Columbia Symplectic Geometry and Gauge Theory Seminar

Previously the Columbia Symplectic Geometry, Gauge Theory, and Categorification (SGGTC) seminar.

From Spring 2024, the new SGGT seminar page is found here.

SGGTC seminar schedule (Fall 2023)

Organizers: Johan Asplund, Shaoyun Bai, Marco Castronovo, Yin Li, Francesco Lin and Chiu-Chu Melissa Liu.

Date Speaker Title and abstract
Sep 1
Zhengyu Zong (Tsinghua) Torus knots in lens spaces, open Gromov–Witten invariants, and topological recursion
Show/hide abstract In 1980s and 1990s, the work of Witten, Gopakumar–Vafa, and Ooguri–Vafa suggests that there is a deep relation between the HOMFLY polynomials of knots and A-model topological strings. In the case of knots in S^3, the colored HOMFLY polynomials of knots are conjectured to be related to the open Gromov–Witten theory of the resolved conifold with Lagrangian boundary condition constructed under the conifold transition. Sometimes this relation is called the large N duality. In the case of torus knots in S^3, one obtains a richer picture by introducing mirror symmetry in the above story. By combining the conjecture of Brini–Eynard–Marino, one obtains a conjecture stating that the following three invariants are equivalent:
(1) the open-closed Gromov–Witten invariants of the resolved conifold;
(2) the Eynard–Orantin invariants of the mirror curve;
(3) the colored HOMFLY polynomial of the knot.
This conjecture is proved by the work of Diaconescu–Shende–Vafa, Borot–Eynard–Orantin, and Fang–Zong. The above equivalence is conjecturally generalized to torus knots in spherical Seifert 3-manifolds in the work of Borot–Brini. The aim of this talk is to prove the equivalence of item (1) and item (2) in the case of torus knots in lens spaces. This is a joint work with Jinghao Yu.
Sep 8 Maxim Jeffs (Simons Center) Fixed point Floer cohomology and the enumerative geometry of nodal curves
Show/hide 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.
Sep 15 Filip Živanović (Simons Center) Symplectic ℂ*-manifolds
Show/hide abstract I will introduce a broad family of open symplectic manifolds admitting pseudoholomorphic ℂ*-actions, which contains many interesting spaces, such are equivariant resolutions of affine singularities, twisted cotangent bundles, semiprojective toric varieties and Higgs moduli. I will explain how one can construct symplectic cohomology of these spaces and as a consequence obtain a filtration on their cohomology rings, which should be thought of as a Floer-theoretic analogue of Atiyah–Bott filtration. This is joint work with Alexander Ritter.
Sep 22 Riccardo Pedrotti (UT Austin) Towards a count of holomorphic sections of Lefschetz fibrations over the disc
Show/hide 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 can 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.
Sep 29 Xujia Chen (Harvard) Kontsevich’s invariants as topological invariants of configuration space bundles
Show/hide abstract Kontsevich's invariants (also called “configuration space integrals”) are invariants of certain framed smooth manifolds/fiber bundles. The result of Watanabe(’18) showed that Kontsevich’s invariants can distinguish smooth fiber bundles that are isomorphic as topological fiber bundles. I will first give an introduction to Kontsevich's invariants, and then state my work which provides a perspective on how to understand their ability of detecting exotic smooth structures: real blow up operations essentially depends on the smooth structure, and thus given a space/bundle X, the topological invariants of some spaces/bundles obtained by doing some real blow-ups on X can be different for different smooth structures on X.
Oct 6 Yao Xiao (Stony Brook) Equivariant Lagrangian Floer theory on compact toric manifolds
Show/hide abstract We define an equivariant Lagrangian Floer theory on compact symplectic toric manifolds for the subtorus actions induced from the defining torus action of the toric manifolds. We prove that the set of pairs (L,b), each consisting of a Lagrangian torus fiber and a weak bounding cochain, which have non-vanishing equivariant Lagrangian Floer cohomology forms a rigid analytic space (over the non-Archimedean Novikov field). The Lagrangian submanifolds which have nonvanishing Lagrangian Floer cohomology are nondisplaceable by T^r-equivariant Hamiltonian diffeomorphisms.
