Videos & Lectures

Modular forms, Modularity Lifting theorem and Fermat last theorem

1. [Lecture] Fermat’s Last Theorem by Liang Xiao B站 Note
2. [Lecture] Modularity Lifting by Patrick Allen B站 Notes on lecture by Wang
3. [Lecture] Modularity lifting theorems by Toby Gee
4. [Lecture] Fermat’s Last Theorem Conference (Summer 1995) The video is not clear enough, you can compare with book
5. [Lecture] Arizona Winter Semester 2021 and 2022 B站
6. [Lecture] An introduction to modular forms by Yitang Zhang (in Chinese)
7. [Talk] A 2020 View of Fermat’s Last Theorem by Kenneth A. Ribet

Galois representations, $p$-adic Hodge theory and $p$-adic Langlands Program

1. [Lecture] A quick overview to p-adic Hodge theory by Liang Xiao
2. [Lecture] Introduction to p-adic Hodge theory by Denis Benois 1 2 3 4
3. [Lecture] 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 2 3 4
4. [Lecture] p-adic Hodge theory and deformations of Galois representations by Eugen Hellmann 1 2 3 4 5
5. [Lecture] An elementary but modern introduction to p-adic Hodge theory by Sean Howe B站
6. [Lecture] $\ell$-adic representations by Adrian Iovita
7. [Lecture] $p$-adic Galois representations
8. [Lecture] p-adic functions, p-adic representations and ($\varphi, \Gamma)$-modules Given by Yangyi Ou as ASARC intensive lectures at KAIST, 2009
9. [Lecture] Padova School on Serre conjectures and the p-adic Langlands program B站 Just have a look first at stuff on London NT study group It starts with Serre’s modularity conjecture, and you can see how it evolved step by step.
10. [Talk] Lifting Galois representations by Toby Gee
11. [Talk] Moduli of Galois Representations by David Savitt
12. [Talk] Serre weight conjectures in higher dimension by Brandon Levin
13. [Talk] Deformations of Galois representations and applications by Emerton Note
14. [Talk] Locally symmetric spaces and torsion classes by Ana Caraiani
15. [Talk] Reciprocity laws for torsion classes by Ana Caraiani
16. [Talk] Locally symmetric spaces, and Galois representations by Peter Scholze 1 2 B站 Recommended to watch B站
17. [Talk] Number of points modulo p when p tends to infinity by Serre
18. [Talk] Modularity of Galois Representations by Chandrashekhar Khare
19. [Talk] Even Galois Representations and the Fontaine-Mazur conjecture by Frank Calegari
20. [Talk] Local ($\ell = p$) Galois Deformation Rings by Ashwin Iyengar
21. [Talk] On local Galois deformation rings by Vytautas Paskunas
22. [Minicourse] An introduction to deformation of Galois Representations by Ehsan Shahoseini 1 2
23. [Talk] Moduli Stacks of Galois Representations by Matthew Emerton
24. [Talk] Moduli stacks of $(\varphi,\Gamma)$-modules by Toby Gee
25. [Lecture] Deformation Theory and Moduli in Algebraic Geometry B站 Note
26. [Talk] Compatible systems of Galois representations of global function fields by Gebhard Böckle
27. [Talk] A step-by-step introduction to p-adic Hodge theory by Jared Weinstein
28. [Talk] Deforming Galois representations by Ashwin Iyengar
29. [Talk] $p$-adic Langlands by Matthew Emerton
30. [Lecture] Franco-Asian Summer School on Arithmetic Geometry B站
31. [Talk] Adjoint Selmer groups for polarized automorphic Galois representations by Patrick Allen
32. [Lecture] Inverse Galois Problem by Tim Dokchitser
33. [Talk] Counting Galois representations by Frank Calegari

Shimura Varieties

1. [Minicourse] Introduction to Shimura varieties by Liang Xiao 1 2 3 4 5 6 7 8 9 10 Note
2. [Minicourse] Counting points on Shimura Varieties by Yihang Zhu
3. [Talk] On torsion in the cohomology of Shimura varieties by Ana Caraiani
4. [Minicourse] Perfectoid Shimura varieties by Ana Caraiani
5. [Minicourse] An example based introduction to Shimura varieties and their compactifications by Kai-Wen Lan 1 2 3

Etale Cohomology, Perverse Sheaves and $p$-adic Cohomology

1. [Lecture] Etale cohomology by Daniel Litt
2. [Talk] Finiteness Theorems for Etale Cohomology of Excellent Schemes by Ofer Gabber Note
3. [Talk] The pro-etale site by Peter Scholze
4. [Minicourse] on perverse sheaves by Williamson
5. [Talk] Etale and crystalline companions by Kiran Kedlaya
6. [Talk] Arithmetic D-modules and existence of crystalline companion by Abe, Tomoyuki

$p$-adic Geometry, Perfectoid Spaces, Prismatic Cohomology and Geometrization of the Local Langlands Program

