Topology, Geometry and Laplacians of Simplicial Complexes

简单复合体是一种自然抽象的数学对象，在数学的多个领域中发挥着突出的作用。它们表现为曲面或更一般的高维空间的三角剖分，这在拓扑学中对于计算不变量（如欧拉特征）很有用。与几何群论有关的简单复合体的另一个重要例子是建筑物。它们最初是由Jacques Tits在Klein's Erlangen program的工作中引入的，并为群论提供了一种非常成功的几何方法。简单复合体作为组合对象，可以作为光滑几何空间的简化模型。它们的组合性质允许对它们的基本群进行显式计算。基本群是描述连续变形下闭合路径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简单复合体的基本群及其性质。简单复合体的另一个吸引人的特性是，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，并将其推广到更高的维度（流形）。在这个项目中，我们的目标是将已知的曲率几何概念推广到这个更丰富的世界。简单复合体的简单性和多功能性的另一个结果是它们作为几何表示在工业设计和医学成像领域的广泛应用。更好地理解它们的数学性质将导致可以在这些应用中使用的改进的处理算法。

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.