Oct 13 Marcelo Atallah (Montréal) Fixed points of small Hamiltonian diffeomorphisms and the flux conjectures
Show/hide abstract The C⁰ flux conjecture predicts that a symplectic diffeomorphism that can be C⁰ approximated by a Hamiltonian diffeomorphism is itself Hamiltonian. We describe how the flux conjecture relates to new instances of the strong Arnol’d conjecture and make progress towards the C⁰ flux conjecture. This is joint work in progress with Egor Shelukhin.
Oct 20 Ilaria Di Dedda (KCL) Type A symplectic Auslander correspondence
Show/hide abstract Fukaya–Seidel categories constitute a powerful and geometric derived invariant of singularities. In this talk, I will motivate and describe the study of certain isolated singularities, whose Fukaya–Seidel categories play an important role in bordered Heegaard Floer theory. Motivated by representation theory, I will relate these singularities to abstract objects associated to algebras of Dynkin type A. I will introduce “type A symplectic Auslander correspondence”, a purely geometrical construction which realises a notable result in the study of representations of these algebras.
Oct 27 Hokuto Konno (University of Tokyo) Families Frøyshov invariant
Show/hide abstract The Frøyshov invariant of a rational homology 3-sphere is an important numerical invariant out of Floer theory, which allows us to generalize Donaldson's diagonalization theorem to 4-manifolds with boundary. We introduce family versions of the Frøyshov invariant and its cousins (called Manolescu's alpha, beta, gamma invariants). Namely, given a smooth family of rational homology 3-spheres, we define numerical invariants out of families (Seiberg–Witten) Floer homotopy types, which give constrains on smooth families of 4-manifolds bounded by the family of 3-manifolds. This is joint work with Hirofumi Sasahira.
Nov 3 Gabriele Benedetti (VU Amsterdam) Rigidity and flexibility of periodic Hamiltonian flows
Show/hide abstract An old problem in classical mechanics is the existence of periodic flows within specific classes of Hamiltonian systems such as geodesic and magnetic flows, and central forces. In recent years, interest in this problem has been revitalized since a series of papers has unveiled a deep relationship between periodic Hamiltonian flows and systolic questions in symplectic and contact geometry. While only trivial examples of periodic flows among magnetic and central systems exist, Zoll and, later, Guillemin have shown that there are many exotic examples among geodesic flows on the two-sphere. Following Guillemin's approach, the goal of this talk is to show how the Nash–Moser implicit function theorem can be applied to construct magnetic flows on the two-torus which are periodic for a single value of the energy. This is joint work with Luca Asselle and Massimiliano Berti.
Nov 10 Dominique Rathel-Fournier (Montréal) Unobstructed Lagrangian cobordism groups of surfaces
Show/hide abstract Lagrangian cobordism is a relation between Lagrangian submanifolds of a symplectic manifold M. Biran and Cornea showed that Lagrangian cobordisms satisfying suitable geometric constraints give rise to algebraic decompositions in the derived Fukaya category DFuk(M). They asked whether any such decomposition in DFuk(M) comes from a Lagrangian cobordism in this way. The goal of this talk is to answer this question in the case where M is a symplectic surface of higher genus. We introduce a cobordism group whose relations are given by unobstructed immersed Lagrangian cobordisms, and explain how to extend Biran and Cornea's framework to this case. We will then sketch the proof that this unobstructed cobordism group is isomorphic to the Grothendieck group of DFuk(M), which gives a positive answer to Biran and Cornea's question at the level of K-theory. The proof builds upon work of Perrier, as well as the computation by Abouzaid of the Grothendieck group of DFuk(M).
