There is a long and fruitful history of interaction between probability and many other fields such as combinatorics and ergodic theory. This proposal will further many of these connections in a number of fields including random simplicial complexes random matrix theory, extremal combinatorics and geometric group theory. In particular this proposal focuses on problems in the following areas: topology of random simplicial complexes, spectrum of random matrices, higher dimensional generalizations of percolation and subshifts of finite type.
Randomness is evident in many aspects of modern life from conflicting political polls to fluctuations in the stock market. Probability theory is the branch of mathematics which tries to quantify randomness. Over the last few decades the language, ideas and results from probability theory have been applied to many different branches of mathematics. This proposal seeks to continue this trend and extend the connections between probability theory and several branches of mathematics. This proposal also will apply probability theory to topology, a branch of mathematics which until recently has had little interaction with probability. The grant has will sponsor talks that will let undergraduates learn about the ways that mathematics has been applied to a variety of fields.
