The topology of simplicial complexes was first investigated by Henri Poincaré in his seminal paper Analysis Situs.
The highly combinatorial nature of simplicial complexes allows for simplicial complexes to be easily stored and manipulated by computers. The recent theories of persistent homology, topological data analysis and computational topology have used simplicial complexes to explore new areas of algebraic topology using computational methods extracted from the underpinning abstract mathematics.
My research concerns the use of the Maslov index and the algebraic theory of surgery to solve the classical problem of finding explicit combinatorial formulae for the signature and rational Pontryagin classes of a triangulated manifold.
Here is my end of first year report and presentation in which I examine the role of the Maslov index and the algebraic eta invariant in finding the signature of a manifold.