Lecture Notes

Lecture notes are courtesy of MIT students and are used with permission.

Spectral Graph Theory
1 Linear algebra review, adjacency and Laplacian matrices associated with a graph, example Laplacians (PDF)
2 Properties of the Laplacian, positive semidefinite matricies, spectra of common graphs, connection to the continuous Laplacian (PDF)
3 Courant-Fischer and Rayleigh quotients, graph cutting, Cheerger’s Inequality (PDF)
4 (Lazy) random walks, their stationary distribution and l2-convergence, normalized Laplacian, conductance, Monte Carlo methods (PDF)
5 Monte Carlo methods continued, approximate DNF counting, approximating the permanent of 0-1 matrices (PDF)
6 Diameters and eigenvalues, expander graphs (PDF)
7 Nonblocking routing networks, local and almost-linear time clustering and partitioning, Lovasz-Simonovits Theorem (PDF)
8 Local and almost-linear time clustering and partitioning (cont.), PageRank, introduction to sparsification (PDF)
9 Sparsification (combinatorial and spectral), effective resistance, matrix pseudoinverses and tail bounds (PDF)
10 Spectral sparsification (cont.), introduction to convex geometry (PDF)
Convex Geometry
11 Polar of a convex body, separating hyperplanes, norms and convex bodies, Banach-Mazur distance, Fritz John’s theorem (PDF)
12 Separating hyperplanes (cont.), Banach-Mazur distance, Fritz John’s theorem, Brunn-Minkowski inequality (PDF)
13 Brunn-Minkowski inequality (cont.), Brunn’s theorem, isoperimetric inequality, Grunbaum’s theorem (PDF)
14 Approximating the volume of a convex body (PDF)
15 Random sampling from a convex body (cont.), grid walk, introduction to concentration of measure (PDF)
16 Concentration of measure and the isoperimetric inequality, Johnson-Lindenstrauss theorem (PDF)
17 Johnson-Lindenstrauss theorem (cont.), Dvoretsky’s theorem (PDF)
Lattices and Basis Reduction
18 Lattices, fundamental parallelepiped and dual of a lattice, shortest vectors, Blichfield’s theorem (PDF)
19 Minkowski’s theorem, shortest/closest vector problem, lattice basis reduction, Gauss’ algorithm (PDF)
20 LLL algorithm for lattice basis reduction, application to integer programming (PDF)
Iterative Methods for Linear Algebra
21 Iterative methods to solve linear systems, steepest descent (PDF)
22 Convergence analysis of steepest descent and conjugate gradients (PDF)
23 Preconditioning on Laplacians, ultra-sparsifiers (PDF)
Multiplicative Weights
24 Multiplicative weights (PDF)
25 Multiplicative weights and applications to zero-sum games, linear programming, boosting, and approximation algorithms (PDF)

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