High Dimensional Expanders 2025

A year long lecture/seminar series on high-dimensional expanders, which will culiminate with a 2-week distinguished lecture series by Prof. Alex Lubotzky in December 2025.
Organizers: Anish Ghosh, , Mahesh Kakde, Mahan Mj

Online Videos of Lectures: YouTube Playlist

Lectures

1. [22 Jan] Basic Spectral Graph Theory I (): Introduction, Administrivia, Graphs, Adjaceny and Laplacian Matrix, Spectral Theorem
   [Notes, YouTube Video, Jupyter notebook (ipnyb)]; Ref: [ Spi (Chap 1-3)]
2. [29 Jan] Basic Spectral Graph Theory II (): Eigenspectrum of the Adjacency Matrix and Laplacian, Graph Colouring and Wilf's theorem, Hall's drawing of graphs
   [Notes, YouTube Video]; Ref: [ Spi (Chap 3-4)]

Tentative Schedule (for remaining lectures):

• Basics from Graph theory. Expansion, Cheeger inequality ().
  1. 2025-01-31 (Fri) 14:00-15:30@TIFR:A201: Cayley Graphs, Random Walk Matrix
  2. 2025-02-03 (Mon) 14:00-15:30@TIFR:A201: Random Walk Matrix, convergence, expander mixing lemma
  3. 2025-02-06 (Thurs) 14:00-15:30@TIFR:A201: Cheeger's Inequalities
  4. 2025-02-10 (Mon) 14:00-15:30@TIFR:A201: Cheeger's Inequalities (contd), Expander graphs
  5. 2025-02-13 (Thurs) 14:00-15:30@TIFR:A201: Expander graphs, Ramanujan graphs
  6. 2025-02-17 (Mon) 14:00-15:30@TIFR:A201: Expander graphs, zig-zag expanders
• Kazhdan Property T, Relative property T, Margulis's construction of expanders.
• LPS Ramanujan graphs I. Bruhat-Tits trees of p-adic groups, arithmetic lattices, orders in quaternionic algebras.
• LPS II. Ramanujan's conjecture, Deligne, Drinfeld.
• LPS III. From Deligne/Drinfeld to optimal spectral gap.
• High dim. Bruhat Tits buildings of p-adic groups.
• Applications of HDXs towards testable codes, PCPs etc

Resources/References

[Din22]

Irit Dinur, a course on High dimensional expanders (HDX), Weizmann Institute, Fall 2022.[YouTube Playlist]

[DV25]

Irit Dinur and Thomas Vidick, Winter school on expansion in groups, combinatorics, and complexity, Weizmann Institute, January 5-8, 2025. [YouTube Channel]

[Ghe23]

Summer School on High-dimensional Expanders, Ghent University, May 22-26, 2023. [YouTube Playlist]

[Lub18]

Alex Lubotzky, High dimensional expanders", Plenary Lecture, ICM, Rio De Janiero 2018. [YouTube Video]

[Lub23]

Alex Lubotzky's Fall 2023 Minerva Mini-course, "High Dimensional Expanders and Their Applications in Mathematics and Computer Science", Priceton. [YouTube Playlist]

[Spi25]

Dan Spielman. Spectral and Algebraic Graph Theory (draft of book), 2025. [ .html ]

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