Syllabus
Theory of determinants and Combinatorics. Trees, Kirchhoff theorem.
Probability and Stochastic matrices. Markov chains. Spectral properties of positive matrices,
Frobenius theorem, spectral gap.
Laplacian and the potential theory on lattices and graphs (Dirichlet forms, Coulomb potentials, random walks). Gaussian measures, Wick formulas.
Heat equation on lattices, path integrals, Feynman Kac formulas.
Discrete Fourier transform. Introduction to waveletts.
Operators on finite dimensional spaces, functions of operators, Laurent series of the resolvent, Jordan normal form, spectral decomposition. Introduction to unbounded operators (and corresponding quadratic forms) on Hilbert space.
Random matrices and their spectra.
Pfaffian. Introduction to the calculus of anticommuting variables.
Linear algebra and statistical physics. Mayer expansion. Determinants of Laplace operators.
Onsager solution of the Ising model.
Annotation
Advanced topics of linear algebra for physicists.. A complement to the basic course of mathematics for physicists.