Columbia Symplectic Geometry and Gauge Theory Seminar

Jan 26: Alex Xu (Columbia) "The Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds with Asymptotically Hyperbolic Ends"

Abstract: One of the features of the Seiberg-Witten equations is that existence of irreducible solutions give rise to apriori estimates for the total scalar curvature. This was used by LeBrun in the late 90s when he constructed the first examples of closed 4-manifolds that satisfy the strict Hitchin-Thorpe inequality yet do not admit any Einstein metrics. In this talk, I'll describe a method of constructing irreducible solutions to the Seiberg-Witten equations on certain finite volume 4-manifolds with asymptotically hyperbolic $T^3$ ends. These 4-manifolds arise as complements of smoothly embedded tori with zero self-intersection in an ambient closed 4-manifold. As an application, this allows us to construct infinitely many examples of finite volume 4-manifolds with $T^3$ ends that do not admit any asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. By extending the construction to the Pin^-(2) monopoles due to Nakamura, we are able to construct examples with signature 0.

Feb 02: Chris Woodward (Rutgers) "Tropical disk counting in almost toric manifolds"

Abstract: In joint work with Sushmita Venugopalan (Chennai), we generalize Mikhalkin's tropical curve counting formula to the case of disks bounding tropical Lagrangians in almost toric manifolds, such as moment fibers or tropical Lagrangians in the ADE Lagrangian configurations in del Pezzo surfaces. This allows, for example, the direct computation of disk potentials of these Lagrangian tori and spheres and allows one to identify split-generators for the Fukaya category of del Pezzo's. (For tori in del Pezzo surfaces, similar results appear after an algebro-geometric detour in Sam Bardwell-Evans, Man-Wai Mandy Cheung, Hansol Hong, Yu-Shen Lin. For cubic surfaces, split generators appear in Sheridan's thesis. A formula of this type was widely expected, as in, for example, Vianna's thesis.)

Feb 09: Yi Wang (Purdue) "A simple chain model of loop spaces and application to the study of Lagrangian submanifolds"

Abstract: It was a proposal of Fukaya that one can use chain level string topology to study a Lagrangian submanifold L, as pseudoholomorphic disks induce loops in L and bubbling of disks is governed by string topology operations. A rigorous construction of chain level loop bracket was worked out by Irie, whose applications include a proof of Audin's conjecture. In this talk, I will describe a simplification of Irie's chain model of loop spaces, which greatly simplifies some technicalities that arise in related work when applying the chain model to symplectic topology, and which is also interesting from a purely algebraic topological point of view. Then I will discuss a variant of the chain model in the S^1-equivariant context, which allows one to lift the Fukaya A_infinity of L to a Maurer-Cartan element in the dg Lie algebra of cyclic invariant chains on the free loop space of L. Applications include the nonexistence of Lagrangian submanifolds in C^{2n} that are homotopy equivalent to a manifold with negative sectional curvature (this slightly generalizes a theorem attributed to Viterbo, under additional assumptions). Part of the work is joint with Irie and Rivera.

Feb 16: Spencer Cattalani (Stony Brook) "Complex Cycles and Symplectic Topology"

Abstract: Among all almost complex manifolds, those which are tamed by symplectic forms are particularly well studied. What geometric properties characterize this class of manifolds? That is, given an almost complex manifold, how can one tell whether it is tamed by a symplectic form? By a 1976 result of D. Sullivan, this question can be answered by studying complex cycles. I will explain what complex cycles are and their role in two recent results, which confirm speculations posed by M. Gromov in 2000 and 1985, respectively. The first is that an almost complex manifold admits a taming symplectic structure if and only if it satisfies a certain bound on the areas of coarsely holomorphic curves. The second is that an almost complex 4-manifold which has many pseudoholomorphic curves admits a taming symplectic structure. This leads to an almost complex analogue of D. McDuff’s classification of rational symplectic 4-manifolds.

Feb 23: Deeparaj Bhat (MIT) "Surgery Exact Triangles in Instanton Theory"

We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil. In particular, we show the Poincaré homology sphere is not an instanton L-space (with Z/2 coefficients), in contrast with Heegaard Floer and monopole Floer theories. Finally, we sketch the proof of the triangle inspired by the Atiyah-Floer conjecture and results from symplectic geometry.

Mar 1: Joseph Breen (Iowa) "The Giroux correspondence in arbitrary dimensions"

The Giroux correspondence between contact structures and open book decompositions is a cornerstone of 3-dimensional contact topology. While a partial correspondence was previously known in higher dimensions, the underlying technology available at the time was completely different from that of the 3-dimensional theory. In this talk, I will discuss recent joint work with Ko Honda and Yang Huang on extending the statement and technology of the 3-dimensional correspondence to all dimensions.

Mar 8: Daniel Pomerleano (UMass Boston) "The Quantum GIT conjecture"

Let X be a Fano variety with G action. The quantum GIT conjecture predicts a formula for the quantum cohomology of "anti-canonical" GIT quotients X//G in terms of the equivariant quantum cohomology of X. The formula is motivated by ideas from 3- dimensional gauge theory ("Coulomb branches") and provides a vast generalization of Batyrev's formula for the quantum cohomology of a toric Fano variety. I will explain the conjecture and then describe joint work in progress with C. Teleman towards proving it. The strategy of proof involves ideas from Varolgunes' theory of relative symplectic cohomology.

