learning course video zone

This is learning course video zone for Langlands group . It collects courses related to Langlands program . Continually updated. 

Modular forms, Modularity Lifting theorem and Fermat last theorem

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

B站链接:https://www.bilibili.com/video/BV17y4y127sG ;一个完整的笔记合集:https://yufanluo.com/wp-content/uploads/2021/01/FLT-by-xiao-liang.pdf

Lecture 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

Lecture 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

Lecture 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

Lecture 5:A friendly introduction to the theory of modular forms https://www.math.arizona.edu/~swc/

Lecture 6:An introduction to modular forms by Yitang Zhang https://www.bilibili.com/video/BV1B4411y7N1?from=search&seid=2629272658836513858

Etale cohomology

Lecture :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

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

Lecture 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

Lecture 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

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

Note:https://www.math.uni-bonn.de/people/scholze/Berkeley.pdf

Lecture 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

Lecture 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

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

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

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

Course homepage: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.

Note:https://www.icts.res.in/event/page/18447

Lecture 9:The topic for this course is prismatic cohomology by kedlaya

https://www.bilibili.com/video/BV1tX4y1G7bM

Course homepage:https://math.ucsd.edu/~kedlaya/math206/

Lecture 10:Simons Lecture Seriesp-adic algebraic geometry by Bhargav Bhatt

  1. https://www.youtube.com/watch?v=zVICcM8PUEI&t=79s&ab_channel=StonyBrookMathematicsStonyBrookMathematics
  2. https://www.youtube.com/watch?v=HyAZFB5niNM&ab_channel=StonyBrookMathematicsStonyBrookMathematics
  3. https://www.youtube.com/watch?v=oEebN8VCHEQ&ab_channel=StonyBrookMathematicsStonyBrookMathematics

Note:http://www.math.stonybrook.edu/Videos/SimonsLectures/PDFs/20210415-Bhatt.pdf

Lecture 11:p-adic Riemann-Hilbert functor and vanishing theorems by Bhargav Bhatt https://www.bilibili.com/video/BV1P541157Np?p=1

Lecture 12:https://www.bilibili.com/video/BV1Gt4y1X7dq?p=1

Geometric Langlands program

Lecture :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

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

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

Higher category theory

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

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

Lecture 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.

Lecture 3:Introduction to Derived Geometry http://bicmr.pku.edu.cn/content/show/70-2449.html

Password:Geometry

Rational points on varieties

Lecture 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

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

Course homepage:https://sites.google.com/view/ma-842-spring-2021/home

Lecture 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

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

Berkeley. Course homepage http://wwwf.imperial.ac.uk/~buzzard/MSRI/ (底下有完整Latex笔记)

Lecture 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
  5. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_d8zm99gi/1_y4s3ep5b

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

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

Lecture 4:Langlands correspondence and Bezrukavnikov’s equivalence by Geordie Williamson https://www.youtube.com/watch?v=9icAHvcVRAc&list=PLtmvIY4GrVv9kMZmf4Fm1mHUGYzTPY8pV&ab_channel=SydneyMathematicalResearchInstitute-SMRI

For notes, see: https://arxiv.org/pdf/2103.02329.pdf

Lecture 5:Relative aspects of the Langlands program by Farrell Brumley https://www.bilibili.com/video/BV19f4y1478E?p=1&share_medium=iphone&share_plat=ios&share_source=GENERIC&share_tag=s_i&timestamp=1615239713&unique_k=yvhg3e

Course homepage:https://www.math.univ-paris13.fr/~brumley/M2-2021.html

Popular science:Modular forms and Galois representations by Sug Woo Shin https://www.youtube.com/watch?v=2cGTHuiRo3E&ab_channel=MathnetKorea

Lecture 6: Sato-Tate distributions https://www.math.arizona.edu/~swc/aws/2016/index.html

Lecture 7: An introduction to the Langlands correspondence by Matthew Emerton

  1. https://www.youtube.com/watch?v=oXQvcNdu-b8&t=520s&ab_channel=MathsVideos
  2. https://www.youtube.com/watch?v=EjewLsTU8Dw&t=508s&ab_channel=MathsVideos

Algebraic Number Theory

Lecture 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)

Course homepage:http://math.ucsd.edu/~kkedlaya/math204a/ (Part I) , http://math.ucsd.edu/~kkedlaya/math204b/ (Part II)

Lecture 2:Algebraic Number Theory II https://homepages.uni-regensburg.de/~spj54141/teaching.html

Iwasawa theory

Lecture 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

Course homepage: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.