Nov 17 Jae Hee Lee (MIT) Quantum Steenrod operations of symplectic resolutions
Show/hide 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 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.
Nov 24 No seminar (Thanksgiving)
Dec 1 Zihong Chen (MIT) Cyclic open closed maps and quantum Steenrod operations
Show/hide abstract Fix a closed monotone symplectic manifold X. The cyclic open-closed map is a map from the cyclic homology of the Fukaya category to the S¹-equivariant quantum cohomology of X. It is a key ingredient in the study of noncommutative Hodge theory on the A-side, and has various applications to enumerative mirror symmetry (cf. Ganatra–Perutz–Sheridan). Using this as a starting point, I will discuss my work in progress on categorifying characteristic p enumerative invariants. Via the cyclic open-closed map, we give a Fukaya-categorical interpretation of quantum Steenrod operations due to Fukaya and Wilkins. I will also explain how this approach allows one to infer information about mod p enumerative invariants of X from certain structures and computations of the quantum connection in characteristic 0 (which are much better understood). If time permits, I will discuss an example computation (intersection of quadrics in ℙ⁵) or potential applications to arithmetic mirror symmetry.
Dec 8 Kevin Sackel (UMass Amherst) Products of locally conformal symplectic manifolds
Show/hide abstract Locally conformal symplectic (LCS) geometry is a variant of symplectic geometry in which the symplectic form is locally only defined up to positive scale. For example, for the symplectization R x Y of a contact manifold Y, translation in the R direction are symplectomorphisms up to scale, and hence the quotient (R/Z) x Y is naturally an LCS manifold. The importation of symplectic techniques into LCS geometry is somewhat subtle because of this ambiguity of scale. In this talk, we define a notion of product for LCS manifolds, in which the underlying manifold of an LCS product is not simply the smooth product of the underlying manifolds, but which nonetheless appears to fill the same role in LCS geometry as the standard symplectic product does in standard symplectic geometry. As a proof of concept, with input from an LCS result of Chantraine and Murphy, we use the LCS product to prove that C⁰ small Hamiltonian isotopies have a lower bound on the number of fixed points given by the rank Morse–Novikov homology. This is a natural generalization of the classical symplectic proof of the analogous result by Laudenbach and Sikorav which uses the graph of a Hamiltonian diffeomorphism in the product manifold. These results are joint work in progress with Baptiste Chantraine.

SGGTC seminar schedule (Spring 2023)

Organizers: Mohammed Abouzaid, Johan Asplund, Marco Castronovo, Yin Li, Francesco Lin and Mike Miller Eismeier.

Date Speaker Title and abstract
Jan 20
Ian Montague (Brandeis) Seiberg–Witten Floer K-Theory and Cyclic Group Actions
Show/hide abstract Given a spin rational homology sphere equipped with a cyclic group action preserving the spin structure, I will introduce equivariant refinements of Manolescu's kappa invariant, derived from the equivariant K-theory of the Seiberg–Witten Floer spectrum. These invariants give rise to equivariant relative 10/8-ths type inequalities for equivariant spin cobordisms between rational homology spheres. I will explain how these inequalities provide applications to knot concordance, obstruct cyclic group actions on spin fillings, and give genus bounds for knots in punctured 4-manifolds. If time permits I will explain how these invariants are related to equivariant eta-invariants of the Dirac operator, and describe work-in-progress which provides explicit formulas for the S1-equivariant eta-invariants on Seifert-fibered spaces.
Jan 31
Umut Varolgunes (Boğaziçi University) Non-archimedean analytic mirrors of symplectic cluster manifolds (part 2)
Show/hide abstract Relative symplectic cohomology defines a sheaf in the base of a Lagrangian torus fibration with singularities (under some conditions on grading datum). I will explain the computation of this sheaf in some examples. I will then use some of these computations to construct non-archimedean analytic mirrors and explain the concrete open and closed string consequences. The focus will be on explaining the general framework and will be conjectural at times. Joint work with Mohammed Abouzaid and Yoel Groman.
