Floer homology can tell whether a knot is fibered, and this has led to proofs that both knot Floer homology and Khovanov homology can positively identify a small handful of knots: the unknot, the trefoils, the figure eight, and the cinquefoil. In this talk, I'll discuss recent work with John Baldwin in which we show for the first time that both invariants can also detect non-fibered knots, including 5_2, and that HOMFLY homology detects infinitely many knots. The key input is a classification of genus-1 knots which are "nearly fibered" from the perspective of knot Floer homology.

Moduli spaces of J-holomorphic curves usually carry a "normal" complex structure coming from the nature of Cauchy-Riemann-type equations. In recent work joint with Guangbo Xu, we take advantage of such a structure to construct abstract perturbations of J-holomorphic curve equations to define integral Gromov-Witten type invariants and integral Hamiltonian Floer homology for general symplectic manifolds. The latter leads to a solution of the Arnold conjecture over the integers, realizing a conjectural picture of Fukaya-Ono proposed in the 90s. I will survey this circle of ideas and perhaps tell you some possibilities going beyond integer-valued invariants.

The traditional rational-valued Gromov-Witten invariants of a symplectic manifold with a circle or torus action can be often be computed using equivariant localization. In this talk, I will discuss a homotopy-theoretic extension of this, for Abouzaid-McLean-Smith’s Morava K-theory Gromov-Witten invariants.

We explain how one can define Gromov Witten invariants of any genus with respect to any generalized cohomology theory in which Chern classes as well as fundamental classes of complex orbifolds make sense.

This talk will focus on the homology concordance group of knots in integer homology 3-spheres bounding integer homology 4-spheres. Using knot Floer homology, we will construct integer-valued homomorphisms from the homology concordance group. This leads to the existence of an infinite rank summand of the quotient group of the homology concordance group modded out by knots in the 3-sphere. This is joint work with Irving Dai, Jennifer Hom, and Matthew Stoffregen.

A long standing question in the study of exotic behavior in dimension four is whether exotic behavior is “stable". For example, in thinking about the four-dimensional h-cobordism theorem, Wall proved that simply connected, exotic four-manifolds always become smoothly equivalent after applying a suitable stabilization operation enough times. Similarly, Hosokawa-Kawauchi and Baykur-Sunukjian showed that exotic surfaces become smoothly equivalent after stabilizing the surfaces some number of times. The question remains, how many stabilizations are necessary, and is one always enough? By considering certain satellite operations, we provide a negative answer to this question in the case of exotic surfaces with boundary. (This draws on joint work with Hayden, Kang, and Park).

We will discuss how certain knot invariants can be homogenized to yield invariants for braids. The idea of this type of homogenization has been around for decades, for example in pioneering work by Gambaudo-Ghys on Levine-Tristram signatures.

By focusing on knot invariants stemming from Heegaard Floer theory, we find a new inequality relating the fractional Dehn twist coefficient with the 4-genus of knots, answering a question of Hubbard-Kawamuro-Ceren Kose-Martin-Plamenevskaya-Raoux-Truong-Turner.

In fact, with the tool of homogenization, this new inequality is readily seen to be one of a family of inequalities, the most famous of which is the slice Bennequin inequality established by Kronheimer-Mrowka and Rudolph.

The meta goal of the talk is to invite the audience to consider what homogenization yields for their favorite (Floer homotopy) invariants.

A few years ago, Matt Stoffregen and I constructed equivariant Khovanov spectra for p-periodic links using Lawson, Lipshitz, and Sarkar's Khovanov stable homotopy type. In this talk, I will sketch our construction and then discuss applications of these spectra in low-dimensional topology. If time permits, I will also suggest further research problems that may require the use of Khovanov spectra or Manolescu-Sarkar's knot Floer spectra.