture is the modest assertion that a regular ring of any characteristic is a splinter, and was finally settled by André in mixed characteristic more than three decades after Hochster's verification of the equal characteristic case using Frobenius techniques. In this talk, I will discuss some recent work on splinter rings in both positive and mixed characteristics. In particular, inspired by recent work of Bhatt on the Cohen-Macaulayness of the absolute integral closure, I will describe a global notion of splinter in the mixed characteristic setting called global +-regularity with applications to birational geometry in mixed characteristic. This is based on joint works arXiv:2103.10525 with Rankeya Datta and arXiv:2012.15801 with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Joe Waldron, and Jakub Witaszek.

Abstract: This is partially a continuation of Tucker's talk, on the other hand no prerequisites from his talk are required. We generalize the theory of globally F-regular pairs from positive to mixed characteristic, which we call global +-regularity, and introduce certain stable sections of adjoint line bundles. This is inspired by recent work of Bhatt on the Cohen-Macaulayness of the absolute integral closure. We will discuss applications of these results to birational geometry in mixed characteristic. Joint work with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek.

Abstract: The canonical models of varieties of general type are broadly understood thanks to the breakthroughs of Hacon-McKernan-Xu. In contrast, there are many open questions concerning canonical models of varieties of intermediate Kodaira dimensions. In this talk, we introduce the definition of Iitaka volume and use it to study the boundedness of canonical models under some natural conditions.

Abstract: The derived category of a variety is a rich and interesting object which contains a significant amount of geometric information. However, analyzing the structure of this category can be difficult when the variety in question is singular. We discuss some recent work of the speaker which extends some results about the derived categories of smooth varieties to the singular setting, in particular, to singular projective curves.

Abstract: The classical Noether-Lefschetz theorem says that for a very general surface S of degree 4 in P^3 over the complex numbers, the restriction map from the divisor class group on P^3 to S is an isomorphism. In this talk, we will show a Noether-Lefschetz result for varieties over fields of arbitrary characteristic. The proof uses the relative Jacobian of a curve fibration, and it also works for singular varieties (for Weil divisors).

Abstract: In my talk I will describe recent developments in classifying algebraic varieties in positive and mixed characteristic. I will start by explaining the background for complex varieties and providing motivation for the study of their arithmetic analogues. The results which will be discussed in my talk are partially based on recent breakthroughs in arithmetic geometry and commutative algebra.

Abstract: In recent years, Fujita-Odaka, Li and Blum-Jonsson developed a valuative criterion for K-stability of Fano varieties. We generalize this criterion to the log Fano cone singularities. We also show the higher rank finite generation conjecture for log Fano cone singularities, which implies the Yau-Tian-Donaldson Conjecture for the Sasakian-Einstein metric.

Abstract: A conjecture of Orlov states that the Rouquier dimension of the derived category of a smooth projective variety is equal to its dimension. We'll discuss the meaning of the conjecture and some things we know about it, and then explain the proof of a weakened version. This weakened version implies a fact predicted by Orlov's conjecture: If X, Y are smooth projective varieties and there is a fully faithful functor from the derived category of X to the derived category of Y, then the dimension of X is at most the dimension of Y.

Abstract: Finding canonical metrics on compact Kähler varieties has been an intense topic of research for decades. A famous result of Yau says that every compact Kähler manifold with non-positive first Chern class admits a Kähler-Einstein metric (when the Chern class is negative this was also independently proved by Aubin). In this talk, I'll present some recent joint works with Hamid Abban, Yuchen Liu and Chenyang Xu on the existence of Kähler-Einstein metrics when the first Chern class is positive and the variety is possibly singular (such varieties are called Fano varieties). I'll focus on two particular aspects: the solution of the Yau-Tian-Donaldson conjecture, which predicts that the existence of Kähler-Einstein metrics on Fano varieties is equivalent to an algebro-geometric stability condition called K-polystability, and a systematic approach (using birational geometry) to decide whether Kähler-Einstein metrics exist on explicit Fano varieties.

Abstract: In this talk, we will discuss the boundedness of Calabi-Yau threefold admitting an elliptic fibration. First, we will survey some ideas to address weak forms of boundedness of varieties admitting an elliptic fibration. Then, we will switch focus to Calabi-Yau varieties and discuss how the Kawamata-Morrison cone conjecture comes in the picture when studying boundedness properties for this class of varieties. To conclude, we will see how this circle of ideas applies to the case of elliptic Calabi-Yau threefolds. This talk is based on work joint with C.D. Hacon and R. Svaldi.

Abstract: The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach to mirror symmetry. This led to the notion of essential skeleton and the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu. In this talk, I will introduce these objects and report on recent results extending the approach of Nicaise-Xu-Yu. This yields new types of non-archimedean retractions. For families of quartic K3 surfaces and quintic 3-folds, the new retractions relate nicely with the results on the dual complex of toric degenerations and on the Gromov-Hausdorff limit of the family. This is based on a work in progress with Léonard Pille-Schneider.

