Rutgers Symplectic Seminar

Spring 2024

The talks will be on Thursday 1:30-2:30 at 705 Hill Center. 

Speaker: Soham Chanda (Rutgers)

Title: Augmentation Varieties and Disk Potential

Abstract: Dimitroglou-Rizell-Golovko constructs a family of Legendrians in prequantization bundles by taking lifts of monotone Lagrangians. These lifted Legendrians have a Morse-Bott family of Reeb chords. We construct a version of Legendrian Contact Homology(LCH) for Rizell-Golovko's lifted Legendrians by counting treed disks. Our formalism of LCH allows us to obtain augmentations from certain non-exact fillings. We prove a conjecture of Rizell-Golovko relating the augmentation variety assoiciated to the LCH of a lifted Legendrian and the disk potential of the base Lagrangian. As an application, we show that lifts of monotone Lagrangian tori in projective spaces with different disk-potentials, e.g. as constructed by Vianna, produce non-isotopic Legendrian tori in contact spheres. The above work is a joint project with Blakey, Sun and Woodward.

Speaker: Amanda Hirschi

Title: A localisation formula for symplectic GW theory

Abstract: I will briefly describe the construction of a global Kuranishi chart for moduli spaces of stable pseudoholomorphic maps and how the construction can be made equivariant whenever the target symplectic manifold carries a Hamiltonian action. Subsequently I will explain how this leads to a combinatorial formula for the equivariant Gromov-Witten invariants of a Hamiltonain torus manifold satisfying some mild assumption. This is partially joint work with Mohan Swaminathan.

Speaker: Maxim Jeffs (Simons Center for Geometry and Physics)

Title: Fixed point Floer cohomology and the enumerative geometry of nodal curves
 Abstract: I'll explain how, for singular hypersurfaces, a version of their genus-zero Gromov–Witten theory may be described in terms of a direct limit of fixed point Floer cohomology groups. This construction is easy to define and very amenable to computation. As an illustration, I'll talk about recent joint work with Yuan Yao and Ziwen Zhao, where we calculate the full ring structure on this direct limit in the case of Dehn twists on curves, giving a direct proof of closed-string mirror symmetry for nodal curves. These calculations involve contributions from counts of many non-trivial holomorphic curves; since curves of genus $g \geq 1$ contain no non-constant rational curves, these types of enumerative invariants are impossible to obtain using classical curve-counting methods. To finish, I'll discuss our ongoing project studying enumerative geometry and mirror symmetry for singular varieties using fixed point Floer cohomology.

Speaker: Basak Gurel (University of Central Florida)

Title: Topological entropy, barcodes and Floer theory

Abstract: Topological entropy is one of the fundamental invariants of a dynamical system, measuring the orbit complexity. In this talk, we discuss a connection between the topological entropy of compactly supported Hamiltonian diffeomorphisms and Floer theory. Namely, we introduce a new invariant associated with the Floer complexes of the iterates of such a diffeomorphism, which we call barcode entropy. We show that barcode entropy is closely related to topological entropy and that these invariants are equal in dimension two. We also touch upon applications of the methods to some other questions in symplectic topology. The talk is based on joint work with Erman Cineli and Viktor Ginzburg.

Speaker: Daniel Pomerleano (UMass Boston)

Title: The quantum connection on a monotone symplectic manifold

Abstract: The small quantum connection on a monotone symplectic manifold M is one of the simplest objects in enumerative geometry. Nevertheless, the poles of the connection have a very rich structure.  After reviewing this background, I will outline a proof that, under suitable assumptions,  the quantum connection of M is of unramified exponential type.  This is joint work (partially in progress) with Paul Seidel.

Speaker: Erkao Bao (University of Minnesota)

Title: Equivariant, obstructed Morse homology

Abstract:  In this talk, we define equivariant Morse homology for closed manifolds equipped with a finite group action. A significant challenge arises from the fact that the moduli spaces of trajectories of the negative gradient are usually not transversely cut out, a manifestation of the conflict between symmetry and genericity. We present two distinct approaches to tackle this issue:

Approach 1: stabilizing the Morse function;

Approach 2: obstruction bundle gluing in the spirit of semi-global Kuranishi structure.

