课程视频区

这是 Langlands group 学习课程视频专区,主题是 Langlands program,收集与主题有关的、由相关专家开设的、有完整视频录像和笔记的课程,并附上群内成员相应的学习评论/意见/笔记,持续更新。目前的相关主题课程包括如下:

Fermat last theorem and Modularity Lifting theorem

课程1: Fermat’s Last Theorem by xiao liang http://bicmr.pku.edu.cn/~lxiao/2020fall/2020fall.htm

一个完整的笔记合集:https://yufanluo.com/wp-content/uploads/2021/01/FLT-by-xiao-liang.pdf

课程2:Modularity Lifting by Patrick Allen https://patrick-allen.github.io/teaching/f20-modularity-lifting.html

B站链接:https://www.bilibili.com/video/BV1mv411k76k

Notes on lecture by Wang https://yufanluo.com/wp-content/uploads/2021/01/Modularity-Lifting-by-Patrick-Allen.pdf

课程3:Modularity lifting theorems by Toby Gee https://www.math.arizona.edu/~swc/aws/2013/index.html

共四讲,后两讲画了很多图,很形象,强推

课程4:Fermat’s Last Theorem Conference (Summer 1995) https://www.bilibili.com/video/BV1rt411R7uM

视频有些模糊,可以对照着书看:https://yufanluo.com/wp-content/uploads/2021/01/标准版.pdf

Etale cohomology

课程:Etale cohomology by Daniel Litt https://www.daniellitt.com/tale-cohomology :

Notes on Lecture I :

  1. 介绍了一下这门课和啥是etale cohomology
  2. (Serre)不存在有理系数或者Q_p系数(p是base field的特征)的weil cohomology theory。这里必须强调一点,注意litt的claim中,那个schemes的base field是F_p的代数闭域!不是F_p! 事实上,这个这个论证对于over包含F_{p^2}的域k都是对的,特别地,对于F_p的代数闭域。但是,对于F_p,这个serre的论证是不对的!所以对于F_p,存在系数在Q_p的晶体上同调论(weil cohomology)与这里serre的论证不矛盾。
  3. 介绍了weil conjecture,并且对1维,即曲线的情形,给出了有理函数部分的几何证明,函数方程部分作为exercise,RH部分没证明。关于这部分,i.e. 曲线的weil conj.的完整证明可参考 https://math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Raskin.pdf
  4. 介绍了一般情形的weil conjecture的有理部分的证明,关键是发展一种系数在特征为0的域的weil cohomology给出zeta函数的上同调解释,其实就是Lefschetz trace formula for l adic cohomology。一旦你承认这样的cohomology的存在性,有理函数部分就是线性代数了。这部分可参考milne的 lectures on etale cohomology section 27。而对于dwork的原始证明,可参看这个https://arxiv.org/abs/2012.01809。多提一句,除了 l adic cohomology这个weil cohomology外,我们还有晶体上同调,也有Lefschetz trace formula ,所以晶体上同调也能证明这个有理函数部分。
  5. 最后介绍了Serre’s Kähler analogue,将在下节课给出大概证明。Litt故意留很多gap,有很多exercise。

$p$-adic geometry, perfectoid spaces,prismatic cohomology and geometrization of the local langlands program

课程1:$p$-adic geometry AWS2007 https://www.math.arizona.edu/~swc/aws/2007/notes.html

课程2:Perfectoid Spaces and the Weight-Monodromy Conjecture by Peter Scholze at IHES, 2011. http://www.ihes.fr/~abbes/CAGA/scholze.html B站链接见https://www.bilibili.com/video/BV1vs411E7wP

这是Scholze 解读他本人的博士论文 perfectoid space http://www.ihes.fr/~abbes/CAGA/PerfectoidSpaces.pdf

课程3:Overview: Perfectoid Spaces and their Applicationshttps://www.youtube.com/watch?v=CiN7X7V-HOE&t=2646s&ab_channel=xuan-gottfriedYANG 和 ICM 2014https://www.youtube.com/watch?v=82nhZ1KtcQs&ab_channel=SeoulICMVOD

前者笔记见:https://yufanluo.com/wp-content/uploads/2021/01/Overview.pdf ;后者笔记见https://www.math.uni-bonn.de/people/scholze/ICM.pdf

课程4:Berkeley lectures on $p$-adic geometry by scholze https://www.bilibili.com/video/BV15s41147ke

讲义下载地址:https://www.math.uni-bonn.de/people/scholze/Berkeley.pdf

课程5:AWS2017,applications of perfectoid space https://www.math.arizona.edu/~swc/aws/2017/index.html

