Speaker: Zhiyuan Li (Stanford University)

Time: Aug. 10, 11,13 (14:00-16:30)

Classroom: 全29

Title: Special cycles on Shimura varieties

Abstract: I will talk about the recent proof of Noether-Lefschetz (NL) conjectures on moduli space of K3 surfaces. The proof will be divided into three parts.

Lecture 1: Noether-Lefschetz conjecture and its generalization

Abstract: In the first lecture, I will give an short introduction to NL conjecture, which describes the Picard group of moduli space of K3 surfaces. This includes the Hodge theory on K3 surface and Noether-Lefschetz theory. As the first goal, I will explain how we relate this conjecture to special cycles on Shimura varieties and more importantly, to automorphic representation theory. This will make use of Matsushima formula and Zucker's work on L^2 cohomology of locally Hermitian symmetric varieties.

Lecture 2: Theta correspondence and special theta lifting

Abstract: Howe's theta correspondence theory is used to study automorphic representations on Howe dual pairs and Kudla-Millson special theta lifting theory is the key ingredient of Kudla program, which studies the modularity of generating series of special cycles on Shimura varieties. In the second lecture, I will briefly explain how to use the theories above to connect special cycle classes with automorphic representations coming from theta correspondence.

Lecture 3: Arthur's theory of classification of automorphic representations
Abstract: Recently, Arthur classifies automorphic representation of orthogonal groups into some discrete packets, called Arthur's packets. In the last lecture, I will briefly explain what these packets are and the general philosophy of how to use Arthur's theory to study cohomology classes on Shimura varieties. Then I will talk about the proof of NL conjecture and its generalizations.