Lecture 5 Notes: Eigen Values
Topics
Diameters and their relationship to 2
Expanders
Butterfly networks
1
Diameters and Eigenvalues
So far, every time weve dealt with eigenvalues, its had something to do with
connectivity. For example, the spectral gap

Lecture 6 Notes: Butterfly Networks
1
Administrivia
There were two administrative announcements made at the beginning of class today.
The first problem set will be posted online tonight. It will be due two weeks from today, on
October 15.
Some people ha

Lecture 2 Notes: Rayleigh Quotient
Todays lecture covers three main
parts:
Courant-Fischer formula and Rayleigh quotients
The connection of 2 to graph cutting
Cheegers Inequality
2
Courant-Fischer and Rayleigh Quotients
The Courant-Fischer theorem give

Lecture 4 Notes: DNF Exponentials
1
Administrivia
Two additional resources on approximating the permanent
Jerrum and Sinclair's original paper on the algorithm
An excerpt from Motwani and Raghavan's Randomized Algorithms
2
Review of Monte Carlo Methods
We

Lecture 3 Notes: Matrix Notation
1
Random walks
Let G = (V; E) be an undirected graph. Consider the random process that starts from some vertex v 2 V
(G), and repeatedly moves to a neighbor of the current vertex chosen uniformly at random.
For t _ 0, and

Lecture 8 Notes: Sparsifiers
At the end of the previous lecture, we began to motivate a technique called Sparsification. In
this lecture, we describe sparsifiers and their use, and give an overview of Combinatorial and
Spectral Sparsifiers. We also define

Lecture 9 Notes: General Covexes
1
Outline
Today well introduce and discuss
Polar of a convex body.
Correspondence between norm functions and origin-symmetric bodies (and see how convex
geometry can be a powerful tool for functional analysis).
Fritz-Jo

Lecture 1 Notes: Intro to Course
2
Review of Lecture 1
All of the following are covered in detail in the notes for Lecture 1:
The definition of LG, specifically that LG = DGAG, where DG is a diagonal matrix of degrees and AGis the adjacency matrix of
gra

Lecture 10 Notes: Hyperplane Theorem
1
Outline
Today well go over some of the details from last class and make precise many details that were
skipped. Well then go on to prove Fritz Johns theorem. Finally, we will start discussing the
Brunn-Minkowski ineq

Lecture 7 Notes: Local Clustering
1
Administrivia
You should probably know that
its hints are also posted there.
Also, today in class there was a majority vote for posting problem sets earlier. Professor Kelner
will post the problem sets from two years a