Abstract: As a generalization of the YTD conjecture for Fano varieties, Collins-Székelyhidi proved a version of the conjecture for log Fano cone singularities. Motivated by the variational approach, we will define a non-Archimedean Monge-Ampère energy in the setting of affine cones, generalizing the one defined for polarized pairs by Boucksom-Hisamoto-Jonsson. We will also see that this agrees with an energy functional defined earlier by Li, which computes the limit slope of the Archimedean Monge-Ampère energy.

Abstract: In recent years, Bridgeland stability conditions have become a central tool in the study of moduli of sheaves and their birational geometry. However, moduli spaces of Bridgeland semistable objects are known to be projective only in a limited number of cases. After reviewing the classical moduli theory of sheaves on curves and surfaces, I will present a new projectivity result for a Bridgeland moduli space on an arbitrary smooth projective surface, as well as discuss how to interpret the Uhlenbeck compactification of the moduli of slope stable vector bundles as a Bridgeland moduli space. The proof is based on studying a determinantal line bundle constructed by Bayer and Macrì. Time permitting, I will mention some work on PT-stability on a 3-fold.

Abstract: One of the best invariants for studying the birational geometry of a variety is its holomorphic forms. In characteristic 0, low degree hypersurfaces (or more generally Fano varieties) do not have any holomorphic forms. For this reason, many problems about birational geometry of these varieties are quite difficult and interesting. E.g. (1) determining if the birational automorphism group is infinite or finite, (2) studying the possible rational endomorphisms of a Fano variety, and (3) understanding the rationality/nonrationality of a Fano variety. Surprisingly, Kollár showed that in characteristic p>0, there are Fano varieties that admit many global (n-1)-forms, and introduced a specialization method for using these forms in characteristic p to control the birational geometry of characteristic 0 Fano varieties. In this talk, we show how this method helps to study problems (1)-(3). This is joint work with Nathan Chen.

Abstract: A longstanding conjecture of T. Fujita asserts that if X is a smooth complex projective variety of dimension n and if L is an ample line bundle, then K_X+mL is basepoint free for m>=n+1. The conjecture is known up to dimension five by work of Reider, Ein, Lazarsfeld, Kawamata, Ye and Zhu. In higher dimensions, breakthrough work of Angehrn, Siu, Helmke and others showed that the conjecture holds if m is larger than a quadratic function in n. We show that for n>=2 the conjecture holds for m larger than n(loglog(n)+3). This is joint work with L. Ghidelli.

Abstract: We explore the moduli space of stable surfaces, where the simplest of questions continue to remain open for almost all invariants. A few such questions: Of the allowable singularities, which ones actually occur on a stable surface? Which of these deform to smooth surfaces? How can we use this knowledge to find divisors in the moduli spaces? Can we develop a stratification of these moduli spaces by singularity type? Our focus will be on cyclic quotient singularities, with an emphasis on discussing concrete visual examples built out of rational, K3, and elliptic surfaces.

Abstract: Given a K3 surface X over a number field k, and a Brauer class A on X, what can we say about the set of primes good reduction of X at which A vanishes? We show that this set contains a set of positive natural density when X is a very general K3 surface. If X is special, this set can have density 0. We use this result to show there exist very general cubic fourfolds, which are conjecturally irrational, that have rational mod p specializations at a set of primes of positive natural density. This is joint work with Sarah Frei and Brendan Hassett.

Abstract: In this talk, I will introduce the so-called "exceptional non-canonical pairs". Although being noncanonical, such pairs are expected to have nice properties. In particular, it is predicted that the set of minimal log discrepancies (mlds) of exceptional non-canonical pairs should satisfy the ascending chain condition (ACC). I will show the relationship of this conjecture with the termination of flips, and the conjecture holds in dimension 3.

Abstract: The Kodaira dimension of smooth projective varieties is an important birational invariant. In this talk, we will discuss some conjectures on the behavior of Kodaira dimension proposed by Popa. We prove an additivity result for the log Kodaira dimension of algebraic fiber spaces over abelian varieties, a superadditivity result for algebraic fiber spaces over varieties of maximal Albanese dimension, as well as a subadditivity result for log pairs over abelian varieties. This is joint work with Mihnea Popa.

Abstract: Enriques categories are generalizations of the derived category of an Enriques surface. Natural examples arise as Kuznetsov components of Gushel-Mukai threefolds and quartic double solids. In this talk, I will discuss a general relation between moduli of stable objects on these categories and on some associated K3 categories. We will discuss applications to the geometry of Fano varieties. This is based on work in progress with Alex Perry and Laura Pertusi.