Feb 3 Masaki Taniguchi (iTHEMS/RIKEN) Concordance Invariants from Equivariant Singular Instanton Theory
Show/hide abstract We introduce a new framework to obtain concordance invariants from equivariant singular instanton theory. As a special case, our invariant recovers Kronheimer–Mrowka's s^# invariant. Moreover, our new description of s^# enables us to show the quasi-additivity of s^#, answering a question of Gong. As a topological application, we produce a wide class of patterns whose induced satellite maps on the concordance group generate infinite rank, giving a partial answer to a conjecture of Hedden and Pinzón-Caicedo. This is joint work with Aliakbar Daemi, Hayato Imori, Kouki Sato and Christopher Scaduto.
Feb 10 Joshua Wang (Harvard) Colored sl(N) homology and SU(N) representations
Show/hide abstract The Khovanov homology of a rational knot or link happens to coincide with the cohomology of its space of SU(2) representations that send meridians to traceless matrices. This coincidence is closely related to the spectral sequence from Khovanov homology to an SU(2) instanton homology defined by Kronheimer and Mrowka. Motivated by a conjectural spectral sequence from colored sl(N) homology to a hypothetical colored SU(N) instanton homology, I'll explain how the colored sl(N) homology of the trefoil agrees with the cohomology of its space of SU(N) representations that send meridians to a conjugacy class associated to the color. This gives the first computation of colored sl(N) homology of a nontrivial knot. I'll also discuss work in progress with Michael Willis on the colored sl(N) homology of (2,m) torus knots and links.
Feb 17 Benjamin Gammage (Harvard) 2-categorical 3d mirror symmetry
Show/hide abstract "3d mirror symmetry" encompasses a range of statements relating symplectic and algebraic invariants of a dual pair of hyperkähler manifolds. In the spirit of the homological mirror symmetry program, we propose that the best statement of (topologically twisted) 3d mirror symmetry is an equivalence between 2-categories of boundary conditions for a pair of 3-dimensional topological field theories: namely, Rozansky–Witten theory and a still under development "Fukaya–Fueter theory". By modeling the Fukaya–Fueter theory by perverse schobers, we establish this "homological 3d mirror symmetry" in the abelian case. This is based on joint work with Justin Hilburn and Aaron Mazel-Gee.
Feb 24 Nicki Magill (Cornell) Symplectic embeddings of Hirzebruch surfaces
Show/hide abstract The four dimensional ellipsoid embedding function of a toric symplectic manifold M measures when a symplectic ellipsoid embeds into M. It generalizes the Gromov width and ball packing numbers. This function can have a property called an infinite staircase, which implies infinitely many obstructions are relevant in determining whether embeddings exist. Based on various work with McDuff, Pires, and Weiler, we will discuss the classification of which Hirzebruch surfaces have infinite staircases. The argument relies on a correspondence between constructing embeddings via almost toric fibrations and finding obstructions via exceptional spheres. The talk will focus on explaining this correspondence.
Mar 3 Angela Wu (LSU) On Lagrangian quasi-cobordisms
Show/hide abstract A Lagrangian cobordism between Legendrian knots is an important notion in symplectic geometry. Many questions, including basic structural questions about these surfaces are yet unanswered. For instance, while it is known that these cobordisms form a preorder, and that they are not symmetric, it is not known if they form a partial order on Legendrian knots. The idea of a Lagrangian quasi-cobordism was first defined by Sabloff, Vela-Vick, and Wong. Roughly, for Legendrians \Lambda_1 and \Lambda_2, it is the smooth composition of a sequence of alternatingly ascending and descending Lagrangian cobordisms which start at \Lambda_1 and end at \Lambda_2. This forms a metric monoid on Legendrian knots, with notion of the minimal genus between any two Legendrian knots. In this talk, I will discuss some new results about Lagrangian quasi-cobordisms, based on some work in progress with Sabloff, Vela-Vick, and Wong.
