Columbia Symplectic Geometry, Gauge Theory, and Categorification Seminar

Jan 30: Yikai Teng (Rutgers University-Newark) "Khovanov homology and exotic planes"

Since the 1980s, mathematicians have discovered uncountably many "exotic" embeddings of R^2 in R^4, i.e., embeddings that are topologically but not smoothly isotopic to the standard xy-plane. However, until today, there have been no direct, computable invariants that could detect such exotic behavior (with prior results relying on indirect arguments). In this talk, we define the end Khovanov homology, which is the first known combinatorial invariant of properly embedded surfaces in R^4 up to ambient diffeomorphism. Moreover, we apply this invariant to detect new exotic planes, including the first known example of an exotic plane that is a Lagrangian submanifold of the standard symplectic R^4.

Feb 6: Amanda Hirschi (Sorbonne University) "On the Donaldson 4-6 problem"

The Donaldson 4-6 question asks how deformation classes of stabilised symplectic forms, i.e. after taking the product with S^2, are related to the underlying smooth structures on the underlying smooth manifold in dimension 4 I will describe one example of a smooth 4-manifold admitting two symplectic forms which remain deformation inequivalent after taking the product with S^2, giving counterexamples to one implication of the conjectured relation. On the other hand, I will explain why two symplectic manifolds, whose stabilisations are deformation equivalent, have the same Gromov-Witten invariants. This is joint work with Luya Wang.

Feb 13: Mihai Marian (University of British Columbia) "An operator on bordered Khovanov homology induced by 2-cabling"

Cabling strongly invertible knots induces an operator on associated 4-ended tangles. In the case of 2-cabling, I will describe the construction of the resulting induced operator on the bordered Khovanov theory of Koteslkiy–Watson–Zibrowius, a theory that assigns immersed curves in a 4-punctured sphere to 4-ended tangles. Finally, I will discuss some of the structure revealed by this operator, in particular, how it relates to a new concordance invariant due to Lewark–Zibrowius.

Feb 20: Daniel Pomerleano (UMass Boston) "Frobenius intertwiners and the p-adic Gamma class"

In recent joint work with Bai and Seidel, we formulated a conjecture regarding the existence of an overconvergent Frobenius structure on the quantum cohomology of Fano manifolds. This candidate structure is constructed from Morita's p-adic Gamma function, and its conjectural overconvergence is intrinsically linked to integrality properties of Givental's fundamental solution. In this talk, I will describe progress toward reinterpreting classical Dwork-type constructions within the framework of symplectic topology. If time permits, I will extend these considerations to the Calabi-Yau case.

Feb 27: Vivek Shende (CQM Syddansk Universitet / UC Berkeley) "Skein traces from curve counting"

Given a 3-manifold M, and a branched cover arising from the projection of a Lagrangian 3-manifold L in the cotangent bundle of M to the zero-section, we define a map from the skein of M to the skein of L, via the skein-valued counting of holomorphic curves. When M and L are products of surfaces and intervals, we show that wall crossings in the space of the branched covers obey a skein-valued lift of the Kontsevich-Soibelman wall-crossing formula. Holomorphic curves in cotangent bundles correspond to Morse flow graphs; in the case of branched double covers, this allows us to give an explicit formula for the the skein trace. After specializing to the case where M is a surface times an interval, and additionally specializing the HOMFLYPT skein to the gl(2) skein on M and the gl(1) skein on L, we recover an existing prescription of Neitzke and Yan.

Feb 27: Yu-Shen Lin (Boston University) "From tropical curves to special Lagrangians"

Given a tropical curve in R^n, a central question in tropical geometry is whether it can be lifted to a holomorphic curve in the corresponding toric variety. For instance, every tropical curve in R^2 admits such holomorphic lifting. However, such liftings may not exist in general when n>2. On the mirror side, Mikhalkin showed that any tropical curve admits a Lagrangian lifting. In this talk, we will show a refinement of the result of Mikhalkin: Every locally planar tropical curve can be lifted to a special Lagrangian in (C^*)2 via a gluing construction. Moreover, we construct a one-parameter family of special Lagrangians whose Gromov-Hausdorff limit collapses to the given tropical curve in the adiabatic limit. This is joint work with S.-K. Chiu and Y. Li.

Feb 27: Isabella Khan (MIT) "A Heegaard Floer perspective on the Z-hat invariant"

Heegaard Floer homology is a powerful and computable 3-manifold invariant which is known to be isomorphic to Némethi’s lattice homology construction for all plumbed 3-manifolds. In this talk, I will discuss progress towards using this isomorphism to express the Z-hat invariant from quantum topology in terms of Heegaard Floer generators, and the possible applications of such a result.