Georgia Institute of Technology College of Sciences

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.

Let P be a set of points and L be a set of lines in the plane, what can we say about the number of incidences between P and L,    I(P, L):= |\{ (p, l)\in P\times L, p\in L\}| ?

The problem changes drastically when we consider a thickening version, i.e. when P is a set of unit balls and L is a set of tubes of radius 1. Furstenberg set conjecture can be viewed as an incidence problem for tubes. It states that a set containing an s-dim subset of a line in every direction should have dimension at least  (3s+1)/2 when s>0. 

We will survey a sequence of results by Orponen, Shmerkin and a recent joint work with Ren that leads to the solution of this conjecture

A major question in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0, m_1$. In the region of the phase space where the massless body is far from the primaries, the problem can be studied as a (fast) periodic perturbation of the 2 Body Problem (2BP), which is integrable.

We prove that the restricted 3BP exhibits topological instability: for any values of the masses $m_0, m_1$ (except $m_0 = m_1$), we build orbits along which the angular momentum of the massless body (conserved along the flow of the 2BP) experiences an arbitrarily large variation. In order to prove this result we show that a degenerate Arnold diffusion mechanism takes place in the restricted 3BP. Our work extends previous results by Delshams, Kaloshin, De la Rosa and Seara for the a priori unstable case $m_1< 0$, where the model displays features of the so-called a priori stable setting. This is joint work with Marcel Guardia and Tere Seara.