中科院数学与系统科学研究院
数学研究所
华罗庚数学重点实验室
华罗庚青年数学论坛
综合报告
时 间：2018.06.25（星期一）
地 点：数学院南楼N913室
9 :30-10 :30
Kazhdan-Lusztig theory of matroids
Botong Wang (Wisconsin-Madison)
Matroids are combinatorial generalizations of configuration of points in vector spaces, or equivalently, hyperplane arrangements. I will discuss two conjectures in matroid theory. The first is a “top-heavy” conjecture by Dowling and Wilson in the 70’s, and the second is some non-negativity conjecture about the Kazhdan-Lusztig polynomial of matroids introduced recently by Elias-Proudfoot-Wakefield. I will explain the proofs of the conjectures in the realizable case (the first conjecture by Huh and myself, and the second by E-P-W). The proof uses Hodge theory of the matroid analogous of the Schubert varieties. I will also talk about some work in progress of extending the proof to the non-realizable case, which is joint with Tom Braden, June Huh, Jacob Matherne and Nick Proudfoot.
The second talk is an introduction to matroids. I will talk about the definition and basic properties of matroids, and how they arise naturally from graphs and vector configurations.
The third talk is on Hodge theory of matroids. I will introduce the Chow rings of matroids, and the work of Adiprasito-Huh-Katz on the Hodge theory of these Chow rings. I will also discuss some log-concavity results as applications.
10:45-11:45
Bimeromorphic Geometry of K ？hler Threefolds
Wenhao Ou （UCLA）
In this series, we will describe bimeromorphic geometry of K？hler threefolds, which is an analogue of birational geometry of projective varieties. The main goal is to generalize Minimal Model Programs (MMP) in the setting of K？hler geometry. The essential theorem in classic MMP is Mori's bend-and-break theorem, which enables us to find rational curves in a projective variety. The proof of this theorem relies on the pure algebraic method, ``reduction modulo p ''. For a K？hler variety however, it is sometimes even difficult to find subvarieties.
A breakthrough in this direction has been made by H？ring and Peternell. They established MMP for non uniruled K？hler threefolds (that is, K？hler threefolds which are not covered by rational curves), and hence proved the existence of minimal models. They also built certain special MMP for uniruled K？hler threefolds, and conclude the existence of Mori fiber spaces. Together with Campana, they proved the abundance theorem for K？hler threefolds as well. Motivated by their ideas, in a recent joint work with Das, we show the log abundance theorem for K？hler threefolds.
The first talk of the series will be introductory. We will start by going through classifications of compact complex curves and surfaces. Then we will review some main theorems in classic MMP, and introduce their conjectural analogues in K？hler geometry. The remaining two talks will focus on the techniques used in the constructions of MMP for K？hler threefolds, and in the proofs of abundance theorems.
下午2:00-3:00
Algebraic dynamics of polynomial endomorphisms of the affine plane
Junyi Xie （CNRS, France）
In these lectures, we study the algebraic dynamics of polynomial endomorphisms of the affine plane. In particular, we introduce the valuation tree at infinity and use it to study the periodic points and periodic curves.
下午3:15-4:15
Recognizing rational homogeneous spaces of Picard number one by its varieties of minimal rational tangents
Qifeng Li (KIAS)
In the series work on deformation rigidity of rational homogeneous spaces of Picard number one, Hwang and Mok develop the theory of varieties of minimal rational tangents (VMRT). The theory of VMRT is a powerful tool to study Fano manifolds of Picard number one. The basic idea is that a large part of the global geometry of a Fano manifold of Picard number can be controlled by its VMRT at a general point. The equivalence of VMRT structure is one of the main problems in this theory. When the VMRT structure is modeled on some well-known manifolds such as G/P_max, the equivalence problem is the question whether we can recognize the manifold by its VMRT’s.
In this lecture series, we will start from basic notions in VMRT theory, and introduce the problem of equivalence of VMRT structures. We will also have a short review on its applications to such as deformation rigidity, classification problems and contact geometry. Then we will give an overview of historical solution of recognizing G/P_max by its VMRT’s. Finally, we will give an introduction to the part of Cartan geometry that is closely related to VMRT theory. This is at present the main machinery for recognizing a (quasi)homogeneous manifold from its VMRT’s.
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