Notes on p-adic Hodge theory:

讲下Scholze里introduction里面的一些精彩部分,给出了他探索weight-monodromy猜想的一些线索,尤其是如何想到定义perfect algebra的,主要原因是cotangent complex controls deformation,有一个结果relatively Frobenius 推出vanishing of cotangent complex; 可以参考 http://www-personal.umich.edu/~stevmatt/perfectoid2.pdf

课程6:Prismatic cohomology by Scholze https://www.bilibili.com/video/BV1GJ41147a7?p=1&share_medium=iphone&share_plat=ios&share_source=COPY&share_tag=s_i&timestamp=1611070488&unique_k=qsqSNZ

A great preview of those lectures was given by Tao’s blog https://terrytao.wordpress.com/2019/03/19/prismatic-cohomology/. 一个很不错的笔记见 http://www.math.columbia.edu/~chaoli/doc/BhattEilenberg.html

课程7:Geometrization of the local Langlands correspondence by scholze, https://www.bilibili.com/video/BV17y4y1B7vy?from=search&seid=11345345554728752100

课程主页见 http://www.math.uni-bonn.de/people/scholze/Geometrization/ Notes by Tony feng :https://www.mit.edu/~fengt/beyond.html

课程8:Workshop on “Perfectoid spaces”https://www.icts.res.in/program/perfectoid2019/talks

课程主页:https://www.icts.res.in/program/perfectoid2019

This goal of the proposed summer school is twofold

  1. Give an advanced introduction to Scholze’s theory. 
  2. To understand the relation between perfectoid spaces and some aspects of arithmetic of modular (or, more generally, automorphic) forms such as representations mod p, and lifting of modular forms, completed cohomology, local Langlands program and special values of L-functions.

相关笔记下载地址:https://www.icts.res.in/event/page/18447

Serre conjecture and $p$-adic Langlands program

课程:School on Serre conjectures and the p-adic Langlands program https://mediaspace.unipd.it/channel/School+on+Serre+conjectures+and+the+p-adic+Langlands+program/119214951

可先看一下去年London NT study group的资料: https://nms.kcl.ac.uk/james.newton/lntsg/lntsg2020.html。这个是从最早的Serre modularity conjecture开始说起,能一步步看到它是怎么演变的。

Geometric Langlands program

课程:Langlangds program by Frenkel https://www.bilibili.com/video/BV1XW411C7GN

第一P是一个overview,第二P到第五P讲了函数域的langlands correspondence,还有是怎么到达几何langlands的,第五P讲了一维的证明,也就是deligne的GCFT。第6P到最后讲复数域上的情形,还有他的langlands correspondence for loop group(他写了一本书)

Stacks and moduli

科普:What is a Stack? https://www.youtube.com/watch?v=91fJ3GTM7Dk&list=LL&index=2&ab_channel=TaylorDupuy

课程:Introduction to stacks and moduli by Jarod Alper https://sites.math.washington.edu/~jarod/math582C.html

Galois Representation

课程1$\ell$-adic representations by Adrian Iovita https://www.bilibili.com/video/BV1cr4y1w71S

课程2:$p$-adic Galois representations https://www.bilibili.com/video/BV1WD4y1R7Kh

Higher category theory

科普:$ \infty $-Category Theory for Undergraduates https://www.youtube.com/watch?v=A6hXn6QCu0k&feature=youtu.be&ab_channel=EmilyRiehl

课程1:Lecture course “Topological Cyclic homology”, WS 2020/21 https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/tc.html

课程2:Introduction to higher category theory by Tobias Dyckerhoff https://www.math.uni-hamburg.de/home/dyckerhoff/inftycat2021/index.html

This course will be an introduction to higher category theory focussing on the approach via $ \infty $-categories.

Rational points on varieties

课程1:Rational points on curves by Henri Darmon https://www.bilibili.com/video/BV1M5411H7f5

This article surveys a few of the highlights in the arithmetic of curves: the proof of the Mordell Conjecture, and the more detailed theorythat has developed around the classes of curves most studied until now by number theorists: modular curves, Fermat curves, and elliptic curves. 讲义下载地址:http://www.claymath.org/library/proceedings/cmip08c.pdf

课程2:Faltings’ proof of the Mordell Conjecture by Jared Weinsteinhttps://www.bilibili.com/video/BV1Bt4y1B7VX

课程主页:https://sites.google.com/view/ma-842-spring-2021/home

课程3:Arizona Winter School 2015: Arithmetic and Higher-Dimensional Varieties https://www.math.arizona.edu/~swc/aws/2015/index.html

