Topology, Geometry and Laplacians of Simplicial Complexes

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.

Simplicial complexes are the most commonly used geometric representations for modelling 3D objects. Even when other representations are used (e.g. splines), they are usually converted to simplicial complexes for visualisation, or for running numerical simulations (e.g. Finite Element Methods).

Areas in the commercial private sector where simplicial complexes are widely used for geometric modelling include animation, computer games and industrial design (Computer Aided Design/Manufacturing).

In the public sector, two typical applications of geometric modelling involving simplicial complexes are the digitisation of cultural heritage and the analysis of medical scans. Museums create digital 3D models of their artefacts for analysis and preservation and for making them accessible to a wider audience though the web. Doctors process 3D scan data from medical devices such as CT, MRI and ultrasound, creating simplicial complex representations which can be automatically analysed.

The theoretical results of the project will lead to a better understanding of the fundamental properties of simplicial complexes and could have a long term impact in ways that we cannot predict at this stage. The applied results can have immediate practical implications, by developing methods for measuring and improving the quality of the models.