Georgia Institute of Technology College of Sciences

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.

I will discuss recent work with K. Honda and Y. Huang on proving the Giroux correspondence between contact structures and open book decompositions. Though our work extends to all dimensions (with appropriate adjectives), this talk will focus on the 3-dimensional proof. I will first recall Giroux’s argument for existence of supporting open book decompositions, formulating it in the language adapted to our proof. The rest of the talk will be spent describing the proof of the stabilization correspondence.

We show that in Cartan-Hadamard manifolds M^n, n≥ 3, closed infinitesimally convex hypersurfaces S bound convex flat regions, if curvature of M^n vanishes on tangent planes of S. This encompasses Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in M^3 with minimal total absolute curvature bound Euclidean convex bodies, as stated by M. Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT(0) spaces, and other techniques from Alexandrov geometry outlined by A. Petrunin, including Reshetnyak’s majorization theorem, and Kirszbraun’s extension theorem.

Contact 3-manifolds arise organically as boundaries of symplectic 4-manifolds, so it’s natural to ask: Given a contact 3-manifold Y, does there exist a symplectic 4-manifold X filling Y in a compatible way? Stein fillability is one such notion of compatibility that can be explored via open books: representations of a 3-manifold by means of a surface with boundary and its self-diffeomorphism, called a monodromy. I will discuss joint work with Andy Wand in which we exhibit first known Stein-fillable contact manifolds whose supporting open books of genus one have non-positive monodromies. This settles the question of correspondence between Stein fillings and positive monodromies for open books of all genera. Our methods rely on a combination of results of J. Conway, Lecuona and Lisca, and some observations about lantern relations in the mapping class group of the twice-punctured torus.