Mar 10 Roger Casals (UC Davis) L-compressible systems and their applications
Show/hide abstract I will discuss progress in our understanding of Legendrian links in the standard Darboux 3-ball, focusing on their Lagrangian fillings and the notion of an L-compressible system. In particular, I will explain how to connect this symplectic geometric problem to the study of cluster algebras and use this connection to prove new results in both fields. This includes the construction of cluster structures for Richardson varieties for any simply-laced simple algebraic Lie group, addressing a conjecture of B. Leclerc, and hints towards an ADE classification for Lagrangian fillings. This is based on recent progress in the study of weaves and their microlocal aspects.
Mar 17 No seminar (Spring break)
Mar 24 Thomas Massoni (Princeton) Anosov flows, non-Weinstein Liouville domains and their wrapped Fukaya categories
Show/hide abstract While Weinstein domains and their symplectic invariants have been extensively studied over the last 30 years, little is known about non-Weinstein Liouville domains. We present a construction in dimension four based on Anosov flows on three-manifolds. The symplectic invariants of these ``Anosov Liouville domains'' constitute new invariants of Anosov flows. The algebraic structure of their wrapped Fukaya categories is in stark contrast with the Weinstein case. We focus on a subcategory $\mathcal{W}_0$ of the wrapped Fukaya category whose objects are in bijection with the simple closed orbits of the flow. Surprisingly, $\mathcal{W}_0$ is not homologically smooth, as it is not finitely split-generated in a maximal way. This talk is mostly based on joint work arXiv:2211.07453 with Kai Cieliebak, Oleg Lazarev and Agustin Moreno.
Mar 31 Soham Chanda (Rutgers) Invariance of Floer cohomology under higher mutation via neck-stretching
Show/hide abstract Pascaleff–Tonkonog defined higher mutations for monotone toric fibers and proved an invariance of disc potential under a change of local system. In this talk, I will define a local version of higher mutations for locally mutable Lagrangians and use neck-stretching to show the invariance of Lagrangian intersection cohomology under a change of local system which agrees with the mutation formula in Pascaleff–Tonkonog.
Apr 7 Ádám Gyenge (Budapest University of Technology and Economics) Blow-ups and the quantum spectrum of surfaces
Show/hide abstract The cup product of ordinary cohomology describes how submanifolds of a manifold intersect each other. Gromov–Witten invariants give rise to quantum product and quantum cohomology, which describe how subspaces intersect in a ”fuzzy”, ”quantum” way. Dubrovin observed that quantum cohomology can be used to define a flat connection on a certain vector bundle called the quantum connection. We verify a conjecture of Kontsevich on the behaviour of the spectrum of the quantum connection under blow-ups for smooth projective surfaces. Joint work with Szilard Szabo.
Apr 14 Adrian Petr (SDU) Legendrian contact homology in circular contactizations and prequantization bundles
Show/hide abstract The Chekanov-Eliashberg DG-algebra is an invariant associated to Legendrians in contact manifolds. It has first been introduced in $R^3$, and then into the general framework of Symplectic Field Theory. In this talk, I will try to explain the relation between Legendrian contact homology in the total space of a circle bundle and Floer theory in the base through a Fukaya-category perspective. Part of this is work in progress with Noémie Legout.
Apr 21 (Online) Yusuf Barış Kartal (Edinburgh) Frobenius operators on symplectic cohomology
Show/hide abstract One can define the Frobenius operators on a commutative ring of characteristic p, and this has generalizations in topological spaces and spectra. A spectrum with a circle action and a compatible Frobenius operator is called a cyclotomic spectrum. The first example is the free loop space, where the Frobenius map sends every loop to its p-fold cover. Another large class of examples arise as the topological Hochschild homology of rings and categories. These two examples suggest that such a structure should also be present on spectral symplectic cohomology, i.e. symplectic cohomology with coefficients in sphere spectrum. In this talk, we discuss the construction of genuine circle actions on the spectral symplectic cohomology, and how one can enhance this to a cyclotomic spectrum. Joint work in progress with Shaoyun Bai and Laurent Cote.
