中科院数学与系统科学研究院

数学研究所

中科院华罗庚数学重点实验室

华罗庚青年数学论坛

综合报告

报告人： 沈俊亮 博士（Yale University）

题  目：The P=W conjecture and hyper-Kähler geometry

时  间：2021.09.24（星期五），09:00-10:00

地  点：数学院南楼N226室 Zoom会议：412 019 4771 密码：mcm1234

摘  要：Topology of Hitchin’s integrable systems and character varieties play important roles in many branches of mathematics. In 2010, de Cataldo, Hausel, and Migliorini discovered a surprising phenomenon which relates these two very different geometric objects in an unexpected way. More precisely, they predict that the topology of Hitchin systems is tightly connected to Hodge theory of character varieties, which is now called the “P=W” conjecture. In this talk, we will discuss recent progress of this conjecture. In particular, we focus on general interactions between topology of Lagrangian fibrations and Hodge theory in hyper-Kähler geometries. This hyper-Kähler viewpoint sheds new light on both the P=W conjecture for Hitchin systems and the Lagrangian base conjecture for compact hyper-Kähler manifolds.

华罗庚青年数学论坛

学术报告

报告人： 沈俊亮 博士（Yale University）

题  目：Cohomology of the moduli of Higgs bundles via positive characteristic

时  间：2021.09.29（星期三），09:00-11:00

地  点：Zoom会议：466 356 2952   密码：mcm1234

摘  要：In this talk, I will explain how the techniques arising from the non-abelian Hodge theory; in positive characteristic provides "consistency checks" of the P=W conjecture, where the latter concerns the non-abelian Hodge theory over the complex numbers. We will focus on two aspects: (1) the Galois conjugation, and (2) the Hodge-Tate decomposition. Based on joint work with Mark de Cataldo, Davesh Maulik, and Siqing Zhang.

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报告人： 沈俊亮 博士（Yale University）

题  目：Kähler differentials, Hodge modules, and Lagrangian fibrations

时  间：2021.10.08（星期五），09:00-11:00

地  点：Zoom会议：412 019 4771 密码：mcm1234

摘  要：In joint work with Qizheng Yin, we proved in 2018 that the dimsneions of the graded pieces of the perverse filtration associated with a Lagrangian fibration of a compact hyper-Kähler variety are matched with the Hodge numbers of the ambient variety. This "Perverse = Hodge" identity suggests a correspondence between coherent sheaves and constructible sheaves for Lagrangian fibrations. In this talk, I will discuss a possible categorification of the connection above, which relates the decomposition theo of a Lagrangian fibration to Kähler differentials of the ambient variety. Based on joint work in progress with Qizheng Yin.