Time & Venue：
12-29 Mon 9:50-11:25 am, Shuangqing C548
12-30 Tue 13:30-15:05, Shuangqing C548
01-08 Thu 9:50-11:25 am, Shuangqing B725
01-09 Fri 9:50-11:25 am, Shuangqing B725
01-12 Mon 9:50-11:25 am, Shuangqing C548
01-13 Tue 9:50-11:25 am, Shuangqing C548
Description：
自Tate在20世纪60年代提出著名的Hodge–Tate分解猜想以来，p进Hodge理论在随后的六十年中经历了深刻而持续的发展，新的思想与工具不断涌现。其中，完美胚环理论既是最引人注目的突破之一，也逐渐成为理解现代p进几何的基础性语言。
本课程将从历史发展的角度出发，阐释完美胚环在p进几何中的角色与地位，并以此为主线介绍p进Hodge理论的基本框架与核心思想。我们将以友好而细致的方式，向研究生乃至高年级本科生展示p进几何中深刻而优美的技术。
更具体地，我们将从Tate如何利用局部类域论计算离散赋值情形下的 Galois 上同调出发，引入完美胚环的概念，并证明若干关键结果，包括倾斜对应、弧拓扑下的上同调下降以及几乎纯性定理。借助这些工具，我们将计算光滑代数簇的基本群上同调，这是过去六十年来p进Hodge理论的核心内容。最后，我们将讨论这些方法向一般（非离散）赋值环的推广，并展望p进Hodge理论未来的发展。
Since Tate proposed the famous Hodge-Tate decomposition conjecture in the 1960s, p-adic Hodge theory has undergone profound and continuous development over the subsequent sixty years, with new ideas and tools constantly emerging. Among these, the theory of perfectoid rings is one of the most striking breakthroughs and has gradually become a foundational language for understanding modern p-adic geometry.
This course starts from a historical perspective to explain the role and status of perfectoids in p-adic geometry, and uses this as a main thread to introduce the basic framework and core ideas of p-adic Hodge theory. We will present the deep and beautiful techniques of p-adic geometry to graduate students and advanced undergraduates in a friendly and detailed manner.
More specifically, we begin with how Tate used local class field theory to compute Galois cohomology in the discretely valued case, then introduce the notion of perfectoids and prove several key results, including the tilting correspondence, cohomological descent in the arc topology, and the almost purity theorem. Using these tools, we compute the cohomology of the fundamental group of smooth algebraic varieties, which has been a central topic of p-adic Hodge theory over the past sixty years. Finally, we discuss the extension of these methods to general (non-discrete) valuation rings and look ahead to the future development of p-adic Hodge theory.
Prerequisite：
代数几何与代数数论的基础知识
Basics in algebraic geometry and number theory
Reference：
Tate, p-divisible groups.
Faltings, p-adic Hodge theory.
Gabber, Ramero, Almost ring theory.
Scholze, Perfectoid spaces.
Bhatt, Morrow, Scholze, Integral p-adic Hodge theory.
Bhatt, Scholze, Prisms and prismatic cohomology.
Abbes, Gros, Tsuji, p-adic Simpson correspondence.
He, The p-adic Galois cohomology for valuation rings.
Target Audience：Undergraduate and graduate students
Teaching Language：Chinese
Bio:
Tongmu He is a postdoctoral instructor at Princeton University, specializing in arithmetic and algebraic geometry, particularly in p-adic Hodge theory.
Registration: https://www.wjx.top/vm/mZH6yCR.aspx#
