We will discuss random walks, probabilities and measures on finite and infinite groups and graphs, their asymptotic behavior, and numerical and asymptotic invariants. Some of the topics included will be Discrete Fourier Transform, Growth, Amenability, T-property of Kazhdan, Discrete Laplace Operator and its Spectrum.

Among the applications we will consider:

1. Construction of graphs that are good expanders and links to Ramanujan graphs.
2. Zeta functions of groups and graphs, in particular, the Ihara Zeta Function.
3. Study of group and graph analogous of Kac's question "Can you hear the shape of a drum?"
4. Adding machines, groups generated by finite automata and actions on trees.
5. Models of Statistical Physics.
6. Boundaries of groups and graphs.
7. Hyperbolic Groups and Spaces. Symbolic Dynamical Systems and Markov Spaces.