The talk is based on joint works with Tyler Lawson and Ke Zhu.


Speaker: Guillem Cazassus (Oxford)

Title : Hamiltonian actions on Fukaya categories

        Abstract:  We will talk about algebraic structures arising in Lagrangian Floer homology in the presence of a Hamiltonian action of a compact Lie group. First, we will show how the Lagrangian Floer complex can be equipped with an A-infinity module structure over the Morse complex of the group, and how this action permits to define equivariant versions of Floer homology. We will then explain how this structure interacts with the structure of the Fukaya category: both can be packaged into (our version of) an A-infinity bialgebra action, giving an alternative answer to a conjecture of Teleman. This should enable one to build an extended topological field theory corresponding to Donaldson-Floer theory.

This is based on two joint work in progress, one with Paul Kirk, Mike Miller-Eismeier and Wai-Kit Yeung, and another with Alex Hock and Thibaut Mazuir.

    Speaker: Semon Rezchikov (IAS and Princeton)

   Title: Equivariant Floer Homology without Transversality

    Abstract: The Cohen-Jones Segal construction assigns, to a smooth flow category, a stable homotopy type, or spectrum; in the case that the smooth flow category comes from a Morse- or Floer- theoretic construction, the homology of this spectrum will agree with the corresponding Morse- or Floer- homology group. However, even when bubbling does not occur, one can only hope to extract a virtually smooth flow category from Floer theory when one can achieve transversality. This makes it very challenging to extend the Cohen-Jones-Segal construction to the equivariant setting, where there are obstructions to achieving transversality using any small perturbation. In this talk I will explain a generalization of the Cohen-Jones-Segal construction which takes as input a virtually smooth flow category, in which the moduli spaces of morphisms are equipped with Kuranishi charts, rather than being smooth manifolds with corners. This construction assigns a spectrum to such a category, and immediately generalizes to the equivariant setting, producing a genuine G spectrum in that case. This G spectrum is generally not free, but can be reasoned about geometrically. I will discuss technical challenges that must be overcome in this construction, as well as potential applications. 


Location: Hill 423 

Date & time:  Friday, 12 April 2024 at 2pm - 3pm. 

Speaker: Zuyi Zhang (Indiana)

Title: Immersed Lagrangian Floer theory, quilts, and bouding cochains

     Abstract: Let F, F1, F2 be closed surfaces and F ↬ F1 × F2, L1 ↬ F1, L2 ↬ F2 be Lagrangian immersions. In this talk, I will first discuss the singularities of the Lagrangian immersion F ↬ F1 × F2. Following this, we will explore how the Lagrangian composition of F ↬ F1 × F2 relates the boundary maps of CF(L1, F ◦L2; F1), CF(L1◦F, L2; F2), and the quilted Lagrangian Floer chain group CFQ(L1, F, L2; F1, F2). Finally, examples will be given to discuss how to construct the bounding cochains corresponding to figure-eight bubblings.

Speaker: Spencer Cattalani (Stony Brook)

Title: 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. 


Fall 2023

The talks will be on Thursday 1:30-2:30 at 705 Hill Center. 


Speaker: Riccardo Pedrotti (UT Austin)

Title: Towards a count of holomorphic sections of Lefschetz fibrations over the disc 

Abstract: Given a positive factorisation of the identity in the mapping class group of a surface S, we can associate to it a Lefschetz fibration over the sphere with S as a regular fiber. Its total space X is a symplectic 4-manifold, so it is a natural question to ask what kind of invariants of X can be read off from this construction: the word in Mod(S) leads to an easy computations of the homology of X and I. Smith pushed this further by providing us a formula for the signature of the total space in terms of this combinatorial construction. Using this as a motivation, I will report on an ongoing joint work with Tim Perutz, aimed at obtaining an explicit formula for counting holomorphic sections of a Lefschetz fibration over the disk, while keeping track of their relative homology classes. By taking the monodromy of the fibration to be isotopic to the identity, we should get a count of sections for a Lefschetz fibration over the sphere, and in particular an invariant of its total space X. Thanks to Taubes’ SW=GW, these invariants should be closely related to the Seiberg Witten invariants of X.  

