Geometry and Topology seminar series

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

For more info contact the organiser: Alex Takeda

7 December: Amanda Hirschi (University of Cambridge)

Title: Global Kuranishi charts and a localisation formula in symplectic GW theory

Abstract: I will briefly describe the construction of a global Kuranishi chart for moduli spaces of stable pseudoholomorphic maps and show how this allows for a straightforward definition of sympletic GW invariants. Subsequently, I will describe how to extend this to the equivariant setting and sketch the proof of a localisation formula for the equivariant GW invariants of a Hamiltonian manifold. This is partially joint work with Mohan Swaminathan.

23 November: Oscar Kivinen (Aalto University)

Title: The Hilb-vs-Quot conjecture

Abstract: The Oblomkov-Rasmussen-Shende conjecture relates the (a, q, t)-graded Khovanov-Rozansky homology of the link of a plane curve singularity to the topology of its punctual Hilbert schemes. While the t=-1 limit is a theorem of Maulik, and much is understood in the q=1 limit, almost no examples incorporating all three gradings exist. In work with Trinh, we introduce a different kind of Quot scheme and show that an analogous conjecture for these Quot schemes does hold for torus knots and (n, nk)-torus links. This motivates a surprising conjecture about how two kinds of Quot scheme on the same curve are related by a motivic substitution of variables, without reference to links or link homology.

9 November: Agustin Moreno (Heidelberg University)

Title: Symplectic geometry of Anosov flows in dimension three

Abstract: In this talk, I will illustrate how symplectic geometry can be leveraged in the study of uniformly hyperbolic dynamics. Namely, every Anosov flow on a three-manifold M has an associated non-Weinstein Liouville domain V=[-1,1]xM, whose invariants (e.g. symplectic cohomology, Rabinowitz-Floer cohomology, wrapped Fukaya category) are homotopy invariants of the flow. Moreover, every orbit of the flow gives an exact Lagrangian cylinder, all of which are nontrivial and independent of each other in the Fukaya category, such that the A_infty category they generate does not satisfy Abouzaid’s generation criterion, in contrast to the Weinstein case. Time permitting, I will discuss the classical algebraic cases, their symplectic invariants, and further results concerning closed Lagrangians.

2 November: Paolo Ghiggini (Université Grenoble-Alpes)

Title: A relative exact sequence for Lagrangian Floer homology

Abstract: Some years ago Chantraine, Dimitroglou-Rizell, Golovko and I defined a
version of Lagrangian Floer homology for Lagrangian cobordisms also
known as Cthulhu homology. I will show that concatenation of cobordisms
induce a long exact sequence in Cthulhu homology. A similar exact
sequence was proved by Cieleibak and Oancea if the negative ends of the
cobordisms are filled.

26 October: David Kern (KTH Stockholm)

Title: Symmetric 2-Segal conditions for shifted cotangent groupoids

Abstract: Algebraic (derived) stacks are generalised spaces that can be presented
as quotients of geometric groupoids. Similarly, shifted symplectic
stacks admit atlases given by symplectic groupoids. In this talk, based
on joint work in progress with Damien Calaque, we describe such an atlas
for shifted cotangent bundles. It is obtained through a careful study of the structure of groupoid
objects as certain Calabi–Yau algebras in spans, and functoriality of
the cotangent stack construction producing Lagrangian correspondences
from spans.

19 October: Georgios Dimitroglou Rizell (Uppsala University)

Title: Restrictions on the augmentation variety from the relative Calabi-Yau structure of the Chekanov-Eliashberg algebra (Part II: bulk deformations)

Abstract: We describe the relative Calabi-Yau property and resulting restrictions on the augmentation variety in the setting of bulk deformations. This is a continuation of the previous talk on the topic.

12 October: Axel Husin (Uppsala University)

Title: Floer theory with generalized local systems

Abstract: A path local system is a dg-functor from the path category of a topological space to the category of chain complexes. We will define Lagrangian intersection Floer theory with coefficients in path local systems, and use it to construct a functor between categories of path local systems defined on transversally intersecting Lagrangians. In the case when the ambient symplectic manifold is the cotangent bundle of a closed manifold and one Lagrangian is closed and exact and the other is the base, a local computation will show that the functor we defined is an equivalence. 

28 September: Georgios Dimitroglou Rizell (Uppsala University)

Title: Restrictions on the augmentation variety from the relative Calabi-Yau structure of the Chekanov-Eliashberg algebra

Abstract: The Chekanov-Eliashberg DGA of a Legendrian in the boundary of a subcritical Weinstein domain with loop space coefficients admits a relative Calabi-Yau structure (as defined by Brav and Dyckerhoff). This is a generalisation of Sabloff duality that holds for its finite dimensional representations. We describe notions of the fundamental class in the general setting of the relative CY-structure, and show how it can be used to give constraints on the dimension of the augmentation variety. The main application is a constraint on the dimension of the variety of augmentations of the Knot contact homology algebra of a smooth knot. This is joint work in progress with Legout.

14 September: Alex Takeda (Uppsala University)

Title: Pre-CY structures and explicit algebraic analogs of string topology

Abstract: I will explain how using the notion of pre-CY structures and a certain
graphical formalism, one can write down chain-level representatives for some
operations on Hochschild homology that resemble those from string topology,
and that we conjecture to be a new algebraic way of calculating the Chas-
Sullivan product and Goresky-Hingston coproduct. It turns out that the natural
tool for assembling all this data is Efimov's definition of the "categorical
formal punctured neighborhood of infinity", which generalizes the category of
perfect complexes near the divisor at infinity in a compactification of an
algebraic variety and is related, on the other side of mirror symmetry, to
Rabinowitz Fukaya categories. This talk is about joint work with Manuel Rivera
and Zhengfang Wang.

