Abstract: In this talk, I will discuss recent progress towards Shokurov's ascending chain condition conjecture (ACC) for minimal log discrepancies (mlds) in dimension 3 for varieties, especially when the mld is close to 1. The talk will be divided into three parts. Firstly, I will briefly introduce the history and introduce the two main results: 1) The 1-gap for threefold mlds is 1/13. In other words, any 12/13-klt threefold is canonical. 2) 5/6 is the second largest accumulation point of threefold mlds. In particular, mlds of 5/6-lc threefolds satisfy the ACC. Secondly, I will discuss the proofs of these results, as they are somewhat elementary but extremely complicated. Finally, I will discuss some ideas towards further advances on the ACC conjecture in dimension 3 and their difficulties. If I have enough time, I will discuss some related questions and connections with complements and moduli of surfaces. The talk is based on joint works with Yujie Luo and Liudan Xiao.

Abstract: In this talk, we will introduce the absolute coregularity of Fano varieties. In a few words, the coregularity measures the difference between the dimension of the variety and the dimension of the dual complexes among Calabi-Yau structures on the variety. For instance, a toric variety has coregularity zero. On the other hand, a Fano variety X with coreg(X)=dim(X) is exceptional. In this talk, we will explain the relation between the complements and coregularity of a Fano variety. For instance, we prove that a Fano variety of coregularity zero admits at most a 2-complement, while a Fano variety of coregularity one admits at most a 6-complement. This is joint work with Fernando Figueroa, Stefano Filipazzo, and Junyao Peng.

Abstract: In this talk, I will introduce the theory of syzygies of Mori fibre spaces. The theory is a generalization of the classical Sarkisov program. As an application, we can show that the second group homology of the second Cremona group is non-trivial.

Abstract: In this talk, I will introduce the theory of syzygies of Mori fibre spaces. The theory is a generalization of the classical Sarkisov program. As an application, we can show that the second group homology of the second Cremona group is non-trivial.

Abstract: In this talk, I will give estimates on the Kodaira dimension for fibrations over abelian varieties, and give some applications. One of the results strengthens the subadditivity of Kodaira dimension of fibrations over abelian varieties.

Abstract: A conic bundle X to Z is a contraction between normal varieties of relative dimension one such that the anti-canonical divisor of X is relatively ample over Z. We prove a conjecture of Shokurov which predicts that, if X to Z is a conic bundle such that X has canonical singularities and Z is Q-Gorenstein, then Z is always 1/2-lc, and the multiplicities of the fibers over codimension 1 points are bounded from above by 2. Both values 1/2 and 2 are sharp. This is achieved by solving a more general conjecture of Shokurov and McKernan on singularities of bases of lc-trivial fibrations of relative dimension 1 with canonical singularities. This is joint work with Jingjun Han and Chen Jiang.

Abstract: This is a survey talk on the work of Alexis Bouthier, Jingren Chi and Xiao Wang on the multiplicative Hitchin fibration and its application to the fundamental lemma for unramified Hecke algebras.

Abstract: Boundedness of n-complements is well-known for varieties of Fano type in arbitrary dimension. Using this special case, I will talk about boundedness of n-complements for a larger type of surface pairs.

Abstract: This talk is an introduction to the finiteness problem of the number of projective models in the birational class. In addition, I will clarify the relation of this problem to the Morrison-Kawamata cone conjecture and state the result for log Calabi-Yau surfaces with klt singularities.

Abstract: In this talk, I will discuss the behavior of positivity, hyperbolicity, and Kodaira dimension under smooth morphisms of complex quasi-projective manifolds. This includes a vast generalization of a classical result: a fibration from a smooth projective surface of non-negative Kodaira dimension to a projective line has at least three singular fibers. Furthermore, I will explain a proof of Popa's conjecture on the superadditivity of the log Kodaira dimension over bases of dimension at most three. These theorems are applications of the main technical result, namely the logarithmic base change theorem.

Abstract: In this talk I will construct an example of a non-associative Moufang loop of point classes on a cubic surface over a local field and describe a class of cubic surfaces over number fields for which I conjecture that Moufang loops associated with them are non-associative. The question about the existence of non-associative loops of point classes on cubic surfaces was stated in Yuri I. Manin’s book “Cubic Forms” about fifty years ago. All required concepts will be recalled.