Apr 28 Matthew Habermann (University of Hamburg) Homological Berglund–Hübsch–Henningson mirror symmetry for curve singularities
Show/hide abstract Invertible polynomials are a class of hypersurface singularities which are defined in an elementary way from square matrices with non-negative integer coefficients. Berglund–Hübsch mirror symmetry posits that the polynomials defined by a matrix and its transpose should be mirror as Landau–Ginzburg models, and an extension of this idea due to Berglund and Henningson postulates that this equivalence should respect equivariant structures. In this talk, I will begin by giving some background and context for the problem, and then explain my recent work on proving the conjecture in the first non-trivial dimension; that of curves. The key input, inspired by the derived McKay correspondence, is a model for the orbifold Fukaya–Seidel category in this context.
May 5 Shira Tanny (IAS) Higher dimensional approaches to strong closing lemmas
Show/hide abstract The strong closing property for Hamiltonian / Reeb flows concerns the ability to create periodic orbits passing through any given open set, via local $C^\infty$ perturbations. This property is established in dimensions 2-3 and is generally open in higher dimensions. I will discuss two approaches to this problem. The first uses the contact homology algebra, based on a joint work with Chaidez, Datta and Prasad. The second is by direct measurements of pseudoholomorphic curves, based on a joint work in progress with Chaidez and inspired by the works of Hutchings and McDuff–Siegel.
May 12 Sebastian Haney (Columbia) Mirrors to lines in P^3
Show/hide abstract We will construct, for any tropical curve in R^n with vertices of valence at most 4, a Lagrangian submanifold of (C*)^n whose moment map projection is a tropical amoeba. For a certain 4-valent tropical curve in R^3, we can modify this construction to produce a cleanly immersed Lagrangian supporting objects in the wrapped Fukaya category which correspond, under mirror symmetry, to generic lines in CP^3. If time permits, we will explain how to use Lagrangian correspondences to see this mirror relation, as well as work in progress using this result to compute open Gromov–Witten invariants for Lagrangians in the quintic threefold mirror to lines.
Jun 8 Alex Pieloch (MIT) Rationally connected symplectic manifolds
Show/hide abstract This talk will have two parts. The first part of the talk will be motivational and historical in nature. We will discuss a collection of open conjectures about properties of monotone symplectic manifolds that are motivated by well-known properties of algebraic Fano varieties. In the second part, we will establish various properties of enumeratively rationally connected symplectic manifolds, that is, symplectic manifolds that have a non-zero Gromov–Witten invariant with two point insertions. We will show that any of these symplectic manifolds satisfies the following: (1) Its fundamental group is finite. (2) Every rational second homology class is represented by a (non-effective) sum of rational holomorphic curves. (3) If the spherical homology class associated to the non-zero Gromov–Witten invariant is holomorphically indecomposable, then the rational second homology has rank one.

SGGTC seminar schedule (Fall 2022)

Organizers: Mohammed Abouzaid, Johan Asplund, Marco Castronovo, Francesco Lin and Mike Miller Eismeier.

Date Speaker Title and abstract
Sep 8 Yash Deshmukh (Columbia) A homotopical description of Deligne–Mumford compactifications
Show/hide abstract In this talk, I will discuss the problem of extending actions of moduli spaces of framed curves (of all genera) to the Deligne–Mumford compactifications of moduli spaces of curves. I will explain the algebraic analog underlying such extensions, namely that the compactified moduli spaces of curves arise from the moduli spaces of framed curves by suitably homotopy trivializing certain circle actions. I will also sketch the relation between such extensions and the problem of relating GW invariants (in all genera) to Fukaya categories, and indicate how our statement improves on results available in the literature.