17 August: Roman Golovko (Charles university)

Title: On a few special Legendrian submanifolds.

Abstract: We will discuss three types of examples of Legendrian submanifolds in high dimensions that
1) realize an arbitrary finitely generated abelian group in linearized Legendrian contact homology,
2) admit an infinite family of diffeomorphic, but not Hamiltonian isotopic exact Lagrangian fillings,
3) admit an augmentation, whose non-geometricity can not be obstructed by the isomorphism of Seidel-Ekholm-Dimitroglou Rizell

21 June: Adrian Petr (University of Southern Denmark)

Title: Legendrian contact homology in prequantization bundles

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.

15 June: Michael Sullivan (UMass Amherst)

Title: Chain homotopy data for Legendrians and Lagrangians

12 June: Álvaro del Pino Gomez (Utrecht university)

Title: Overtwistedness of maximally non-integrable distributions of rank 2

Abstract: A distribution is maximally non-integrable if its sheaf of tangent vector fields has "as many non-trivial Lie brackets as possible". This condition defines a partial differential relation on the space of distributions that turns out to be rather non-trivial to tackle.

In this talk I will focus on distributions of rank 2. In this case, the maximal non-integrability condition is beyond the applicability of the classic h-principle approaches due to Gromov. However, the 3-dimensional situation (i.e. for 3-dimensional contact structures) was addressed successfully by Eliashberg in 1989 using a so-called removal of singularities approach. He proved that there is a class of 3-dimensional contact structures, called overtwisted, that can be completely classified in homotopical terms.

In 2018, Vogel and myself proved an analogous statement in dimension 4 (i.e. for so-called Engel structures). The proof we gave recovers Eliashberg's statement as well, suggesting that Engel and contact overtwistedness are concrete incarnations of a more general phenomenon.

In this talk I will discuss the following conjecture: maximally non-integrable distributions of rank-2 admit an overtwisted class that satisfies the complete h-principle. My goal will be to outline a strategy of proof generalising the 4-dimensional approach. The crucial technical step is a flexibility statement for submanifolds transverse to a bracket-generating distribution.

This is on-going work with F. ter Haar and F.J Martínez-Aguinaga.

26 May: Inanc Baykur (UMass Amherst)

Title: Exotic 4-manifolds with signature zero

Abstract: We are going to talk about our recent construction of small symplectic 4-manifolds with signature zero, which include the smallest closed simply-connected exotic 4-manifolds with signature zero known to date. Our novel examples are derived from explicit Lefschetz fibrations, with spin and non-spin monodromies. Joint work with N. Hamada.

11 May: Georgios Dimitroglou Rizell (Uppsala university)

Title: A relative Calabi-Yau structure for Legendrian contact homology and applications to the augmentation variety

Abstract: We show how Sabloff duality in linearized Legendrian contact homology can be generalised to a relative Calabi-Yau structure of the Legendrian contact homology DGA, as defined by Brav and Dyckerhoff. We also discuss the generalised notion of the fundamental class and give applications, including constraints on the dimension of the augmentation variety. The structure is established through the acyclicity of a version of Rabinowitz Floer Homology for Legendrian submanifolds with coefficients in the Chekanov-Eliashberg DGA. This is joint work in progress with Legout.

4 May: Tobias Ekholm (Uppsala university)

Title: Counting bare curves VI

27 April: Lukas Nakamura (Uppsala university)

Title: A metric on the contactomorphism group of an orderable contact manifold.

13 April: Tobias Ekholm (Uppsala university)

Title: Counting bare curves V

Abstract: (This is a follow-up of the talks which were given in the last few months). 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.

30 March: Olof Bergvall (Mälardalens university)

Title: Point sets in projective spaces and moduli spaces

23 March: Jack Smith (Cambridge)

Title: Hochschild cohomology of the Fukaya category via Floer cohomology with coefficients

Abstract: The Hochschild cohomology of the Fukaya category is an important symplectic invariant, and plays a central role in the generation criterion of Abouzaid and Sheridan.  Unfortunately, it is usually very difficult to calculate directly.  I will describe new results relating it to a more geometric and computable object, namely Floer cohomology of a Lagrangian with enriched coefficients, and discuss some applications. Perhaps surprisingly, matrix factorisations make an important appearance.

9 March: Nick Kuhn (University of Oslo)

Title: Degenerations of stable sheaves on fibered surfaces

23 February: Jian Qiu (Uppsala university)

Title: Quantization via branes and minimal resolution

Abstract: I want to explain in this talk the framework of quantization using branes, as developed by Gukov-Witten and later refined by Gaiotto-Witten. This approach embeds the symplectic manifold to be quantized into a bigger space (of double the dimension) as a brane, and the open strings ending on the brane play the role of the operators acting on the Hilbert space.
The open strings are expected to be quantized via the A-model, while in practice the open string wave functions are identifiable as holomorphic functions on the larger space and therefore many algebraic approach can be applied for its quantization and bypass the difficult A-model. I will apply Kontsevich’s deformation quantization to deform the algebra of operators. This approach would have been impractical without further tools such as the minimal bi-algebra resolution and some recently developed techniques for deformation of quiver algebras.
I will run you through some examples, such as the quantization of Kleinian singularity, in the A_1 case, the quantization gives the Verma module of Usu(2) and integrality condition for the arameters, which Is unusual for deformation quantization. Further examples include T^*CP^2, which is useful in the context of generalized Kahler structure.
Future applications include the quantization of character varieties, