Rutgers Symplectic Seminar

Talks will be on Friday 12:30-1:30PM at Hill 525, located in Busch Campus of Rutgers University.

• September 26

Speaker: Yash Deshmukh (IAS)

Title: Categorical and geometric enumerative invariants

Abstract: Costello introduced categorical enumerative invariants for a smooth and proper Calabi--Yau category equipped with a splitting of the non-commutative Hodge--de Rham spectral sequence. I will discuss an extension of this to a chain-level CohFT structure on the Hochschild homology of the category. This construction will be universal in a precise sense, and I will explain how to exploit this to compare the categorical CohFT of  a Fukaya category with the geometric CohFT of the underlying symplectic manifold whenever a chain-level open-closed Gromov--Witten CohFT satisfying suitable properties exists.

• October 10

Speaker: Roman Krutowski (UCLA)

Title: Hecke algebras via Morse theory of loop spaces

Abstract: Higher-dimensional Heegaard Floer homology (HDHF) is defined by extending Lipshitz's cylindrical reformulation of Heegaard Floer homology from surfaces to arbitrary Liouville domains. The HDHF also serves as a model for Lagrangian Floer homology of symmetric products.

In this talk, I will present a Morse-theoretic model allowing for computations of the HDHF A_\infty-algebra of k cotangent fibers in the cotangent bundle of a smooth manifold. We apply this model to get an explicit computation of this A_\infty-algebra for the cotangent bundle of the 2-dimensional sphere. The result of this computation produces a differential graded algebra which may be regarded as the derived HOMFLYPT skein algebra of the sphere.

• October 17 (NO Talk)

• October 24

Speaker: Gary Guth (Stanford)

Title: Naturality of Real Heegaard Floer homology

Abstract: In recent years, there has been a flurry of interest in gauge theoretic invariants of manifolds equipped with an involution; in particular, such invariants have been used to detect exotic RP^2-knots (Miyazawa) and to settle the non-sliceness of cables of the Figure 8 knot (Kang-Park-Taniguchi). For 3-manifolds with an involution, there is a Heegaard Floer analogue of these invariants, developed in joint work with Manolescu. However, to develop the 4-dimensional aspects of the theory, it is necessary to first show the real Heegaard Floer homology groups are natural. In the talk, we will review the key ingredients in the proof, highlighting the subtleties that arise in the equivariant setting. This is joint work in progress with C. Manolescu. 

• October 31 (TBA)

• November 7 (TBA)

• November 14 (NO Talk)

• November 21

Speaker: Yao Xiao (Rutgers)

Title: Floer theory on certain multiplicity-free spaces

Abstract: We will discuss certain examples of multiplicity-free spaces and the Lagrangian Floer theory of some Lagrangian tori inside them. 

• November 28 (Thanksgiving break, NO Talk)

• December 5 

Symplectic New York Meetings at Columbia University

09:00-10:00 Bagels&Coffee

10:00-12:00 Ivan Danilenko Quiver Hecke algebras from Floer homology in Couloumb branches

12:00-02:00 Lunch

02:00-04:00 Elise LePage Aganagic's invariant is Khovanov homology

04:00-05:00 Cake&Tea

• December 12

Speaker: Jason Liu (U of Illinois)

Talks will be on Thursday 2:00-3:00PM at Hill Center, located in Busch Campus of Rutgers University.

• January 23

Speaker: Renato Vianna (São Paulo)

Title: Open-string Quantum Lefschetz formula

Abstract: Let Y be a symplectic divisor of X, omega. In the Kahler setting, Givental's Quantum Lefschetz formula relates certain Gromov-Witten invariants (encoded by the G function) of X and Y. Given an Lagrangian L in (Y, omega|Y), we can lift it to a Lagrangian L' in neighbourhood NY subset X. We will introduce the notion of the potential of a Lagrangian, which encodes information of Maslov index 2 J-holomorphic disks with boundary on it. We will discuss the conditions in which the potential for L' relates with the potential for L according to a lifting formula. In particular, this formula involves counts J-holomorphic spheres with certain tangency on Y (part of relative Gromov-Witten invariants). It generalizes a formula that can be extracted from Biran-Khanevski, under some more restrictive assumptions on Y. As applications, we recover some Lefschetz formulas appearing in the work of Coates-Corti-Galkin-Kasprczyk and show the existence of infinitely many Lagrangian tori in CP^n, Quadrics, Cubics, among other symplectic manifolds. This is joint work with Luis Diogo, Dmitry Tonkonog and Weiwei Wu. 