Mar 22: Oliver Edtmair (UC Berkeley) "Systoles of convex energy hypersurfaces"

Hofer-Wysocki-Zehnder proved that every strictly convex energy hypersurface in R^4 possesses a disk-like global surface of section. They asked whether a systole, i.e. a periodic orbit of least action, must span such a disk-like global surface of section. In my talk, I will give an affirmative answer to this question, based on joint work in progress with Abbondandolo and Kang. I will explain how this result can be used to obtain a sharp symplectic embedding result for convex domains in R^4. Moreover, I will explain how this relates to the strong Viberbo conjecture on the equivalence of normalized symplectic capacities.

Mar 27: Yoel Groman (Hebrew U) "Relative symplectic cohomology and quantitative deformation theory"

Consider a Liouville domain D embedded in a closed symplectic manifold M. To D one can associate two types of Floer theoretic invariants: intrinsic ones like the wrapped Fukaya category which depend on D only, and relative ones which involve both D and M. It is often the case that the intrinsic invariant is amenable to computation. On the other hand, the relative invariants are important, at least in SYZ mirror symmetry, as one can reconstruct the global Floer theory by a local to global principle. Thus it is a fundamental question if the relative invariants can be understood as a deformation of the intrinsic invariant. It turns out the question needs to be approached quantitatively. By shrinking the Liouville domain, the answer is often positive. This circle of ideas is at the heart of a program joint with Mohammed Abouzaid and Umut Varolgunes for a general approach to homological mirror symmetry. I will discuss a work in progress on the application of this circle of ideas to the reconstruction problem in mirror symmetry via relative symplectic cohomology.

Mar 29: Thomas Guidoni (Sorbonne) "Integrable hierarchies for Real Gromov-Witten invariants"

Witten conjecture (Kontsevich theorem) relates the Gromov-Witten invariants of a point to the KdV hierarchy. It is now understood as a particular case of a general phenomenon: the generating series of Gromov-Witten invariants of a target with semi-simple quantum cohomology always gives rise to a solution of an integrable hierarchy of equations.
In this talk, I will present several integrability properties of the Real Gromov-Witten invariants of P^1. For instance, the restriction of their generating series to the stationnary variables is a solution of the KdV hierarchy. These results are obtained using a localization formula and computations in the semi-infinite wedge space.
If time permits, I will describe a reconstruction formula for the genus 0 Real Gromov-Witten invariants. In the case of P^1, it provides an extension of the integrability result to the non-stationnary variables in genus 0.

Apr 05: Roman Krutowski (UCLA) "Heegaard Floer symplectic cohomology and generalized Viterbo's isomorphism theorem"

In recent years several groups of authors introduced various invariants that are based on Lagrangian Floer homology of a symmetric product of a symplectic manifold. In this talk, I will introduce Heegaard Floer symplectic cohomology (HFSH), an invariant of a Liouville domain M which mimics symplectic cohomology of the k-th symmetric product of M. This invariant can also be regarded as a deformation of the k-th symmetric version of symplectic cohomology, obtained by counting curves of higher genus. I will also introduce a multiloop Morse complex and show that for cotangent bundles this complex computes HFSH. This is a joint work with Tianyu Yuan.

Apr 12: Eric Zaslow (Northwestern) "Legendrians and Quantum Groups"

I will describe work in progress relating the two subjects, joint with Roger Casals, Honghao Gao, Lenhard Ng, Linhui Shen and Daping Weng.

Apr 19: Mohan Swaminatan (Stanford) " Constructing smoothings of stable maps"

For positive n, g and d, the moduli space of degree d holomorphic maps from non-singular genus g curves to CP^n is irreducible and smooth of the expected dimension (provided d > 2g-2). However, not every point in the stable map compactification lies in the closure of this smooth open subset, i.e., not every stable map is smoothable. I will talk about a sufficient condition for smoothability (valid in all genera), based on joint work with Fatemeh Rezaee.

Apr 26: Jiakai Li (Harvard) "(Real) monopole Floer homology for webs and foams"

Webs are embedded trivalent graphs in the 3-sphere and foams are singular cobordisms between webs. In this talk, I will present a construction of monopole Floer homology for webs that is functorial under foam cobordisms, based on Kronheimer and Mrowka’s monopole Floer homology. The ingredients include real and orbifold Seiberg-Witten theory, which are necessary for dealing with Klein-four symmetries. The talk will focus on the setup of the theory and compare it with the instanton counterpart J^♯.

May 3: Hang Yuan (Northwestern) "Unobstructedness in Lagrangian Floer theory and non-archimedean geometry"

This talk explores the problem of disk obstructions in Lagrangian Floer theory. Before, the concept of bounding cochains is introduced by Fukaya-Oh-Ohta-Ono and Kontsevich to address this problem. However, the uncertain existence of bounding cochains complicates both calculations and practical applications. We aim to propose a new notion of unobstructedness that not only builds on traditional bounding cochains but also incorporates certain aspects of non-archimedean geometry. This new framework of unobstructedness is shown to be a connected condition within the space of embedded Lagrangian submanifolds. For example, by a result from Rizell-Goodman-Ivrii, we can show that any Lagrangian torus in R^4, CP^2, or S^2 x S^2 always admits a bounding cochain.

May 10: Andrey Smirnov (University of Carolina at Chapel Hill) "Towards arithmetic mirror symmetry"

There exists a well-known analogy between integral representations of special functions appearing in mirror symmetry and finite sums in number theory. This connection is explained in terms of Frobenius automorphism for quantum differential equations over p-adic fields. In my talk I will review this construction and explain its generalization to p-adic q-difference equations.