Columbia Symplectic Geometry, Gauge Theory, and Categorification Seminar

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

The SGGTC seminar normally meets in Room 307 on Fridays 11:30 am–12:30 pm in the Mathematics Department.

SGGTC seminar schedule (Fall 2022)

Note: Non-standard seminar dates or locations are marked with ※.
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 R^4. 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 CP^2 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->C^N), 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.

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Previous semesters: 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.