Seminar series within Geometry and Topology

The seminars are usually held on Thursdays at 13:15 in room Å64119 and on Zoom.

For more info contact the organisers: Côme Dattin and Noémie Legout.

Upcoming seminars

PREVIOUS TALKS

12 January: Noah Porcelli (Imperial College London)

Title: Framed bordism of exact Lagrangians via Floer homotopy

Abstract: Lagrangian Floer theory is a useful tool for studying the structure of the homology of Lagrangian submanifolds. In some cases, it can be used to detect more- we show it can detect the framed bordism class of certain Lagrangians and in particular recover a result of Abouzaid--Alvarez-Gavela--Courte--Kragh about smooth structures on exact Lagrangians in cotangent bundles of spheres. The main technical tool we use is Large's recent construction of a stable-homotopical enrichment of Lagrangian Floer theory.
This is based on joint work-in-progress with Ivan Smith.

8 December: Thomas Kragh (Uppsala university)

Title: Seiberg-Witten Floer Homotopy Types II: Family Floer theory and twisted spectra

Abstract: I will start by discussing the example that was discussed at length last time during Alice's talk, and provide some variations and context (relating it to some notions of family Floer theory in cotangent bundles, background classes, partially wrapping, and micro-local support - hoping that each audience member will find at least one of these relations provides some insight into such examples and why they are interesting). I will then describe how to construct somewhat explicit twisted spectra out of finite dimensional approximations and Conley indices in the Seiberg-Witten (SWF) setting (this setting is actually much easier than the cotangent bundle). I will then talk about other twisted categories than that of twisted spectra. In particular I will describe how S^1-equivariant homotopy theory naturally arises in this context and how we actually construct a "twisted parametrized S1-equivariant spectrum" representing SWF (I will not assume much knowledge about any of these stable categories - and I will try and keep most of the general discussion at a heuristic level). This is joint work with A. Hedenlund and S. Behrens.

1 December: Alice Hedenlund (Uppsala university)

Title: Seiberg-Witten Floer Homotopy Types I: Twisted Spectra

Abstract: Seiberg–Witten theory has played a central role in the study of smooth low-dimensional manifolds since their introduction in the 90s. Parallel to this, Cohen, Jones, and Segal asked the question of whether various types of Floer homology could be upgraded to the homotopy level by constructing (stable) homotopy types encoding Floer data. In 2003, Manolescu constructed Seiberg-Witten Floer spectra for rational homology 3-spheres, and in particular used these to settle the triangulation conjecture. A precursor to Manolescu's Seiberg-Witten Floer spectra, the Bauer-Furuta invariant, was moreover used by Furuta to make significant progress on the "11/8 Conjecture" which deals with what quadratic forms are realisable as the intersection form on a smooth 4-manifold.

This is the first talk of a two-part series reporting on joint work in progress with S. Behrens and T. Kragh. In this talk I will give an expository introduction to twisted spectra, which are essential for constructing Floer homotopy types in the situation where our infinite-dimensional manifold is "non-trivially polarised".  Roughly, one could think of twisted spectra as arising as global sections of a bundle whose fibre is the (infinity-)category of spectra. I will also explain how these mathematical objects naturally appear in Seiberg-Witten Floer theory. Next week, Thomas will go into the more concrete constructions of twisted spectra from Seiberg-Witten Floer data using finite-dimensional approximation and Conley index theory.

24 November: Tobias Ekholm (Uppsala university)

Title: Counting bare curves III

Abstract: This is a follow up of the talk "Counting bare curves II".

17 November: Tobias Ekholm (Uppsala university)

Title: Counting bare curves II

Abstract: This is a follow up of the talk "Counting bare curves I" which was held on October 20th.

10 November: Two talks

Marco Golla (Nantes university)

Title: Alexander polynomials and symplectic curves in CP^2.

