Topology, Geometry and Laplacians of Simplicial Complexes

单纯形复形是自然抽象的数学对象，在数学的许多领域中起着重要的作用。它们表现为曲面的三角剖分或更一般的高维空间，这在拓扑学中用于计算欧拉特征等不变量。与几何群论有关的单纯复形的其他重要例子是建筑物。他们首先介绍了雅克山茨在他的工作克莱因的埃尔兰根计划，并提供了一个非常成功的几何方法群论。单纯复形作为组合对象，可以作为光滑几何空间的简化模型。它们的组合性质允许明确计算其基本群。基本群是描述连续变形下闭路等价性的基本代数工具。在这个项目中，我们的目标是更好地理解单纯复形的基本群及其性质。单纯复形的另一个有吸引力的性质是，它们可以被赋予额外的几何结构，并成为通往比经典曲面更丰富的世界的门户，并将其推广到更高维（流形）。在这个项目中，我们的目标是推广已知的几何概念的曲率到这个更丰富的世界。单纯复形的简单性和多功能性的另一个结果是它们在工业设计和医学成像领域中广泛用作几何表示。更好地理解它们的数学特性将导致改进的处理算法，可以在这些应用中使用。

Simplicial complexes are natural abstract mathematical objects which play a prominent role in several fields of mathematics. They appear as triangulations of surfaces or more general higher dimensional spaces, which are useful in topology for the computation of invariants like the Euler characteristic. Other important examples of simplicial complexes, in connection with geometric group theory, are buildings. They were first introduced by Jacques Tits in his work on Klein's Erlangen program and provide a very successful geometric approach to group theory. Being combinatorial objects, simplicial complexes can serve as simplified models of smooth geometric spaces. Their combinatorial nature allows explicit computations of their fundamental groups. Fundamental groups are a basic algebraic tool to describe the equivalence of closed paths under continuous deformations. In this project, we aim to obtain a better understanding of the fundamental groups of simplicial complexes and their properties. Another attractive property of simplicial complexes is that they can be endowed with additional geometric structures and become gateways to a much richer world than the classical surfaces and their generalisation to higher dimensions (manifolds). In this project, we aim is to generalize known geometric concepts of curvature into this richer world. Another consequence of the simplicity and versatility of simplicial complexes is their wide use as geometric representations in the fields of industrial design and medical imaging. The better understanding of their mathematical properties will lead to improved processing algorithms that can be used in these applications.