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Explore discrete Morse theory through this comprehensive lecture that combines topology and combinatorics in an accessible, geometric approach. Begin with fundamental concepts before diving into three cutting-edge research projects: the complex of discrete Morse functions on graph families, merge trees with cycle information, and discrete Morse theory applications to open simplicial complexes. Learn how gradient vector fields on simplicial complexes form their own simplicial structures and discover methods for determining homotopy types through collaborative research with undergraduate students. Examine how merge trees can be enhanced beyond connectedness to include cycle information, with solutions to realization problems for merge trees induced by discrete Morse functions. Investigate the theoretical framework for discrete Morse theory on open simplicial complexes, where traditional structure is modified but analogous results remain achievable. Gain insights into current research directions and potential applications through work conducted with collaborators from Ursinus College, Dioscuri, and University of Florida, making this advanced mathematical topic accessible to researchers and students interested in applied algebraic topology.
