Combinatorial Ergodic theory

摘要：DMS-0244994: Byrna KraPenn州立大学提出在遍历理论和可微动力学的几个领域进行研究，重点是遍历理论在数论和组合学中的应用。主要研究领域是“非常规”遍历平均的研究，研究函数平均的遍历二定理，即使系统是遍历的，极限也不一定是常数。研究者建议研究多项式平均值，交换变换的平均值和沿其他递归序列的平均值。其他的研究是基于她对圆映射的研究，包括映射的可微性、旋转数的算术性质和不变极限集的维数之间的联系。该项目研究系统的长期行为，其动力学太复杂或太混乱，无法在局部理解。动态系统被用来模拟自然科学和社会科学中的问题，它们的研究导致了对被模拟现象演变的理解。该项目的教育部分旨在为宾夕法尼亚州立大学的研究生和本科课程做出贡献，专门设计了一些课程，在早期阶段将研究融入学习过程。2003年秋季，PI将在MASS（数学高等研究学期）项目中开发和教授一门本体动力学和拉姆齐理论课程，这是一个长期存在的项目，目标是在全国竞赛中选出有才华的四年级本科生。拓展部分旨在使宾夕法尼亚州立大学数学系成为一个对女性学习和工作更有吸引力的地方，创建了一个关于女性问题的焦点小组，并支持本科生女性的研究项目。

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.