RTG Seminar

Michael Usher (U Georgia)
Interlevel persistence and Floer-Novikov theory


Classical finite-dimensional Morse theory analyzes the relations between the sublevel sets of a generic smooth function on a compact smooth manifold. This leads to an algebraic structure---nowadays called (by some) the sublevel persistence module---analogues of which can be constructed and used to great effect in settings such as symplectic Floer theory or in Novikov's Morse theory for S^1-valued functions even when some of the geometric ingredients from the classical context are not present.  In the classical case, it can be useful to consider homologies not just of sublevel sets but of interlevel sets (preimages of general intervals, including singletons); however, in the Floer-theoretic context it is not so obvious what the analogue of the homology of an interlevel set is.  I will explain a general algebraic framework for obtaining interlevel persistence-type barcodes from the sorts of complexes that arise in Floer or Novikov theory by using a structure analogous to Poincaré duality.  In the Novikov context this recovers known invariants from a different perspective, while in the Floer case it strictly refines existing constructions.

This follows RTG Colloquium talk on Wednesday 3:30-4:30.

Event Date: 
November 4, 2021 - 2:30pm to 3:30pm
214 MLH
Calendar Category: 
Seminar Category: 
Representation Theory