Abstract: Libgober defined Alexander polynomials of (complex) plane projective curves and showed that it detects Zariski pairs of curves: these are curves with the same singularities but with non-diffeomorphic complements. He also proved that the Alexander polynomial of a curve divides the Alexander polynomial of its link at infinity and the product of Alexander polynomials of the links of its singularities. We extend Libgober's definition to the symplectic case and prove that the divisibility relations also hold in this context. This is work in progress with Hanine Awada.

Renato Vianna (Federal university of Rio de Janeiro)

Title: Lagrangian fibrations on smoothing of algebraic cone.

Abstract: This is joint work with Santiago Achig-Andrango. Given a lattice polytope Q in R^n, we can consider its cone C(Q), in R^n+1, given by taking the rays from the origin passing through Qx{1} in R^nxR.

Being a rational polyhedral cone, we associate an affine toric algebraic variety Y, which is often singular. Altaman shows how to construct deformations of Y from a Minkowiski decomposition Q = M_1 + … + M_k of Q. Gross constructs a special Lagrangian fibration out of this data, but with general Lagrangian fibre equals to T^n x R (rather than T^n+1). Auroux describes in his work how to obtain a singular Lagrangian fibration  with a generic fibre a torus T^n out of a auxiliary complex (symplectic) fibration with a fibrewise preserving T^n-1 action.

In his work, Achig-Andrango imposes conditions in the Minkowiski decomposition to ensure that Altman’s deformation gives a smoothing of Y and constructs a complex fibration in the generic fibre Y_e, with genral fibre (C^*)^n and singular fibres with local model associated to each term M_i of the Minkowiski decomposition. Using that as an auxiliary complex fibration, Achig-Angrango constructs a singula Lagrangian fibration, studies its monodromies and show it can be represented by a convex base diagram with cuts, whose image in R^n+1 is the dual of the cone C(Q). Moreover, the fibration contains a one parameter family of monotone Lagrangian fibres, whose superpotential can be obtained under wall-crossing formulas and described in terms of the Minkowiski decomposition of Q.

We will present Achig-Andrango’s work and time permiting, discuss some further developments.

27 October: Georgios Dimitroglou-Rizell (Uppsala university)

Title: Positive loops of non-Legendrians, and C^0 contactomorphisms.

Abstract: We show that any n-dimensional non-Legendrian submanifold of a 2n+1-dimensional contact manifold admits a small positive loop that preserves parametrisations, which is generated by a non-negative global contact isotopy. The existence of the positive loop is then used to show that a homeomorphism that arises as the C0-limit of contactomorphisms has the property that the image of a Legendrian submanifold is again a Legendrian, under the assumption that the image is smooth. This is joint work in progress with M. Sullivan.

20 October: Two talks

Tobias Ekholm (Uppsala university)

Title: Counting bare curves I

Abstract: In a series of talks we will discuss how to count bare holomorphic curves with Lagrangian boundary conditions in a Calabi-Yau 3-fold. Here bare means that the curves does not have any symplectic area zero components. The main application and motivation for this is the construction of invariant skein valued counts of open curves. In this first talk we will explain  why the usual approaches to Gromov-Witten counts does not work to define skein valued counts and present one of the key technical results for bare counts: we finds a class of perturbations for the Cauchy-Riemann equations that has the property that if a ghost bubbles forms in the limit of sequence of bare curves then it leaves a trace (e.g. vanishing complex derivative) on the limiting positive area curve on which it is attached. In later talks we will construct a perturbation scheme using polyfold language with perturbations of this kind that gives rise to solutions of the perturbed Cauchy-Riemann equations that admit forgetful maps. The talks report on joint work with Vivek Shende.

Yuan Yao (University of California, Berkeley)

Title: Computing Embedded Contact Homology in the Morse-Bott Setting using Cascades

Abstract: I will first give an overview of ECH. Then I will describe how to compute ECH in the Morse-Bott setting a la Bourgeois. I will discuss some classes of examples where this approach works. Finally I will sketch the gluing results that allow us to compute ECH using cascades.

