Calendar

Abstract:

I will explain how to use bordered Floer homology to obtain a practical, algebraic computation of knot Floer homology. This is joint work with Zoltán Szabó.

Abstract:

Castelnuovo’s bound is a very classical result in algebraic geometry. It asserts a sharp bound on the genus of a curve of degree d in n-dimensional projective space. It is an interesting question to ask whether analogues of Castelnuovo’s bound hold in almost complex geometry. There is a direct analogue in dimension four. In dimension at least eight genus bounds can be established for generic almost complex structures. These results leave open the case of dimension six.

Bryan and Panharipande introduced the notion of super-rigidity of an almost complex structure. They also speculated that this condition might hold for a generic almost complex structure (compatible with a fixed symplectic structure). It had been believed for a long time that super-rigidity will play an important role in the proof of the Gopakumar–Vafa conjecture. However, it turned that Ionel and Parker’s recent proof of this conjecture did not make use of it. Nevertheless, super-rigidity has important consequences. I will present one of these consequences, namely, a genus bound for index zero pseudo-holomorphic curves. This is joint work with Aleksander Doan and, heavily, relies on work by De Lellis, Spadaro, and Spolaor and ideas of Taubes.

There has been a lot of progress towards establishing Bryan and Pandharipande’s super-rigidity conjecture in the work of Wendl. In fact, based on his ideas, Aleksander Doan and I have developed an abstract framework for equivariant transversality/Brill–Noether type questions. Wendl’s work shows that the super-rigidity conjecture holds provided generic real Cauchy-Riemann operators satisfy an easy to state analytic condition. I will explain what this condition means and discuss a few cases in which this condition (or versions of it hold).

Abstract:

A Weinstein manifold is a symplectic manifold which admits a Lagrangian skeleton, and associated to any Weinstein manifold is its wrapped Fukaya category, a powerful algebraic invariant. One important case is that the wrapped category of C^n is trivial. The talk will discuss a partial converse: if X is any Weinstein 6-manifold which is contractible, admitting an arboreal skeleton so that the wrapped category is inductively collapsing, then X is symplectomorphic to C^3. A large portion of the talk will be defining the notion of inductively collapsing: this is a purely algebraic condition, but it depends on a presentation of the wrapped category, which itself comes from a chosen Lagrangian skeleton.

Abstract:

Given a 3-manifold Y, what are the possible definite intersection forms of smooth 4-manifolds with boundary Y? Donaldson’s theorem says that if Y is the 3-sphere, then all such intersection forms are standard integer Euclidean lattices. I will survey some new progress on this problem, for other 3-manifolds, that comes from instanton Floer theory.