Combinatorics in geometry and algebra with applications to the natural sciences

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Mathematically, the research focuses on interactions between combinatorics, geometry, and algebra. The projects are centered around specific applications to natural sciences, including evolutionary biology, medical imaging, dynamics of chemical reactions, algorithms and computational complexity, and theoretical high-energy physics. More specifically, the projects include: (1) algorithms for finding centroids in spaces of phylogenetic trees, and general gradient optimization methods for piecewise differentiable functions on polyhedral spaces, with applications to evolutionary biology and medical imaging, and to computational complexity of convex polyhedral spheres; (2) applications of commutative algebra of binomials to dynamics and stability of mass action kinetics in chemistry; (3) efficient algorithms for combinatorial game theory using connections to commutative algebra, based on techniques involving generating functions; and (4) commutative algebra and algebraic geometry of algebraic varieties, with large group actions, arising in high-energy physics.Interactions between modern combinatorics, geometry, and algebra have become increasingly rich in recent years. This research will explore and deepen these connections at their interfaces with a number of other disciplines, including biology, chemistry, physics, computer science, statistics, and operations research. The specific mathematical projects focus on computational geometry and optimization; the algebra of polynomials; and smooth geometry in the presence of symmetry. Despite the range of the proposed intellectual activities in mathematics and its applications, they are drawn together by unifying themes in geometry and algebra. By supporting personnel (undergraduates, graduate students, and postdoctoral researchers), this project will have substantial positive effects on education through research in the aforementioned areas. Fostering ties between these areas, among personnel as well as in scientific discovery, is an explicit, central objective.