1. [Lecture] $p$-adic geometry AWS2007
2. [Lecture] Perfectoid Spaces and the Weight-Monodromy Conjecture by Peter Scholze at IHES, 2011 B站 This is Scholze’s interpretation of his own doctoral dissertation perfectoid space
3. [Lecture] Overview: Perfectoid Spaces and their Applications, Note and ICM 2014,Note
4. [Lecture] Berkeley lectures on $p$-adic geometry by scholze Note
5. [Lecture] AWS2017,applications of perfectoid space
6. [Lecture] Prismatic cohomology by Scholze A great preview of those lectures was given by Tao’s blog a great Note
7. [Lecture] Geometrization of the local Langlands correspondence by scholze Course homepage Notes by Tony feng
8. [Lecture] Workshop on “Perfectoid spaces” Course homepage Note
9. [Lecture] prismatic cohomology by kedlaya Course homepage Note
10. [Lecture] Simons Lecture Series：p-adic algebraic geometry by Bhargav Bhatt 1 2 3 Note
11. [Lecture] p-adic Riemann-Hilbert functor and vanishing theorems by Bhargav Bhatt
12. [Lecture] 四部曲
13. ICM2018 Perfectoid spaces and the homological conjectures
14. [Minicourse] Bun_G, shtukas, and the local Langlands program
15. [Talk] Geometrization of the local Langlands correspondence by Laurent Fargues
16. [Lecture] Fourier transform and the geometrization of local Langlands by Arthur-César Le Bras 1 2 3 4 5
17. [Talk] The Fargues-Fontaine curve and local Langlands by Arthur-César Le Bras
18. [Talk] Overview of the Fargues–Fontaine curve by Gabriel Dospinescu
19. [Talk] Vector bundles on the Fargues-Fontaine curve by Gabriel Dospinescu
20. [Workshop] Workshop on Geometrization of Local Langlands Correspondence video

$L$-functions

1. [Lecture] Arithmetic of Elliptic curves and special values of L-functions
2. [Lecture] Analytic Aspects of L-functions and Applications to Number Theory

Langlands Program

1. [Lecture] Automorphic Forms and the Langlands Program by Kevin Buzzard at the Summer Graduate School of MSRI Course homepage (You can find the Latex notes below)
2. [Lecture] The local Langlands correspondence and local-global compatibility for $GL_2$ by Sug Woo Shin 1 2 3 4 5
3. [Lecture] Representation theory and number theory by Benedict Gross Note
4. [Lecture] Langlands correspondence and Bezrukavnikov’s equivalence by Geordie Williamson Note
5. [Lecture] Relative aspects of the Langlands program by Farrell Brumley Course homepage
6. [Lecture] Sato-Tate distributions
7. [Lecture] An introduction to the Langlands correspondence by Matthew Emerton 1 2
8. [Lecture] Motives and L-functions by Frank Calegari 1 2 Note
9. [Popular science] Modular forms and Galois representations by Sug Woo Shin
10. [Talk] The local Langlands conjecture by Richard Taylor
11. [Talk] Local Langlands correspondence for $GL_n$ over $p$-adic fields by Michael Harris
12. [Lecture] Between electric-magnetic duality and the Langlands program by David Ben-Zvi
13. [Lecture] IHES Summer School on the Langlands Program
14. [Workshop] Langlands correspondence for function fields-explain Vincent Lafforgue’s ground breaking work B站
15. [Mini course]Introduction to Langlands-Shahidi Method by Dongming She B站

Algebraic Number Theory and Class Field Theory

1. [Lecture] An introduction to algebraic and analytic number theory by Kedlaya Part I Part II Course homepageI II
2. [Lecture] Algebraic Number Theory II by Johannes Sprang
3. [Lecture] Algebraic Number Theory by Howe
4. [Lecture] Class field theory standpoint and its so different three fundamental generalisation by Ivan Fesenko

Geometric Langlands Program

1. [Lecture] Langlangds program by Frenkel P1:overview, P2-P5: langlands correspondence for function fields; P5: Deligne’s proof of GCFT; P6: the case of over complex number and langlands correspondence for loop group
2. [Talk] Geometric class field theory
3. [Talk] Classical and geometric Langlands over function fields By Dennis Gaitsgory
4. [Workshop] Towards the proof of the geometric Langlands conjecture B站

Stacks and Moduli

1. [Lecture] What is a Stack?
2. [Lecture] Introduction to stacks and moduli by Jarod Alper
3. [Lecture] Moduli spaces in algebraic geometry by Ravi Vakil

Higher Category Theory

1. [Popular science] $ \infty $-Category Theory for Undergraduates
2. [Lecture] Lecture course “Topological Cyclic homology”, WS 2020/21
3. [Lecture] Introduction to higher category theory by Tobias Dyckerhoff
4. [Lecture] Introduction to Derived Geometry Password：Geometry

Rational Points on Varieties

1. [Lecture] Rational points on curves by Henri Darmon Note
2. [Lecture] Faltings’ proof of the Mordell Conjecture by Jared Weinstein Course homepage
3. [Lecture] Arizona Winter School 2015: Arithmetic and Higher-Dimensional Varieties
4. [Talk] Introduction to the Mordell conjecture by Xinyi Yuan

Iwasawa theory

1. [Lecture] Iwasawa theory by Ted Chinburg 1 2 Course homepage
2. [Lecture] Arizona Winter School 2018: Iwasawa Theory

Ramification Theory and Geometry

1. [Lecture] Arizona Winter School 2012: Ramification and Geometry

Condensed Mathematics

1. [Popular science] Condensed Mathematics
2. [Lecture] Masterclass in Condensed Mathematics Bilibili Note
3. [Seminar] Condensed Seminar
   • Who cares about pyknosis
   • Pyknotic sets
   • Pyknotic abelian groups
   • Animating pyknotic sets (AKA pyknotic spaces)
4. [Seminar] Joint Seminar TUM/UR: Condensed/Pyknotic Mathematics
5. [Seminar] Condensed Mathematics (in Chinese)
6. [Lecture] Lecture (Summer 2022): Condensed Mathematics and Complex Geometry by Peter Scholze and Dustin Clausen B站

Deligne–Lusztig Theory

1. [Lecture] Introduction to Deligne Lusztig Theory