Expanders: constructions and their applications - Winter/Summer Semester (2015-16)

Time: Mon 14:00-17:30 (with a 30 min break 15:00-15:30)
Location: G77 (TIFR)
Instructors : and
Homepage: http://www.tcs.tifr.res.in/~prahladh/teaching/2015-16/expanders/

Course Announcement


  1. [Jan 15: Prahladh] Introduction, motivation and examples
    Administrivia; 3 motivating examples: superconcentrators, error-correcting codes and error-reduction of one-sided error randomized algoithms; spectral connection; examples of expanders.
    Ref: [HLW (Sec. 1), DH (Lec. 1), Vad (Sec. 4.1.1 and 4.3.1)]

  2. [17 Feb: Anish] Outline of Math component of course

  3. [22 Feb: Lovy Singhal] Tutorial: Introduction to Haar measure, Finite dimensional representation of groups.

  4. [29 Feb: Anish] Kazhdan's property (T)

  5. [7 Mar: Anish] SL(2,R) does not have property (T)

  6. [11 Mar: Lovy Singhal] Tutorial: Unitary representations.

  7. [14 Mar: Prahladh] Expanders -- the spectral connection
    Spectral expansion implies vertex expansion, expander mixing lemma, Cheeger's inequalities.
    Ref: [DH (Lec. 2), , Vad (Sec. 4.1.1 and 4.1.2), Tre (Lec. 3-4)]

  8. [18 Mar: Prahladh] Error-Correcting Codes
    Linear time decodable error-correcting codes: Sipser-Spielman Codes.
    Ref: [DH (Lec. 6), Gur, GI, SS, Zem]

  9. [28 Mar: Prahladh] Random walks on expanders
    Random walks in (general) undirected graphs, undirected connectivity is in randomized logspace (Aleliunas et. al.), Pseudo-random generators: AKS generator.
    Ref: [DH (Lec. 3-4), Vad (Sec. 4.2)]

  10. [4 Apr: Prahladh] Zig-Zag expanders
    Operations on graphs: squaring, tensoring, replacement product, zig-zag product; Construction of expanders using zig-zag product.
    Ref : [Vad (Sec 4.3.2-4.3.3)]

  11. [7 Apr: Lovy Singhal] Tutorial: p-adic numbers.
    Ref: [Rob (Sec I.2.4)]

  12. [8 Apr: Lovy Singhal] Tutorial: Iwasawa Decomposition and Dual spaces.
    Ref: [Bha (Lec. 1 and 7)]

  13. [11 Apr: Anish] SL(2,R)⋉R^2 and SL(3,R) have property (T)

  14. [18 Apr: Anish] SL(3,Z) has property(T) and construction of expander families from lattices.

  15. [25 Apr: Prahladh] Zig-zag and beyond
    Undirected connectivity in logspace, lossless expanders via lossless conductors.
    Ref: [HLW (Sec 10), Kop (Lecture 11-12)]

  16. [2 May: Prahladh] Bipartite Ramanujan expanders for all degrees - I
    2-lifts of graphs; eigen spectrum of lifts; matching polynomials and its properties.
    Ref: [MSS, Spi (Lec. 25-26 )]

  17. [9 May: Prahladh] Bipartite Ramanujan expanders for all degrees - I
    Interlacing polynomials and their properties.
    Ref: [MSS, Spi (Lec. 22 and 26)]


The links below point to the actual location of papers at the author's homepages or other electronic repositories and the papers are not reposted locally.

  1. [Bha] R. Bhatia, Notes on Functional Analysis, Texts and Readings in Mathematics, 50, Hindustan Book Agency 2009.
  2. [BHV] B. Bekka, P. de la Harpe, A. Valette, "Kazhdan’s property (T)", New Mathematical Monographs, 11, Cambridge University Press, 2008.
  3. [Bre] E. Breuillard, "Expanders graphs, property (tau) and approximate groups", PCMI Lecture Notes, Summer School, Utah, Jul 2012.
  4. [DH] C. Dwork, P. Harsha, "CS369E: Expanders in Computer Science", Lecture notes for a course on expanders at Stanford University, Spring 2005.
  5. [Gur] Venkatesan Guruswami. Error-correcting codes and Expander graphs SIGACT News, 35(3): 25-41, September 2004.
  6. [GI] Venkatesan Guruswami, Piotr Indyk: "Linear time encodable/decodable codes with near-optimal rate", IEEE Trans. Information Theory 51(10): 3393-3400 (2005).
  7. [HLW] S. Hoory, N. Linial, A. Wigderson, "Expander graphs and their applications", Bulletin of the American Mathematical Society, vol. 43, no. 4, pp. 439-561, 2006.
  8. [Kop] S. Kopparty, "Topics in Complexity Theory and Pseudorandomness", Rutgers, Spring 2013.
  9. [Laf] V. Lafforgue, "Un renforcement de la propriété (T) [French]" (Translation: Strengthening of the property (T)), Duke Math. J., 143(3):559--602, 2008. (see also: B. Liao, "Strong Banach property (T), after Vincent Lafforgue", arXiv:1212.4646 and A. N. Aaserud. "Lafforgue's new proof that SL(3,R) has Kazhdan's property (T)")
  10. [MSS] Adam W. Marcus, Daniel A. Spielman, Nikhil Srivastava, Interlacing families I: Bipartite Ramanujan graphs of all degrees, Annals of Mathematics, 182 (1), pp. 307-325. 2015.
  11. [Sha] Y. Shalom, The algebraization of property (T), International Congress of Mathematicians Madrid 2006.
  12. [Rob] A. M. Robert, A Course in p-adic Analysis, Graduate Texts in Mathematics, 198, Springer 2000.
  13. [Spi] Dan Spielman, "AM 561/CS 662: Spectral Graph Theory", Yale, Spring 2015.
  14. [SS] Michael Sipser, Daniel A. Spielman: Expander codes. IEEE Transactions on Information Theory 42(6): 1710-1722 (1996).
  15. [Vad] S. Vadhan, "Pseuorandomness (survey/monograph)", Foundations and Trends in Theoretical Computer Science: Vol. 7: No. 1-3, pp 1-336.
  16. [Tre] L. Trevisan, "CS294: A course on Expanders", UC Berkeley, Spring 2016.
  17. [Zem] Gilles Zemor: "On Expander Codes", IEEE Transactions on Information Theory 47(2):835-837 (2001).
  18. [Zim] R. Zimmer, "Ergodic Theory and Semisimple Groups", Monographs in Mathematics, Volume 81, Springer 1984.

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