Combinatorics and Topology of Simplicial Complexes

研究单纯复形的拓扑学和组合学。 这包括工作的频谱离散拉普拉斯经营者类似于拉普拉斯经营者的黎曼流形，有关问题的拓扑结构的复交换代数，以及拓扑结构的某些偏序集的细分凸多面体。 研究者研究被称为单纯复形的数学对象。 这些对象是在人们想把日常的物体或形状，如汽车零件或生物分子，用计算机表示出来时产生的。许多重要的问题都围绕着理解或剪裁这些物体的形状，而单纯复形给出了一个具体的方法来做到这一点。 研究者的部分工作可以被认为是对这些物体的形状施加限制，如果我们知道有多少块是用来建造它们的话。

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.