Abstract: In studying Higgs bundles, one is naturally led to consider C*-families of flat connections. One fruitful approach to studying the asymptotic behavior of such a family is a procedure known as the "exact WKB method", at least for sufficiently generic Higgs bundles. I will describe how these results can be generalized to the most degenerate case of nilpotent Higgs bundles and how this sheds light on a conjecture by Simpson concerning the moduli space of flat connections.

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 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 ([Hancon-Liu21]), Contraction theorem ([Xie22]), Existence of flips ([Liu-Xie22]), some terminations ([Tsa-Xie23]). If time permits, I will give the sketch of the proofs, and say something about Koll\'ar's gluing theory, which is a key ingredient in our proofs. As an interesting corollary, we show that glc singularities are Du Bois.

Abstract: The birational automorphism group is a natural birational invariant associated to an algebraic variety. In this talk, we study the specialization homomorphism for the birational automorphism group. As an application, building on work of Kollár and work of Chen–Stapleton, we show that a very general n-dimensional complex hypersurface X of degree ≥ 5⌈(n+3)/6⌉ has no finite order birational automorphisms. This work is joint with N. Chen and D. Stapleton.

Abstract: In this talk, I will discuss recent progress on the minimal model program for foliations. Part of this talk is based on joint works with Yujie Luo, Fanjun Meng, and Lingyao Xie.