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: , , ,

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)]
3. [31 Jan] Basic Spectral Graph Theory III (): Cayley graphs, Random walk matrix
   [Notes, YouTube Video]
4. [3 Feb] Basic Spectral Graph Theory IV (): Random Walk Matrix (irreducibility, reversibility, eigenspectrum, convergence), Expander Mixing Lemma
   [Notes, YouTube Video]
5. [6 Feb] Basic Spectral Graph Theory V (): Cheeger's Inequalities
   [Notes, YouTube Video]
6. [10 Feb] Basic Spectral Graph Theory VI (): Cheeger's Inequalities (Contd), Expander Graphs: vertex and spectral expansion, examples of expander constructions
   [Notes, YouTube Video]; Ref: [Vad (Chap 7)]
7. [13 Feb] Basic Spectral Graph Theory VII (): Spectral expansion implies vertex expansion; Expander application: error reduction, hitting-set lemma
   [Notes, YouTube Video]; Ref: [Vad (Chap 7)]
8. [17 Feb] Basic Spectral Graph Theory VIII (): Zig-Zag expanders
   [Notes, YouTube Video]; Ref: [Vad (Chap 7)]
9. [20 Feb] Basic Spectral Graph Theory IX (): Zig-Zag expanders (contd)
   [Notes, YouTube Video]; Ref: [Vad (Chap 7)]

Tentative Schedule (for remaining lectures):

• Basics from Graph theory. Expansion, Cheeger inequality ().
• Alex Lubotzky's Minerva Mini-Course "HDXs and Their Applications in Mathematics and Computer Science" [YouTube Playlist] (Moderated by )
  • 14:00-15:30, every Friday starting 28 Feb 2025
• 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", Princeton. [YouTube Playlist]

[Spi25]

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

[Vad]

Salil Vadhan. Pseudorandomness, Found. Trends Theor. Comput. Sci., 7(1-3):1–336, 2012. [ .html | doi ]

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