Title: Morse inequalities on singular spaces.
Abstract: The classical Morse inequalities are a well understood result, which has various proofs using combinations of topology, algebra, and analysis. Witten’s proof of the Morse inequalities revolutionized the subject, leading the way to generalizations in various directions, corresponding to different operators to infinite dimensional spaces and singular spaces. I will explain a new proof which uses ideas from both the topological and analytical proofs, using the best of both worlds. I will explain some conjectural results on how these should extend to a wider class of singular spaces. This includes classes of algebraic varieties (real and complex) and moduli spaces such as that of Riemann surfaces with the Weil Peterson metric.
