Discrete Morse theory

Morse theory studies smooth surfaces or manifolds by looking at generic real-valued functions defined on them. Morse theory was developed in 1927 by Marston Morse; it represents "the most important single contribution to mathematics by an American mathematician", according to Fields laureate Steven Smale. Discrete Morse theory is a way to simplify a triangulated surface/manifold (or even an arbitrary simplicial complex) maintaining some of its topological properties, like homotopy and homology groups. The theory was developed by Forman in 1998, and unlike the smooth counterpart, it is quite elementary. In this short course, we plan to cover (1) basic definitions, (2) relations between smooth and discrete Morse vectors, (3) obstructions from elementary knot theory and (4) computational approaches.