Combinatorial and Algebraic Aspects of Varieties

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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. This proposal addresses several specific problems that span a wide range of mathematics including topology, algebraic geometry, representation theory, statistics, computer science and combinatorics. The central theme is to facilitate computation and understanding in these areas. The four areas of research include varieties with combinatorial structures, k-Schur functions, branched polymers, and combinatorial/statistical algorithms for analyzing ordered data.The work proposed will have a broad impact in several areas of pure math, theoretical physics, computer science and statistics. All of the proposed work will have a computational focus which will further develop algorithms and proof techniques. All of the proposed work will have a human impact component through teaching and mentoring of undergraduates, graduate students and postdocs doing research. In terms of education, the PI has initiated a service learning course to address problems in non-profit organizations and small businesses in our community through mathematics.