13 October: Matthew Habermann (Hamburg university)

Title: Homological Berglund--Hübsch--Henningson mirror symmetry for curve singularities

Abstract: Invertible polynomials are a class of hypersurface singularities which are defined 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.

29 September: Wenyuan Li (Northwestern university)

Title: Lagrangian cobordism functor in sheaf theory

Abstract: Given a Lagrangian cobordism between Legendrian submanifolds in the unit cotangent bundles, we construct a functor between the categories of sheaves on the base manifold with singular support on the Legendrians. This provides a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies enhanced with loop space coefficients. The construction works for singular exact Lagrangian cobordisms between singular Legendrians as well. I will then deduce some new obstructions to Lagrangian cobordisms between high dimensional Legendrians. If time permits, I will discuss immersed exact Lagrangian cobordisms based on future work in progress by constructing the cobordism functor from a different perspective.

22 September: Filip Strakoš (Uppsala university)

Title: DGA-homotopy criteria for augmentations of CE-algebras

Abstract: In a suitably generic situation the differential of Chekanov-Eliashberg algebra can be augmented. The set of all DGA-homotopy classes of augmentations is then a Legendrian isotopy invariant, and so is its cardinality. We will use linearized and bilinearized Legendrian contact homology to derive DGA-homotopy criteria for connected and disconnected Legendrian submanifolds respectively. In particular, we will see that while in the connected case the DGA-homotopy criterion descends to the chain level, this is no longer true in the disconnected case. Then we will apply the results to the question of geography of the bilinearized Legendrian contact homology generalizing results of Bourgeois and Galant.

23 June: Yash Deshmukh (Columbia university)

Title: Homotopically extending actions of properads of Riemann surfaces

Abstract: A result of G. Drummond-Cole relates the framed little disks operad to the genus zero Deligne-Mumford operad. In this talk, I will discuss the higher genus, multiple input-output analog of this result. I will explain that extending an action of the properad of Riemann surfaces with parameterized boundaries to an action of the properad of nodal Riemann surfaces is equivalent, homotopically, to providing the data of trivializations of certain operations coming from the moduli spaces of annuli. In the process, I will also introduce a new partial compactification of the moduli spaces of Riemann surfaces which is relevant to the study of symplectic cohomology, and describe a version of the above statement concerning extension of actions to these partial compactifications.

16 June: Luis Diogo (Uppsala university)

Title: The Floer cohomology of Lagrangian tori in the cotangent bundle of the 2-sphere

Abstract: One of the simplest symplectic manifolds that one can consider is the cotangent bundle of the 2-sphere, and we will address the problem of classifying its monotone Lagrangian tori as objects in the Fukaya category. Specifically, we show that the Floer cohomology algebras of such Lagrangians are very similar to those of the classical Entov-Polterovich tori.  

9 June at 10:30 in room Å4004: Cecilia Karlsson (University of Borås)

Title: Chekanov-Eliashberg DGA for Weinstein handle attachments in higher dimensions

Abstract: The symplectic homology of a Weinstein manifold is encoded in the Chekanov-Eliashberg DGA of the attaching spheres of the top index handles. In this talk I describe a geometric set-up where we can calculate the Chekanov-Eliashberg DGA of the attaching spheres from Legendrians in 1-jet spaces. Since the latter is well-studied this simplifies the calculations. If time permits I will use these techniques to calculate the singular homology of the free loop space of CP^2.
This is a generalization of similar work by Ekholm-Ng to higher dimensions.

