Combinatorics and Topology of Simplicial Complexes

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The investigator studies the topology and combinatoricsof simplicial complexes. This includes work on the spectraof discrete Laplace operators analogous to Laplace operators on Riemannian manifolds, questions relating the topology of thecomplexes to commutative algebra, and the topology of certainpartially ordered sets of subdivisions of convex polytopes. The investigator studies mathematical objects calledsimplicial complexes. These objects arise when one wantsto take everyday objects or shapes, such as automobile partsor biological molecules, and represent them in a computer.Many important problems revolve around understanding or tailoringthe shape of such objects, and the simplicial complexes givea concrete way to do this. Part of the investigator's workmay be thought of as placing constraints on the shapes of theseobjects, if we know how many pieces are used to build them.