Back to the home page of A.Grigor'yan
12.10.2009 - 05.02.2010
Linear Algebra, Analysis I, Probability I (elementary probability theory only)
A discrete Laplace operator on ZN and on an arbitrary locally finite graph. Weighted graphs and weighted Laplace operator. The maximum principle and the solvability of the Dirichlet problem. Definition of a Markov kernel and a Markov chain. Transition probability. Reversibility. Random walk on a weighted graph as a reversible Markov chain. The notion of ergodicity of Markov chains.
The boundaries of the spectrum and the simplicity of the bottom eigenfunction. The variational characterization of the eigenvalues. The first eigenvalue l1 of the Laplace operator . The convergence to equilibrium for a random walk (Perron-Frobenius theorem). The rate of convergence to the equilibrium. Eigenvalues on products.
Isoperimetric inequalities and Cheeger's inequality. Estimating l1 from below via diameter. Estimating lk from above via distances and measures. The concentration phenomenon. Expansion rate.
Definition of the Dirichlet Laplacian (with boundary condition). Cheeger's inequality for the Dirichlet eigenvalues. Solving the Dirichlet problem by iterations. Isoperimetric inequality on Cayley graphs of finitely generated groups. Isoperimetric inequality in ZN.
The notion of the heat kernel (the transition density) of a random walk. On-diagonal upper bounds of heat kernels via Faber-Krahn inequality. On-diagonal lower bounds of the heat kernel via volume function and l1. The type problem: recurrence versus transience of a random walk. The Nash-Williams test for recurrence. An isoperimetric test for transience. Polya's theorem on the type of a random walk in ZN. Estimates of the return probabilities on Cayley graphs . The type problem on Cayley graphs.