Speaker: Marco Castronovo (Columbia)

Title: Dubronvin dynamics

Abstract: The cohomology ring of a compact symplectic manifold can be deformed by counting pseudo-holomorphic spheres, and this categorifies by considering disks. I will raise a question, make a proposal, do a calculation, and prove a theorem. Question: why do we focus on certain points in the space of deformations? Proposal: study cohomological/categorical properties that hold for generic deformation parameters. Calculation: the eigenvalues of Dubrovin’s operator are distinct for all parameters of P^2, and coalesce for non-generic parameters of Gr(2,4). Theorem: the Dubrovin spectrum contains the curvatures of nontrivial objects in the Fukaya category.

No talk (SCGP workshop)


Speaker: Han Lou (University of Georgia)

Title: On the Hofer Zehnder Conjecture for Semipositive Symplectic Manifolds

Abstract: Arnold conjecture says that the number of 1-periodic orbits of a Hamiltonian diffeomorphism is greater than or equal to the dimension of the Hamiltonian Floer homology. In 1994, Hofer and Zehnder conjectured that there are infinitely many periodic orbits if the equality doesn't hold. In this talk, I will show that the Hofer-Zehnder conjecture is true for semipositive symplectic manifolds with semisimple quantum homology. This is a joint work with Marcelo Atallah. 


Speaker: Yao Xiao (Stony Brook)

Title: Equivariant Lagrangian Floer theory on compact toric manifolds

Abstract: We define an equivariant Lagrangian Floer theory on compact symplectic toric manifolds for the subtorus actions. We prove that the set of Lagrangian torus fibers (with weak bounding cochain data) with non-vanishing equivariant Lagrangian Floer cohomology forms a rigid analytic space. We can apply tropical geometry to locate such Lagrangian torus fibers in the moment map.  These Lagrangian submanifolds are nondisplaceable by equivariant Hamiltonian diffeomorphisms.  

Speaker: Jae Hee Lee (MIT)

Title: Quantum Steenrod operations of symplectic resolutions

Abstract: I will consider the quantum connection of symplectic resolutions, which is of interest in representation theory and more recently in symplectic topology. I will explain the relationship of the quantum connection in positive characteristic with the quantum Steenrod power operations of Fukaya and Wilkins. The relationship provides a geometric understanding of the p-curvature of such connections, while also allowing new computations for quantum Steenrod operations, including the case of the Springer resolution.

Speaker: Soham Chanda (Rutgers)_

Title: TBA

Abstract: TBA


Fall 2022

Time : Fridays 11am EST

Location : Zoom ( Contact organizers for link to zoom meeting)


Speaker : Cheuk Yu Mak

Title  :  Lagrangian links quasimorphisms and non-simplicity of Hameomorphism group

Abstract : We will explain the construction of a sequence of homogeneous quasimorphisms of the area preserving homeomorphism group of the disc using Lagrangian Floer theory for links. This sequence of quasimorphisms has asymptotically vanishing defect, so it is asymptotically a homomorphism. We will then explain how studying the subleading asymptotic of these quasimorphisms enable us to show that the Hameomorphism group is not the smallest normal subgroup of the area preserving homeomorphism group. This is a joint work with Daniel Cristofaro-Gardiner, Vincent Humilière, Sobhan Seyfaddini and Ivan Smith. 


Speaker :  Amanda Hirschi

Title: A construction of global Kuranishi charts for Gromov-Witten moduli spaces of arbitrary genus 

Abstract: Symplectic Gromov-Witten invariants have long been complicated by the fact that delicate local-to- global arguments were required in their construction. In 2021 Abouzaid-McLean-Smith gave the first construction of global charts for general Gromov-Witten moduli spaces in genus zero. I will describe a generalisation of their construction for stable maps of higher genus and discuss possible applications. This is joint work in progress with Mohan Swaminathan.