Combinatorial Ergodic theory

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Abstract: DMS-0244994: Byrna KraPenn State UniversityKra proposes research in several areas of ergodic theory anddifferentiable dynamics, with an emphasis on applications of ergodictheory to number theory and combinatorics. The major area of research isthe study of ``non-conventional'' ergodic averages, studying ergodictheorems for averages of functions where even if the system is ergodic,the limit need not be constant. The investigator proposes studyingpolynomial averages, averages for commuting transformations and averagesalong other sequences of recurrence. Additional lines of research arebased on her work on circle maps, including connections between thedifferentiability of the map, the arithmetic properties of its rotationnumber and the dimension of its invariant limit set.The project studies the long term behavior of systems whose dynamics istoo complicated or too chaotic to be understood locally. Dynamicalsystems are used to model problems in natural and social sciences andtheir study leads to an understanding of the evolution of the modeledphenomena. The educational portion of the project aims to contribute tothe graduate and undergraduate programs at Penn State University, withspecial courses designed to inject research into the learning process atan early stage. In fall 2003, the PI will develop and teach a course ontopological dynamics and Ramsey Theory in the MASS (Mathematics AdvancedStudies Semesters) program, a long-standing program aimed at talented fourth-year undergraduates chosen in national competition. The outreach portion aims to make theMathematics Department at Penn State a more attractive place for women tostudy and work, with the creation of a focus group on women's issues andsupport of research projects for undergraduate women.