Sep 16 Rohil Prasad (Princeton) Generic equidistribution of periodic orbits for area-preserving surface diffeomorphisms
Show/hide abstract In this talk, I will explain why a generic area-preserving diffeomorphism of a closed surface has a sequence of periodic orbits which equidistribute in the surface. The proof uses several formal properties of spectral invariants from periodic Floer homology, along with a variational argument inspired by works of Marques–Neves–Song and Irie on generic equidistribution for minimal hypersurfaces and three-dimensional Reeb flows, respectively.
Sep 20 Juan Muñoz-Echániz (Columbia) Topology of families of contact structures on 3-manifolds and Floer homology
Show/hide abstract The contact invariant of a contact 3-manifold, defined by Kronheimer and Mrowka, is an element in the monopole Floer homology of the 3-manifold canonically attached to the contact structure. I will discuss a generalisation of this to an invariant of families of contact structures and discuss its applications to the topology of the space of contact structures and contactomorphisms. Time permitting, I will also discuss work in progress on detecting certain non-trivial contactomorphisms given by Dehn twists on spheres.
Sep 30 Sam Bardwell-Evans (Boston) Scattering diagrams from holomorphic discs in log Calabi–Yau surfaces
Show/hide abstract In this talk, we construct special Lagrangian fibrations for log Calabi–Yau surfaces and scattering diagrams from Lagrangian Floer theory of the fibers. These scattering diagrams recover the algebro-geometric scattering diagrams of Gross–Pandharipande–Siebert and Gross–Hacking–Keel, allowing us to relate open Gromov–Witten invariants to log Gromov–Witten invariants. The argument relies on a holomorphic/tropical disc correspondence to control the behavior of holomorphic discs. This talk is based on joint work with Man-Wai Mandy Cheung, Hansol Hong, and Yu-Shen Lin.
Oct 7 No seminar (Krichever Conference)
Oct 14 Wenyuan Li (Northwestern) Duality and spherical adjunction for microlocal sheaves
Show/hide abstract Consider the topological Fukaya category of a cotangent bundle with a singular Legendrian stop given by microlocal sheaves. We study the sheaf theoretic cap and cup functors coming from microlocalization along the stop and its left adjoint. Our result says that, when we have a full stop or a swappable stop, the sheaf theretic functors form a spherical adjunction, so that twists/cotwists are the monodromy/wrap-once functors. On the other hand, we provide an example which is not full or swappable such that spherical adjunction fails. Moreover, we prove that the wrap-once functor is the inverse Serre functor when restricting to the proper subcategories. We will explain how these results follow from the sheaf theory counterpart of the Sabloff duality exact sequence and Poincaré duality for Fukaya–Seidel categories. This is joint work in preparation with Christopher Kuo.
Oct 21 Ipsita Datta (IAS) Lagrangian cobordisms, enriched knot diagrams, and algebraic invariants
Show/hide abstract We introduce new invariants to the existence of Lagrangian cobordisms in R4. These are obtained by studying holomorphic disks with corners on Lagrangian tangles, which are Lagrangian cobordisms with flat, immersed boundaries. We develop appropriate sign conventions and results to characterize boundary points of 1-dimensional moduli spaces with boundaries on Lagrangian tangles. We then use these to define (SFT-like) algebraic structures that recover the previously described obstructions. This talk is based on my thesis work under the supervision of Y. Eliashberg and on work in progress joint with J. Sabloff.
Oct 28 Kai Hugtenburg (Edinburgh) The cyclic open-closed map, u-connections and R-matrices
Show/hide abstract This talk will review recent progress on obtaining Gromov–Witten invariants from the Fukaya category. A crucial ingredient is showing that the cyclic open-closed map, which maps the cyclic homology of the Fukaya category of X to its S1-equivariant quantum cohomology, respects connections. Along the way we will encounter R-matrices, which were used in the Givental–Teleman classification of semisimple cohomological field theories, and allow one to determine higher genus Gromov–Witten invariants from genus 0 invariants. I will then present some evidence that this approach might extend beyond the semisimple case. Time permitting, I will also explain work in progress on obtaining open Gromov–Witten invariants from the Fukaya category.