• January 30

Speaker: Jean Gutt (IAS and Toulouse)

Special Time and Location: 4-5PM, Hill 260

Title: Ekeland-Hofer capacities as coming from positive S^1 equivariant symplectic homology

Abstract: Associated to a star-shaped domain in R^{2n} are two increasing sequences of capacities: the Ekeland-Hofer capacities and the so-called Gutt-Hutchings capacities. I shall recall both constructions and then present the main theorem that they are the same. This is joint work with Vinicius Ramos.

• February 20

Speaker: Kenny Blakey (MIT)

Title: Bounding Lagrangian intersections using Floer homotopy theory

Abstract: I will describe a new lower bound on the number of intersection points of a Lagrangian pair, in the exact setting, using Steenrod squares on Lagrangian Floer cohomology which are defined via a Floer homotopy type.

• February 27

Speaker: Filip Zivanovic (Simons Center for Geometry and Physics)

Title: Symplectic cohomology construction via C*-actions

Abstract: In a series of papers with Alexander Ritter, we construct a symplectic cohomology for open non-convex symplectic manifolds that admit pseudo-holomorphic C*-actions. Out of this construction, we get a filtration on ordinary cohomology of these manifolds, which in practice is explicitly computable, using Morse-Bott-Floer spectral sequences. I will show concrete examples of ADE resolutions and low-dimensional Higgs bundles. If time permits, I will comment on the comparison with other known filtrations for the latter spaces.

• March 6

Speaker: Kyler Siegel (University of Southern California)

Title: Ellipsoids and scattering diagrams

Abstract: A central open problem in higher dimensional quantitative symplectic geometry is to understand when one ellipsoid can be squeezed into another by a Hamiltonian flow. In principle symplectic field theory (SFT) provides substantial tools for tackling this question, but the SFT of ellipsoids is still not well-understood. In this talk, I will discuss an ongoing project which applies ideas from log Calabi-Yau mirror symmetry in order to understand the SFT of ellipsoids in terms of an algebraic object called scattering diagrams. This in turn allows us to construct new families of curves using powerful combinatorial tools from the theory of cluster algebras. 

• April 10

Speaker: Sebastian Haney (Columbia)

Title: Infinity inner products and open Gromov-Witten invariants

Abstract: The open Gromov-Witten (OGW) potential is a function defined on the Maurer-Cartan space of a closed Lagrangian submanifold in a symplectic manifold with values in the Novikov ring. From the values of the OGW potential, one can extract so-called open Gromov-Witten invariants, which count pseudoholomorphic disks with boundary on the Lagrangian. Standard definitions of the OGW potential only allow for the construction of OGW invariants with values in the real or complex numbers. In this talk, we will present a construction of the OGW potential which gives invariants valued in any field of characteristic zero. The main algebraic input for our construction is a homotopy cyclic inner product on the (curved) Fukaya A-infinity algebra, which generalizes the notion of a cyclically symmetric inner product and is determined by a proper Calabi-Yau structure on the Fukaya category.

• April 17

Speaker: Yefeng Shen (Oregon)

Title: Quantum spectrum and Gamma structure for standard flips

Abstract: In his 1998 ICM talk, Dubrovin conjectured that a Fano variety X has semisimple quantum cohomology if and only if the derived category D(X) possesses a full exceptional collection. Using Gamma structures, this conjecture has been revisited, refined, and generalized to nonsemisimple cases in the last decade. 

In this talk, we investigate the quantum cohomolohy for projective bundles, blow-ups, and standard flips and its connection to derived categories. By restricting the quantum cohomology to a fiber curve direction, we obtain a decomposition of QH(X) into asymptotic Gamma classes, indexed by the spectrum of quantum multiplication of the first Chern class of X. Using a sequence of reductions and asymptotic expansions of Meijer G-functions, we show this decomposition is compatible with the semi-orthogonal decompositions of projective bundles, blow-ups, and standard flips studied by Orlov and Belmans-Fu-Raedschelders. 

The talk is based on work joint with Mark Shoemaker. If time permits, I will also talk about our recent progress on weighted flips.

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

• February 1

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.

• February 12 (Monday): 

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.

• February 22 

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.

• February 29

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.

• March 7

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.

• March 21

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.

• April 4

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.

• April 11

    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. 

• April 12

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.

• April 18

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. 

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

• October 12

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.  

• October 26

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.

• November 2

No talk (SCGP workshop)

• November 9

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. 

• November 16

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.  

• November 30

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.

• December 7

Speaker: Soham Chanda (Rutgers)_

Title: TBA

Abstract: TBA

Time : Fridays 11am EST

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

• Sep 23

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. 

• Sep 30

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.