CAREER: Discrete Structures in Continuous Contexts

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The unifying idea behind the investigator's research is to isolate combinatorial structures that govern or arise from continuous data. Projects based on this idea range from computational geometry of polyhedra to equivariant algebraic geometry. First examples include a variety of open problems concerning the computational complexity of shortest paths on convex polyhedral spheres, as well as the deeper geometry controlling this complexity. The polyhedral methods underlying the metric combinatorics that arises in this context have potential applications to graph-coloring questions, to the construction of cycles representing characteristic classes on piecewise linear manifolds, and to algorithmic questions in computational biology. The central issues shed fundamental light on the nature of convexity and polyhedrality in combinatorics and topology. In other projects under the umbrella of extracting discrete phenomena from continuous objects, the investigator and his colleagues study the combinatorics of degenerations of algebraic varieties, often in the presence of Lie group actions. The irreducible components in such degenerations can index summands in formulas from combinatorics and representation theory, thereby giving geometric explanations for the formulas' positivity. The degenerations themselves are defined using techniques from symbolic computational algebra. Many systems in nature and throughout mathematics involve continuously varying quantities, such as volume or temperature. Surprisingly, finding order in such systems is sometimes equivalent to isolating finite or discrete pieces of information---analogous to identifying the solid, liquid, and gas phases of a material---from the underlying continuous framework. In the investigator's research, the revealed discrete structures produce abstract mathematical rewards of deep aesthetic and theoretical significance. However, concrete applications, especially in the form of exact algorithmic calculations by computer, often arise as byproducts of characterizing continuous systems by way of discrete data. The investigator uses his research as a context in which to foster a vertically integrated mentoring environment, modeled on interactive advising programs more common in laboratory sciences, aimed at involving undergraduate, graduate, and postdoctoral students in their development as teachers as well as researchers.