1. Introduction to K-Stability（刘雨晨 授课）

2. Introduction to partial differential equations（楚健春 授课）

3. Short course on Higgs bundles and local systems（黄鹏飞 授课）

主要面向本科生，低年级研究生和青年学者。时间安排在七月份下旬到八月中旬。

三门课程都会有录屏，并提供讲义，每节课后会及时发布在网页上。后续关于课程的信息更新均会相应地发在网页上。

组织者：李琼玲  陈省身数学研究所 特聘研究员 qiongling.li@nankai.edu.cn

参加在线课程注意事项：用全名，学生用全名+学校+年级的格式参加。

1. Introduction to K-stability

主讲人：刘雨晨  美国西北大学 助理教授

课程时间：7月 25（周一）， 27（周三）， 29（周五）

                8月 8（周一）， 10（周三）， 12 （周五）

               共六次，上午9-11

课程地点：腾讯会议 ID: 889 6874 6728

课程介绍：

K-stability was first introduced by Tian to characterize the solution of the Kähler-Einstein problem on Fano varieties. In the last decade, a purely algebraic geometric study of K-stability has prospered, based on the birational classification theory of varieties centered around the minimal model program. As one of the most important consequences, the K-moduli theory for Fano varieties has been established using purely algebraic methods.  

课程安排：

In this lecture series, we will give an overview of the recent progress in the algebraic theory of K-stability. In the first part, we will discuss Fujita-Li's valuative criterion, and introduce the alpha-invariant and delta-invariant. In the second part, we will discuss the construction of K-moduli spaces from purely algebraic methods, in particular focusing on openness and properness. In the third part, we will discuss methods to check K-stability for explicit Fano varieties, and introduce the wall crossing framework for K-moduli spaces of log Fano pairs.

参考资料：

[1] Chenyang Xu: K-stability of Fano varieties: an algebro-geometric approach. EMS Surv. Math. Sci. 8 (2021), no. 1-2, 265-354. arXiv:2011.10477.

[2] Harold Blum, Yuchen Liu, Chenyang Xu: Openness of K-semistability for Fano varieties. Duke Math. J., to appear. arXiv:1907.02408.

[3] Yuchen Liu, Chenyang Xu, Ziquan Zhuang: Finite generation for valuations computing stability thresholds and applications to K-stability. Ann. of Math., to appear. arXiv:2102.09405.

[4] Yuchen Liu, Chenyang Xu: K-stability of cubic threefolds. Duke Math. J. 168 (2019), no. 11, 2029-2073. arXiv:1706.01933.

[5] Kenneth Ascher, Kristin DeVleming, Yuchen Liu: Wall crossing for K-moduli spaces of plane curves. arXiv:1909.04576.

[6] Hamid Abban, Ziquan Zhuang: K-stability of Fano varieties via admissible flags. Forum Math. Pi, to appear. arXiv:2003.13788.

主讲人介绍: 

Yuchen Liu is currently an Assistant Professor in the math department at Northwestern University. He received his Ph.D. in mathematics from Princeton University in 2017 under the supervision of Professor János Kollár. His research area is algebraic geometry -- Fano varieties, K-stability, moduli spaces, and singularities. He was awarded the 2022 Sloan Research Fellowship in mathematics. 

2. Introduction to partial differential equations

授课教师： 楚健春  北京大学  研究员

课程时间：7月18（周一）， 20（周三）， 22（周五）， 26（周二）， 28（周四）

                   8月1（周一）， 3（周三）， 5（周五）， 9（周二）， 11（周四）

                共十次，上午10-12

课程地点:  腾讯会议 ID: 733 4355 7888

课程介绍：

Partial differential equation is an equation for an unknown function of two or more independent variables that involves partial derivatives. The study of partial differential equations is not only very active, but also has deep connections with other research fields in mathematics and physics. In this mini-course, we will focus on the theory of linear second order elliptic partial differential equations. Some specific topics include Laplace's equation, Poisson's equation, the maximum principle, the Schauder estimates and Dirichlet problem.

预备知识：

Multivariable calculus, linear algebra and functional analysis.

