Linear Algebra and Combinatorics

Apprentice Program - Summer 2013 REU

Instructors: László Babai and Madhur Tulsiani

Schedule (6/24 - 7/14)
Lectures: MTTh 10-12, Ryerson 277
Problem Session: W 10-12, Ryerson 277
Office Hours: MWTh 12-1, Ryerson 154

Schedule (7/15 - 7/28)
Lectures: MTThF 9:30-12, Ryerson 277
Problem Session: W 9:30-12, Ryerson 277
Office Hours: M-F 12-1, Ryerson 162 ("Theory Lounge")

The course will develop the usual topics of linear algebra and illustrate them on (often striking) applications to discrete structures. Emphasis will be on creative problem solving and discovery.

The basic topics include determinants, linear transformations, the characteristic polynomial, Euclidean spaces, orthogonalization, the Spectral Theorem, Singular Value Decomposition. Application areas to be highlighted include spectral graph theory (expansion, quasirandom graphs, Shannon capacity), random walks, clustering high-dimensional data, extremal set theory, and more.

Resources

Lectures

• 6/24: Vector spaces, linear independence and bases
[Notes]
• 6/25: Lagrange interpolation, secret sharing, matrices, row and column rank
[Notes]
• 6/27: Matrices and linear maps, rank-nullity theorem, systems of linear equations
[Notes]
• 7/1: Permutations and determinants
[Notes]
• 7/2: Computing determinants, inverse of a matrix
[Notes]
• 7/5: Eigenvalues and eigenvectors of a matrix, characteristic polynomial of a matrix, unitary matrices
[Notes]
• 7/8: Gram-Schmidt orthogonalization, Schur's theorem, spectral theorem for normal matrices, eigenvalues and eigenbasis for Hermitian matrices, adjacency matrices of graphs
[Notes]
• 7/9: Eigenvalues and eigenvectors of graphs, Rayleigh quotients
[Notes]
• 7/11: Random walks on graphs, diffusion matrices, stationary distributions and convergence times
[Notes]
• 7/15: Laplacian and normalized Laplacian matrices for a graph, eigenvalues and connectivity, expansion of a graph, Cheeger's inequality
• 7/16: Random variables and expectation, eigenvalues of Laplacian as a continous relaxation of expansion, proof of Cheeger's inequality
[Notes] (for lectures on 7/15 and 7/16)
• 7/17: Problems, review: three miracles of linear algebra, inverse matrix, bilinear and quadratic forms, orthogonal and semidefinite matrices, application of the Spectral Theorem: conic sections
[Notes] (Updated on 7/19)
• 7/18: Good characterizations: 2-colorability, planarity. Topological subgraph, Kuratowski's Theorem. Girth. Chromatic number. The Erdös - de Bruijn intersection theorem, Generalized Fisher inequality, incidence matrix, linear algebra method in extremal combinatorics.
[Notes]
• 7/19: Extremal graph theory, Hoffman-Singleton theorem, Eventown/Oddtown
[Notes]
• 7/22: Expander mixing lemma, Singular Value Decomposition, applications to finding a best-fit subspace and low-rank approximations of matrices
[Notes]
• 7/23: Rank of ATA, Euclidean spaces, function spaces, spaces of polynomials, Gram-Schmidt orthogonalization revisited, volume of parallelepiped, isometry, Gram matrix, volume
[Notes]
• 7/24: Problem session. Number theory: 3 squares, 4 squares theorems, congruence, extremal graph theory, largest eigenvalue vs degrees, when is a triangular matrix normal, Schur's theorem, matrix-tree theorem, Perron-Frobenius theorem
[Notes]
• 7/25: Polynomials of matrices. Limits of sequences of matrices. Diagonalizable matrices are dense. Cayley--Hamilton theorem. The minimal polynomial. Orthogonal and symmetric transformations in Euclidean space. Invariant subspaces.
[Notes]
• 7/26: Vector spaces over finite fields. Perp, totally isotropic subspaces. Oddtown, Eventown solved. Finite Markov Chains revisited: nonnegative matrices, Perron-Frobenius Theorem via Brouwer's Fixed Point Theorem. Applications of SVD to machine learning, data analysis. Spectral Theorem: direct proof.
[Notes]