9 June at 13:15 in room Å64119: Martin Bäcke (Uppsala university) Master's thesis presentation

Title: DG-algebra computations for singular Legendrians

Abstract: We compute the homology of the Chekanov-Eliashberg algebra in the boundary of a Weinstein surface, also known as the internal algebra of Ekholm-Ng. This algebra appears as a subalgebra of the Chekanov-Eliashberg algebra of singular Legendrian skeleta of Weinstein hypersurfaces, introduced by Asplund-Ekholm. We show a result relating the algebras arising from different choices of handle decomposition of the surface, and obtain new examples of Legendrians with computable homology.

12 May: Sylvain Douteau (Stockholm university)

Title: Stratified homotopy theory and invariants of embeddings

Abstract: The study of stratified spaces - and more recently, of their associated homotopy theory - was originally understood as a way of dealing with singularities. This is illustrated by Whitney's historical theorem, which states that all analytic varieties over R or C can be stratified in such a way that the strata are smooth manifolds that glue together properly. However, stratifications can also be used to encode structures on smooth objects, such as embeddings. In this context, stratified homotopy theory can be used to obtain relevant invariants. In this talk, I will present an overview of the homotopy theory of stratified spaces and of its applications to embeddings, by investigating the case of knots

5 May: Thibaut Mazuir (IMJ-PRG, Paris)

Title: Higher algebra of A-infinity algebras in Morse theory

Abstract: In this talk, I will introduce the notion of n-morphisms between two A-infinity algebras. These higher morphisms are such that 0-morphisms correspond to standard A-infinity morphisms and 1-morphisms correspond to A-infinity homotopies. The set of higher morphisms between two A-infinity-algebras then defines a simplicial set which has the property of being a Kan complex. The combinatorics of n-morphisms are moreover encoded by new families of polytopes, which I call the n-multiplihedra and which generalize the standard multiplihedra.

Elaborating on works by Abouzaid and Mescher, I will then explain how this higher algebra of A-infinity algebras naturally arises in the context of Morse theory.

28 April: Johan Rydholm (Uppsala university)

Title: Homological mirror symmetry for A_n resolutions and cyclic plumbings

Abstract: We show a homological mirror symmetry equivalence between cyclic plumbings, that is, Milnor fibres of type A_n with a conic removed, and the resolutions of A_n singularities. This is done by symplectically realizing the Ginzburg algebra of the quiver with extended A_n form, using the surgery formula, and showing that this algebra is formal.

21 April: Georgios Dimitroglou-Rizell (Uppsala university)

Title: Floer homology for monotone Weinstein skeleta

Abstract: We discuss the connections between refined potentials, Floer homology of singular monotone Lagrangians, and deformations of the Hochschild cocomplex. This is joint work in progress with P. Ghiggini.

31 March, exceptionally in room Å80101: Daniel Kaplan (Hasselt University, Belgium)

Title: DG algebras associated to plumbed cotangent bundles

Abstract: Given a graph, one can build a 4-dimensional symplectic manifold by plumbing cotangent bundles of 2-spheres according to the graph. Etgu--Lekili established that the wrapped Fukaya category of such a manifold is equivalent to the category of dg-modules for the dg multiplicative preprojective algebra. This motivates the question of whether such dg algebras are formal (i.e., quasi-isomorphic to their homology as dg-algebras). If yes, then the wrapped Fukaya category is equivalent to the category of modules for the multiplicative preprojective algebra. In the first half of the talk, I will sketch this motivation but then leave the setting of geometry/topology to discuss a purely algebraic approach to formality developed in joint work with Travis Schedler (following work of Etingof--Ginzburg on non-commutative complete intersections, which itself utilizes older work of Anick). We prove the formality of these dg algebras if the graph is connected with a cycle. We conjecture that formality holds if the graph is connected and not ADE Dynkin, by analogy to the additive dg-preprojective algebra (also called the Ginzburg dg algebra with zero potential). In the second half of the talk, I will explain how to employ Bergman's Diamond Lemma for ring theory to establish this formality. By focusing on small examples, and side-stepping technical details, I hope to make the talk accessible to both geometers and algebraists. 

