Combinatorics and Topology of Simplicial Complexes

研究者研究简单复合体的拓扑学和组合学。这包括在黎曼流形上类似于拉普拉斯算子的离散拉普拉斯算子的谱上的工作，与交换代数的复形拓扑有关的问题，以及凸多面体细分的某些部分有序集的拓扑。研究者研究被称为简单复合体的数学对象。当人们想要将日常物品或形状（如汽车零件或生物分子）用计算机表示出来时，这些物体就会出现。许多重要的问题都围绕着理解或调整这些物体的形状，而简单的复合体提供了实现这一目标的具体方法。研究者的部分工作可能被认为是对这些物体的形状施加限制，如果我们知道它们是用多少块来建造的。

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.