The theory of Markov chains found applications in diverse areas mathematics, physics, engineering and economics. Markov chains can be modeled using stochastic matrices, which are square matrices with non-negative entries whose rows sum up to one. This allows one to use techniques from analysis and linear algebra to solve questions in probability.
In this project we will attempt to resolve problems on Markov chains arising from research in operator algebras. We will survey the classification of finite and infinite irreducible stochastic matrices, their cyclic decomposition and convergence theorems. We will explain the state-of-the-art on the problems at hand, and try to improve known results. For Markov chains on finite state space, some programming / mathematics software experience will be useful
Math 447 and Math 416. Math 461 is not required, but it is recommended.