参考文献：

[1] D. Gilbarg and N.S. Trudinger Elliptic partial differential equations of second order Classics in Mathematics. Springer-Verlag, Berlin, 2001.

[2] Q. Han A basic course in partial differential equations. Graduate Studies in Mathematics, 120. American Mathematical Society, Providence, RI, 2011.

[3] Q. Han and F.H. Lin Elliptic partial differential equations, Second Edition, Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.

主讲人介绍：

Dr. Jianchun Chu is currently an assistant professor at Peking University. He received his Ph.D. in mathematics from Peking University in 2017 under the supervision of Professor Gang Tian. His research focuses on differential geometry and partial differential equations.

3. Short course on Higgs bundles and local systems 

授课教师：黄鹏飞  海德堡大学  博士后

课程时间：7月25（周一）， 26（周二）， 29（周五）

                  8月1（周一）， 5（周五），8（周一）， 9（周二）， 12 （周五）

                 共八次，下午3:30-5:30

课程地点：腾讯会议 ID: 702 6172 3165

课程介绍：

Higgs bundles and local systems, are two core objects in nonabelian Hodge theory. The classical nonablian Hodge theory (means the base is a projective variety, or more general a compact Kähler manifold), is known as a theory dealing with the passage from Higgs bundles to local systems. Such correspondence, especially at the level of categories, is mainly based on the fundamental work of Donaldson, Corlette, Hitchin, and Simpson, as well as the well-known Riemann–Hilbert correspondence. More precisely, over a compact Kähler manifold, we have a categorical correspondence between (poly)stable Higgs bundles of fixed rank and vanishing all the Chern classes, and (semi)simple local systems of the same rank, and (semi)simple flat connections of the same rank.

课程安排：

This short course is a basic course aimed on the introduction to this theory in the categorical point of view. I will divide it into the following parts (not mean each part will take 1 course):

.    (1)  Overview of the course;

.    (2)  Complex geometry (Kähler manifolds, vector bundles, cohomologies, Hodge theory);

.    (3)  Affine GIT (algebraic groups, group actions on affine varieties, quotients and affine GIT);

.    (4)  Betti spaces (local systems, irreducibility, constructing the moduli space via affine GIT);

.    (5)  De Rham spaces (flat connections, Riemann–Hilbert correspondence, (pluri-harmonic) metrics);

.    (6)  Dolbeault spaces (Higgs bundles, (pluri-)harmonic metrics, harmonic bundles and the correspondence);

.    (7)  Proof on the existence of (pluri-)harmonic metrics;

.    (8)  Other topics (Hitchin morphism, C∗-action, λ-connections and twistor spaces, etc) 

参考文献：

Preliminaries on Kähler geometry:

(1) D. Huybrechts, Complex geometry : an introduction, Universitext, Springer-Verlag Berlin Heidelberg, 2005.

(2) C. Voisin, Hodge theory and complex algebraic geometry I, Cambridge studies in advanced mathematics 76, Cambridge University Press, 2002.

Introduction to geometric invariant theory:

(1) P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute lectures, 1978.

References on nonabelian Hodge theory:

(1)  S. K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. Lon- don Math. Soc. 55, 127-131 (1987).

(2)  K. Corlette, Flat G-bundles with canonical metrics, Jour. Diff. Geom. 28, 361-382 (1988).

(3)  N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 1, 59-126 (1987).

(4)  C. T. Simpson, Constructing of variations of Hodge structure using Yang-Mills theory and applications to uniformization, Jour. Amer. Math. Soc. 1, 867-918 (1987).

(5)  C. T. Simpson, Higgs bundles and local systems, Inst. Hautes E ́tudes Sci. Publ. Math. 75, 5-95 (1992).

(6)  Q. Li, An introduction to Higgs bundles via harmonic maps, SIGMA Symmetry, Integrability and Geometry: Methods and Applications 15, 035, 30 pages (2019).

主讲人介绍：

Pengfei Huang is currently a postdoc at Heidelberg University, he mainly  works on nonabelian Hodge theory, especially the geometry of moduli  spaces and the applications.