An introduction to Langlands program

课程1:Automorphic Forms and the Langlands Program by Kevin Buzzard at the Summer Graduate School of MSRI https://www.bilibili.com/video/BV1YW411L7Zr

Berkeley. 课程主页见 http://wwwf.imperial.ac.uk/~buzzard/MSRI/ (底下有完整Latex笔记)

课程2:The local Langlands correspondence and local-global compatibility for $GL_2$ by Sug Woo Shin

  1. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_134l1psy/1_47ritgx2
  2. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_134l1psy/1_0ijb4ug0
  3. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_134l1psy/1_g6ortsdt
  4. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_134l1psy/1_nt9wgc2

课程3:Representation theory and number theory by Benedict Gross https://www.bilibili.com/video/BV1d441147Lz?p=1

笔记见 http://www.math.columbia.edu/~chaoli/docs/EilenbergLectures.html#sec10

Algebraic Number Theory

课程1:An introduction to algebraic and analytic number theory by kedlaya https://www.bilibili.com/video/BV12X4y1K7Wn (Part I), https://www.bilibili.com/video/BV1HT4y1K7RU (Part II)

课程主页见:http://math.ucsd.edu/~kkedlaya/math204a/ (Part I) , http://math.ucsd.edu/~kkedlaya/math204b/ (Part II)

Iwasawa theory

课程1:Iwasawa theory by Ted Chinburg https://www2.math.upenn.edu/~ted/720F18/hw-720SchedTab.html, B站链接 https://www.bilibili.com/video/BV1854y1p7gohttps://www.bilibili.com/video/BV1tV411q7UM

课程主页见:https://www2.math.upenn.edu/~ted/720F18/info.html#texts In this course I will give an overview of Iwasawa theory and of some aspects of the theory of modular forms and Galois representations. The eventual goal of the course is to describe some current research topics having to do with higher codimension Iwasawa theory.

课程2:Arizona Winter School 2018: Iwasawa Theory https://www.math.arizona.edu/~swc/aws/2018/index.html

Ramification Theory and Geometry

课程1:Arizona Winter School 2012: Ramification and Geometry https://www.math.arizona.edu/~swc/aws/2012/index.html

An introduction to $p$-adic Hodge theory

课程1:A quick overview to p-adic Hodge theory by Xiao liang https://www.bilibili.com/video/BV17y4y127sG?p=14

课程2:Introduction to p-adic Hodge theory by Denis Benois

  1. https://www.youtube.com/watch?v=zk8_hbE-w3o&feature=youtu.be&ab_channel=InternationalCentreforTheoreticalSciences
  2. https://www.youtube.com/watch?v=hqjDJshJmkw&feature=youtu.be&ab_channel=InternationalCentreforTheoreticalSciences
  3. https://www.youtube.com/watch?v=3n_sNGKgqgg&feature=youtu.be&ab_channel=InternationalCentreforTheoreticalSciences
  4. https://www.youtube.com/watch?v=YDBUfgsld1k&feature=youtu.be&ab_channel=InternationalCentreforTheoreticalSciences

课程3:Phi-gamma modules and p-adic Hodge theory 1-2 by Gabriel Dospinescu https://www.youtube.com/watch?v=GU4huGMbxbw&ab_channel=MathematicsvideoVAROQUIHerv%C3%A9

Condensed Mathematics

科普演讲:Condensed Mathematics https://www.youtube.com/watch?v=pzq1FvmEjaM&feature=youtu.be&ab_channel=GraduateMathematics

课程:Masterclass in Condensed Mathematics https://www.bilibili.com/video/BV1YD4y1X74T?p=1&share_medium=android&share_source=more&bbid=PAhsW2pYYQNnA2AGegZ6infoc&ts=1610875564950

课程笔记:https://www.math.uni-bonn.de/people/scholze/Condensed.pdf

相关报告:Joint Seminar TUM/UR: Condensed/Pyknotic Mathematics

  1. Condensed Seminar: Who cares about pyknosis https://www.msri.org/seminars/24809
  2. Condensed Seminar: Pyknotic sets https://www.msri.org/seminars/24915#description
  3. Condensed Seminar: Pyknotic abelian groups https://www.msri.org/seminars/24919
  4. Condensed Seminar: Animating pyknotic sets (AKA pyknotic spaces) https://www.msri.org/seminars/24750

主页:http://www.mathematik.uni-regensburg.de/cisinski/condensed.html

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