This proposal outlines an ambitious research program attacking problems in three areas: the affine Grassmannian, fundamentals of permutations, and representation theory inspired by complexity theory. The central theme is to facilitate computation and understanding in these areas through study of very specific posets, Weyl groups, generating functions, and partitions controlling the micro level structures which we often overlook on the first glimpse of the subjects.
At the heart of algebraic combinatorics is the philosophy that every aspect of mathematics can be made more precise, more concrete and more computationally feasible by identifying key combinatorial structures. Borrowing a term from analysis, this proposal is about ``micro local mathematics''. All of the work proposed will have a broad impact in several areas of pure math and theoretical computer science. All of the proposed work will have a computational focus which will further develop computer proof techniques. All of the proposed work will have a human impact component. The PI has a strong track record of mentoring students at all levels and junior faculty. This grant will greatly enhance the vertically integrated research environment in the Combinatorics Group at the University of Washington and give the group the resources needed to attack these hard problems.
During the three years of this grant, the PI, along with her collaborators, and students, addressed several specific problems that span a wide range of mathematics including topology, algebraic geometry, representation theory and combinatorics. The central theme of our work is to facilitate computation and understanding in these areas through study of very specific posets, Weyl groups, generating functions, and partitions controlling the micro level structures which we often overlook on the first glimpse of the subjects. The main opportunities for research training via this grant are aimedat students and postdocs. The PI advised/collaborated on mathematics research with 6 graduate students working toward the Ph.D's, namelyKurt Luoto, Andrew Crites, Mark Contois, Matt Korson, Austin Roberts,and Brendan Pawlowski; 2 postdocs David Anderson and Sami Assaf; and14 undergraduates, Lee Spires, Natalie Hobson, Kun Ling (2011), AdamMosher (2010), Eva Fornaeus, Alexander Leaf, Hannah Manuel, JohnPardo, and Zhen Wei (REU 2010) Henry Kvinge (2009-2010), Ruth Davidson(2008-2009), Chris Fox and Morgan Eichwald (2009), and Jonathan Weed(2008). The major findings of this grant have been or are being written up for publication in 6 research papers which are available on the Math ArXiv website. The Discrete Mathematical Modeling class that the PI teaches alsoserves for research training and development for undergraduates. The main focus of this course is a service-learning project where studentsintensely study one real world problem inspired by some of thechallenges faced by non-profit organizations, government agencies,small businesses, or universities. The projects are done in teams andthe final results are presented in both a poster session and a written document. Most of the problems that the students find are more challenging to solve than any homework problem from a textbook. The students and the PI worked together to find some angle for attack.This is very much like mathematics research. The enrollment has grown to where in spring 2010, 76 students in total worked on 25 different service projects. The PI is actively involved in outreach programs including UW's Mathday, the Pacific Science Center, the Summer Institute in Math at UW, and the Montly Math Hour. The PI will be speaking at the Association of Women in Mathematics 40th Anniversary and the Fourteenth Annual Nebraska Conference for Undergraduate Women in Mathematics in January 2012.