Penn State University
Kra proposes research in several areas of ergodic theory and differentiable dynamics, with an emphasis on applications of ergodic theory to number theory and combinatorics. The major area of research is the study of ``non-conventional'' ergodic averages, studying ergodic theorems for averages of functions where even if the system is ergodic, the limit need not be constant. The investigator proposes studying polynomial averages, averages for commuting transformations and averages along other sequences of recurrence. Additional lines of research are based on her work on circle maps, including connections between the differentiability of the map, the arithmetic properties of its rotation number and the dimension of its invariant limit set.
The project studies the long term behavior of systems whose dynamics is too complicated or too chaotic to be understood locally. Dynamical systems are used to model problems in natural and social sciences and their study leads to an understanding of the evolution of the modeled phenomena. The educational portion of the project aims to contribute to the graduate and undergraduate programs at Penn State University, with special courses designed to inject research into the learning process at an early stage. In fall 2003, the PI will develop and teach a course on topological dynamics and Ramsey Theory in the MASS (Mathematics Advanced Studies Semesters) program, a long-standing program aimed at talented fourth-year undergraduates chosen in national competition. The outreach portion aims to make the Mathematics Department at Penn State a more attractive place for women to study and work, with the creation of a focus group on women's issues and support of research projects for undergraduate women.
