INTRODUCTION TO ERGODIC THEORY AND DYNAMICS

MATH 807

Treats the qualitative behavior of continuous and discrete dynamical systems, including Hamiltonian systems of differential equations. Typical topics include periodic and almost periodic solutions, the fixed point theorem of Poincare and Birkhoff, invariant curves and KAM theory, celestial mechanics, and chaotic behavior.

MATH/STAT 733

An introduction to measure theoretic probability and stochastic processes.Topics include foundations, independence, zero-one laws, laws of large numbers, convergence in distribution, characteristic functions, central limit theorems, random walks, conditional expectations. Familiarity with basic measure theory (e.g.MATH 629or721) or concurrent registration inMATH 721is strongly recommended.

MATH 858

An introduction to ergodic theory and dynamics on the moduli space of translation surfaces with applications to counting problems in rational billiards and on flat surfaces. Covers dynamical techniques inspired by homogeneous dynamics including the use of Margulis functions to establish quantitative recurrence results; the use of random walks, inspired by work of Benoist and Quint, to prove measure rigidity statements; and the use of Zimmer's theory of the algebraic hull to prove finiteness results.

MATH 415

An introduction to nonlinear dynamical systems including stability, bifurcations and chaos. The course will give underlying mathematical ideas, but emphasize applications from many scientific fields.

MATH 531

A rigorous introduction to probability theory at an advanced undergraduate level. Only a minimal amount of measure theory is used, in particular, the theory of Lebesgue integrals is not needed. It is aimed at math majors and Master's degree students, or students in other fields who will need probability in their future careers. Gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and discusses some classical results with proofs (DeMoivre-Laplace limit theorems, the study of simple random walk on the one dimensional lattice, applications of generating functions).