Nov 4 Agniva Roy (Georgia Tech) Symplectic handlebodies, non-loose knots, and embedding problems
Show/hide abstract We will discuss a construction of closed symplectic handlebodies using ideas laid out by Gay in 2000. This allows us to explicitly understand embeddings of symplectic rational balls in CP2 that were earlier understood only through almost toric fibrations. This is joint work with John Etnyre, Hyunki Min, and Lisa Piccirillo.
Nov 11 (Online) Noah Porcelli (Imperial College) Lagrangian monodromy and string topology
Show/hide abstract One concrete question in symplectic topology is: what diffeomorphisms f of a fixed Lagrangian L can be extended to Hamiltonian diffeomorphism of the ambient symplectic manifold? We will study this in the case when L is exact, and derive strong constraints on the diffeomorphism f, using methods from string topology to extend work of Hu, Lalonde and Leclercq.
Nov 18 (Online) Sukjoo Lee (Edinburgh) Mirror P=W conjecture, extended Fano/LG correspondence, and d-semistable degeneration of Calabi–Yau varieties
Show/hide abstract Mirror P=W conjecture, introduced by Harder–Katzarkov–Przyjalkowski, is a refined Hodge number symmetry between mirror log Calabi–Yau manifolds U and Y. In case that U has a good compactification (X,D) where X is Fano and D is an anti-canonical divisor, one can understand this conjecture from mirror symmetry for the Fano pair (X,D). Its mirror candidate is a hybrid Landau–Ginzburg (LG) model (Y,h:Y->CN), a multi-potential analogue of an ordinary LG model. We will review this story for the first part of the talk. In the second part, we will discuss one of the applications, a topological mirror construction of d-semistable Calabi–Yau varieties.
Nov 25 No seminar (Thanksgiving)
Dec 2 Minh Lam Nguyen (WUSTL) An abelian gauge-theoretic variant of the Seiberg–Witten equations for multiple-spinors
Show/hide abstract We consider a variant of the Seiberg–Witten equations for multiple-spinors. The moduli space of solutions to our generalized Seiberg–Witten equations in the setting of Kähler surfaces has a direct relation with ASD connections of holomorphic vector bundle. We also construct an invariant that detects a certain notion of stability of SU(n)-holomorphic vector bundles.
Dec 9 (Online) Orsola Capovilla-Searle (UC Davis) On Newton polytopes of augmentations of the Legendrian DGA
Show/hide abstract In joint work with Roger Casals we provide a new application of Newton polytopes to the classification of Lagrangian fillings of Legendrian submanifolds in the standard contact (2n + 1) sphere. In particular, we show that Newton polytopes can be used to distinguish infinitely many distinct Lagrangian fillings of Legendrian links in the standard contact 3-sphere and higher dimensional Legendrian spheres in the standard contact (2n + 1) sphere up to Hamiltonian isotopy. We provide the first examples of Legendrian links with infinitely many distinct non-orientable exact Lagrangian fillings.
Dec 16 Umut Varolgunes (Boğaziçi University) Non-archimedean analytic mirrors of symplectic cluster manifolds
Show/hide abstract Relative symplectic cohomology defines a sheaf in the base of a Lagrangian torus fibration with singularities (under some conditions on grading datum). I will explain the computation of this sheaf in some examples. I will then use some of these computations to construct non-archimedean analytic mirrors and explain the concrete open and closed string consequences. The focus will be on explaining the general framework and will be conjectural at times. Joint work with Mohammed Abouzaid and Yoel Groman.

Previous semesters: Fall 2023, Spring 2023, Fall 2022, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007.