24 March: Russell Avdek (Uppsala University)

Title: Relative RSFT via planar diagrams

Abstract: I’ll describe work in progress on a new version of Legendrian rational SFT which generalizes Legendrian contact homology (LCH). It reformulates Ekholm’s RSFT using ideas from Chas-Sullivan's chord diagram formalism for string topology and Hutchings’ ``q-variables only’’ version of closed-orbit RSFT. The result is a differential graded algebra (DGA) with a special filtration which is used to enhance the usual augmentation theory. Basic computations show that LCH and our RSFT contain very different qualitative information. If time permits, I’ll demo software which computes augmentations and bilinearizations of the new RSFT for links in R3.

17 March: Fabio Gironella (Humboldt University)

Title: Liouville orbifold fillings of contact manifolds.

Abstract: The topic of the talk will be Floer theories on Liouville orbifolds with smooth contact boundary. More precisely, I will describe the construction, which only uses classical transversality techniques, of a symplectic cohomology group on such symplectic orbifolds. Time permitting, I will then describe how to deduce the existence, in any odd dimension at least 5, of a pair of contact manifolds with no exact symplectic (smooth) cobordisms in either direction. This is joint work with Zhengyi Zhou.

10 March: Angela Wu at 3pm (note the unusual time!) (Louisiana State University)

Title: Obstructing Lagrangian concordance for closures of 3-braids

Abstract: Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. In this talk I'll show that no Legendrian knot which is both concordant to and from the unstabilized Legendrian unknot can be the closure of an index 3 braid except the unknot itself. The proof uses a variety of techniques in contact and symplectic geometry, from open book decompositions and Weinstein handle diagrams, to symplectic homology and the Legendrian contact homology DGA.

3 March at 2pm (note the unusual time!): Honghao Gao (Michigan State University)

Title: Obstructions to exact Lagrangian fillings

Abstract: We study Legendrian knots and their exact Lagrangian fillings. Viewed as cobordisms, exact Lagrangian fillings functorially induce morphisms between moduli spaces between Legendrian invariants. Algebraic structures over these moduli spaces can be used to define obstructions to exact Lagrangian fillings. In this talk, I will report a joint work with Dan Rutherford, where we construct an obstruction from the algebraic geometry of the augmentation variety, and compare it with previously known obstructions arising from A-infinity algebras.

17 February: Paolo Ghiggini (Nantes University)

Title: Speculations on singular Lagrangian submanifolds and homological mirror symmetry of CP^3.

10 February: Oliver Leigh (Uppsala University)

Title: r-Spin Hurwitz numbers via Stable Maps with Divisible Ramification

Abstract: Hurwitz numbers enumerate smooth covers of the projective plane. Classically, one also imposes a condition requiring all ramification to be of order 1. There are many beautiful and deep results related to classical Hurwitz numbers. This includes the ELSV formula, a link to Gromov-Witten theory, links to mathematical physics and links to topological recursion.     

A natural question to ask is: How many of these results hold when the condition "all ramificaiton is order 1" is replaced with "all ramificaiton is order r"? In this talk we will answer this quesion using the theory of stable maps with divisible ramification. This will include links to r-spin theory via Zvonkine’s r-ELSV formula and a discussion of the subtleties arising in this situation.

3 February at 15:30 (note the unusual time!): Orsola Capovilla-Searle (UC Davis)

Title: Infinitely many planar exact Lagrangian fillings and symplectic Milnor fibers

Abstract: We provide a new family of Legendrian links with infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. This family of links includes the first examples of Legendrian links with infinitely many distinct planar exact Lagrangian fillings, which can be viewed as the smallest Legendrian links currently known to have infinitely many distinct exact Lagrangian fillings. As an application we find new examples of infinitely many exact Lagrangian spheres and tori 4-dimensional Milnor fibers of isolated hypersurface singularities with positive modality.

Past seminars from previous years

Last modified: 2023-01-23