Tuesday, April 24 at 9:00 AM
Many applications involving shapes and data not only require analyzing and processing their geometries, but also associated topologies. In the past two decades, computational topology, an area rekindled by computational geometry has emphasized processing and exploiting topological structures of shapes and data. The understanding of topological structures in the context of computations has resulted into sound algorithms and has also put a thrust in developing further synergy between mathematics and computations in general. This talk aims to delineate this perspective by considering some of the recent developments in topological data analysis including
(i) sparsification of data
(ii) mapper methodology
(iii) optimal homology cycle computations, and
(iv) discrete Morse theory.
For each of the covered topics, we will give the necessary backgrounds in topology, state some of the key results, and indicate open questions/problems. The hope is that the talk will further stimulate the interest in tying together topology with computation.