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

Ramification Theory and Geometry

Lecture 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, Galois representations and $p$-adic Langlands program

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

Lecture 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

Lecture 3:An overview of the theory of p-adic Galois representations by Jared Weinstein $(\varphi,\Gamma)$-modules and p-adic Hodge theory by Gabriel Dospinescu

  1. https://www.youtube.com/watch?v=1KBKSYTWxBA&t=16s
  2. https://www.youtube.com/watch?v=o1oIOqPWd8k&ab_channel=MathematicsvideoVAROQUIHerv%C3%A9
  3. https://www.youtube.com/watch?v=GU4huGMbxbw&ab_channel=MathematicsvideoVAROQUIHerv%C3%A9
  4. https://www.youtube.com/watch?v=gkPKbrPt9xU&ab_channel=MathematicsvideoVAROQUIHerv%C3%A9

Lecture 4:p-adic Hodge theory and deformations of Galois representations by Eugen Hellmann

  1. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_d8zm99gi/1_t7jpqb5e
  2. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_d8zm99gi/1_0750vb3a
  3. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_z85tr239/1_bq0e6fqy
  4. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_z85tr239/1_qv4betgv
  5. https://mediaspace.unipd.it/playlist/dedicated/119214951/1_z85tr239/1_dewzkyby

Lecture 5:An elementary but modern introduction to p-adic Hodge theory by Sean Howe https://www.math.utah.edu/~howe/6370/

B站链接:https://www.bilibili.com/video/BV1Bv411a7Gw?t=2759

lecture1-2是motivication and overview,lecture 3-17:从p进数的构造开始,基本上是local field的内容,不超过serre local field那本书,Lecture18-23 讲了一些galois representation的基本定义啥的,来自几何的Galois rep,所以介绍椭圆曲线,tate module,一些例子,Lecture 24— p adic divisible group,tate theorem,B-ring. 完整的笔记合集见(105MB):

Lecture 6$\ell$-adic representations by Adrian Iovita https://www.bilibili.com/video/BV1cr4y1w71S

Lecture 7:$p$-adic Galois representations https://www.bilibili.com/video/BV1WD4y1R7Kh

overview:

  1. Lecture 1:很不错的动机介绍;后小部分开始正课:证明C_p是代数闭域,所以需要Krasner lemma;最后引出Ax sense Tate theorem,并做了部分证明准备。
  2. Lecture 2:给出Ax sen tate thm完整证明;给出几个例子;计算;介绍group cohomology;并且证明G_Q_p的C_p半线性表示等价类和group cohomology H^1(G_Q_p,GL_n(C_p))有一个双射。引出下节课想要证明sen的定理:H^1 (G_Q_p,GL_n(C_p))=H^1(Gal(K_infinity/Q_p),GL_n(K_infinity))
  3. Lecture3:这里follow B站视频的序号,P3是空的。
  4. Lecture 4—5:修正lecture 1一处typo。继续lecture2最后提到的sen的定理的证明。基本上是fontaine 欧阳毅 的书 section3-2的prop 3.16的证明。
  5. Lecture 6:fontaine的书section 3.2.4
  6. Lecture 7:fontaine的书 section 3.5.3 C_p admissible,Hodge tate representation,B admissible
  7. Lecture 8:继续hodge tate,witt vector,tilts,untilt,A_inf
  8. Lecture9:定义B_dR^+,B_dR。de rham representation。
  9. Lecture 10:前面继续de rham rep,接着开始B_cris的动机,构造
  10. Lecture11:继续cris rep。然后B_st的构造,semistable rep。
  11. Lecture 12—13:继续B_st。weil deligne modules for de rham representations。

Lecture 8:p-adic functions, p-adic representations and (varphi, Gamma)-modules Given by 欧阳毅 as ASARC intensive lectures at KAIST, 2009. https://www.bilibili.com/video/BV1VW411w7HG?t=13

Lecture 9: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。

Condensed Mathematics

Popular science speech::Condensed Mathematics https://www.youtube.com/watch?v=pzq1FvmEjaM&feature=youtu.be&ab_channel=GraduateMathematics

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

Note: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

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

lecture notes in Chinese: https://www.bananaspace.org/wiki/%E8%AE%B2%E4%B9%89:%E5%87%9D%E8%81%9A%E6%80%81%E6%95%B0%E5%AD%A6

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