Aperiodic Order, Dynamical Systems, Operator Algebras and Topology
Abstract:
We construct an action of $\mathbb Z^k$ on a compact zero-dimensional space obtained from a higher graph $\Lambda$ satisfying a mild assumption generalizing the construction of the Markov shift associated to a nonnegative integer matrix. The stable Ruelle algebra $R_s(\Lambda)$ is shown to be strongly Morita equivalent to $C^*(\Lambda)$. Hence $R_s(\Lambda)$ is simple, stable and purely infinite, if $\Lambda$ satisfies the aperiodicity condition.
Let $p$ be a prime. The main subject of my talks is the estimation of exponential sums over an arbitrary subgroup $G$ of the multiplicative group ${\mathbb Z}^*_p$:
$$S(a, G) = \sum_{x\in G} \exp(2\pi iax/p), a \in \mathbb Z_p.$$
These sums have numerous applications in additive problems modulo $p$, pseudo-random generators, coding theory, theory of algebraic curves and other problems.
These notes are for a course I gave while on sabbatical at UBC. The topics covered are: stochastic differential equations, solving PDEs using probability, Harnack inequalities for nondivergence form elliptic operators, martingale problems, and divergence form elliptic operators.
This course presupposes the reader is familiar with stochastic calculus; see the notes on my web page for Stochastic Calculus, for example. These notes for the most part are based on my book Diffusions and Elliptic Operators, Springer-Verlag, 1997.