**Class Hours:** Monday 12-2, Sprinzak 115

**Avi's Office hours:** Wednesday 9-10:30, Ross 215

**Course Requirements:** 50% - lecture notes , 50% - exercises

**TA / Lecture notes coordinator:** Shlomoh Hoory
(shlomoh@cs.huji.ac.il).

**Bulletin Board:**

**Contents:**

**Get all the notes together:** As a pdf file, as a ps.gz file (0.5MB) or
a ps file (2.3MB).

Get a .zip file of all individual .ps files: notes.zip

The notes were written using LaTeX, borrowing from a template of Oded Goldreich. Instructions for writing notes (ps file). A template for writing notes (.tex file).

(A first approximation of the topics to be covered, not completely in order)

**Lecture 1 - Introduction, History and Motivation:**
Superconcentrators for lower bounds (Valiant).
Probabilistic existence of expanders (Pinsker).
Explicit constructions (Margulis) (from Property T).
Amplification without extra bits (Karp, Pippenger and Sipser).
**Scribe:??**

**Algebraic graph theory:** Spectrum of the adjencancy matrix. 2nd
eigen-value. Trace and Girth (Alon- Boppana).
**Links:** Noga Alon

**Notions of expansion:** Vertex expansion,
Edge expansion, Conductance, 2nd e-value, last
e-value, connections to isoperimetric inequalities, Alon's theorem
(and comparison with Jerrum-Sinclair)

**Properties of expanders:** Independent set,
Chromatic number, diameter, mixing time, small sets are
sparser, expander mixing lemma, expansion by d/2. Kahale's
theorem of tightness. Girth (later for LPS).

**Continuous connections:** Manifolds, Laplacian, Cheeger -
how to build manifolds from sequences of graphs
and vice versa (maintaining e-value bounds).
Hyperbolic manifolds? The upper half plane. Approx manifolds by
the graph on fundamental domains and group action.

**Abelian Groups and Cayley graphs:** Fourier transform
on Abelian groups. Cayley graphs of Abelian
groups - computing the eigen values from the Fourier coefficients. Limits on
expansion of Abelian groups. Connection to Linear codes and
epsilon biased sets (Alon - Roichman).

**``Abelian'' consturctions:** Gaber-Galil, discrete Fourier proof by
Jimbo-Marouka (simplified by Boppana). Symmetrization conjecture (Linial)

**Non-Abelian groups and Cayley graphs:** Basic
representation theory. Property T and its connection to eigen-values.
Margulis Theorem - expansion in SL_n(p), n>2.
Graphs of groups acting on sets. Property Tau and its connection
to eigen-values. Selberg's theorem and
Expansion in SL_2(p). Expansion for all gen sets of SL_2(Z) - is
expansion a property of the group? (Lubotzky-Weiss).
Elementary proof of Selberg (Sarnak-Xu).

**LPS,Margulis Ramanujan graph construction.**

**Zig-Zag theorem & elementary construction:** Expanders
as entropy waves. Linear algebra proof. Elementary
construction. Achieving lambda d^{3/4} (and perhaps improvement
to d^{2/3}) (still far from Ramanujan).

**Zig-Zag and Semi-direct product:**
iterative const of Cayley expanders (Alon-Lubotzky-Wigderson ,
Meshulam-Wigderson).

**Applications - Error correcting codes:**
Introduction to Error Correcting Codes.
Erasure codes. Shanon's Theorems. Gilbert Varshamov bound.
Statement of MRRW. Capacity achieving codes for erasures
from expanders (Alon-Luby). Gallager, Tanner, Sipser-Speilman,...
ECC from (lossless) expanders

**Min-entropy zigzag and lossless expanders:**
(Capalbo-Reingold-Vadhan-Wigderson)

**Applications - finding disjoint paths in expanders:**
Peleg-Upfal, Arora-Leighton-Maggs, Frieze-??(FOCS'01). **Links:**
David Peleg
Sanjeev Arora

**Applications - derandomization:**
Expander walks (Ajtai-Komlos-Szemeredi, Cohen-Wigderson,
Impagliazzo-Zuckerman). Weak sources and Extractors
**Links:** David Zuckerman

**Applications - small space derandomization:**
Expanders as discrepancy sets for rectangles (and low
communication comp protocols). Impagliazzo-Nisan-Wigderson version
of Nisan's generator. **Links:** Russell
Impagliazzo Noam Nisan

**Combinatorial applications:**
Universal graphs for all const deg trees (Friedman-Pippenger).
Limits on embeddings (+ semi-definite
programming representation of L2 embeddings and connections to
e-value) (Linial-London-Rabinovich).
...

**Applications - data structures:** Distributed memories (Upfal-Wigderson).
One bit membership queries (Miltersen et al).

A list of references is available as a .bib file.

Book: G. Davidoff, P. Sarnak, A. Vallete Elementary Number Theory, Group Theory and Ramanujan Graphs. [ pdf file (1MB) ps file (1.7MB) ]

Page maintained by Boaz Barak

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