Georgia Institute of Technology College of Sciences

Let ${\mathcal F}L^q_s ({\mathbf R}^2)$ denote the set of all tempered distributions $f \in {\mathcal S}^\prime ({\mathbf R}^2)$ such that the norm $ \| f \|_{{\mathcal F}L^q_s} = (\int_{{\mathbf R}^2}\, ( |{\mathcal F}[f](\xi)| \,( 1+ |\xi| )^s )^q\, d \xi )^{ \frac{1}{q} }$ is finite, where ${\mathcal F}[f]$ denotes the Fourier transform of $f$. We investigate the spectral synthesis for the unit circle $S^1 \subset {\mathbf R}^2$ in ${\mathcal F}L^q_s ({\mathbf R}^2)$ with $1\frac{2}{q^\prime}$, where $q^\prime$ denotes the conjugate exponent of $q$. This is joint work with Prof. Sato (Yamagata University).

The Acyclic Edge Coloring Conjecture, posed independently by Fiam\v{c}ik in 1978 and Alon, Sudakov and Zaks in 2001, asserts that every graph can be properly edge colored with $\Delta+2$ colors such that there is no bicolored cycle. Over the years, this conjecture has attracted much attention. We prove that the conjecture holds asymptotically, that is $(1+o(1))\Delta$ colors suffice. This is joint work with Michelle Delcourt and Luke Postle.

In 2016, Hutchings introduced a knot filtration on the embedded contact homology (ECH) chain complex in order to estimate the linking of periodic orbits of the Reeb vector field, with an eye towards applications to dynamics on the disk. Since then, the knot filtration has been computed for certain lens spaces by myself, and the "action-linking" relationship has been studied for generic contact forms on general three-manifolds by Bechara Senior-Hryniewicz-Salomao. In joint work with Jo Nelson, we study dynamics on surfaces with one boundary component by computing the knot filtration on the ECH chain complex of positive torus knots in S^3. This requires us to understand the contact form as both a prequantization orbibundle and as a periodic open book with positive fractional Dehn twist coefficient. We will focus on the latter point of view to describe how the computation works and the prospects for extending it to more general open books.

I will survey recent progress toward Khovanov homology for links in general 3-manifolds based on categorification of $q$-series invariants labeled by Spin$^c$ structures. Much of the talk will focus on the $q$-series invariants themselves. In particular, I hope to explain how to compute them for a general 3-manifold and to describe some of their properties, e.g. relation to other invariants labeled by Spin or Spin$^c$ structures, such as Turaev torsion, Rokhlin invariants, and the "correction terms'' of the Heegaard Floer theory. There are many problems to work on in this relatively new research area. If time permits, I will outline some of them, and, in the context of plumbed 3-manifolds, comment on the relation to lattice cohomology proposed by Akhmechet, Johnson, and Krushkal.