From Floer homology to spectral theory, and hyperbolic geometry- Francesco Lin (Columbia)
Abstract: In the first part of the talk, I will review some spectral theory of three manifolds, and discuss its relation with Floer homology, and in particular L-spaces. In the second part of the talk, I will discuss how spectral theory on a hyperbolic three manifold can be understood in terms of natural geometric quantities. This is joint work with M. Lipnowski.
On the monopole Lefschetz number of finite order diffeomorphisms- Jianfeng Lin (UC San Diego)
Abstract: Let K be a knot in an integral homology 3-sphere Y, and Σ the corresponding n-fold cyclic branched cover. Assuming that Σ is a rational homology sphere (which is always the case when n is a prime power), we give a formula for the Lefschetz number of the action that the covering translation induces on the reduced monopole homology of Σ. Our formula is motivated by a Witten-style conjecture relating the two gauge theoretic invariants of homology S1 cross S3s (the Furuta-Ohta invariant and the Casson-Seiberg-Witten invariant). As applications, we give a new obstruction (in terms of the Jones polynomial) for the branched cover of a knot in S3 being an L-space and we define a new set of knot concordance invariants in terms of the monopole Lefschetz numbers of covering translations on the branched covers. This is a joint work with Danny Ruberman and Nikolai Saveliev.
L-spaces in instanton Floer homology- Steven Sivek (Imperial)
Abstract: Framed instanton homology $I^\#(Y)$ is a gauge-theoretic invariant which appears to coincide with the hat version of Heegaard Floer homology. Inspired by the notion of a Heegaard Floer L-space, we say that a rational homology sphere Y is an “instanton L-space" if the rank of $I^\#(Y)$ is as small as possible, namely $|H_1(Y)|$. In this talk I’ll summarize what is known about instanton L-spaces, especially those which arise as Dehn surgeries on knots in $S^3$, and what this tells us about fundamental groups, A-polynomials of knots, and the L-space conjecture. Various parts of this are joint with Antonio Alfieri, John Baldwin, Irving Dai, and Raphael Zentner.