turkmath.org

Computational topology is an area between topology and computer science that applies topological techniques for problems in data and shape analysis. One of the techniques used in this area is the discrete Morse theory developed by Robin Forman as a discrete analogue of Morse theory. This theory gives a way of studying the topology of discrete objects via critical cells of discrete Morse functions. In this talk, we will first briefly mention Morse theory. Then we will talk on discrete Morse theory which will be followed by my thesis problem: composing and decomposing perfect discrete Morse functions (the most suitable functions for combinatorial and computational purposes) on connected sums of triangulated manifolds. In this thesis, we prove that one can compose perfect discrete Morse functions on connected sums of manifolds in any dimensions. On decomposing a given perfect discrete Morse function on a connected sum, our method works in dimensions 2 and 3.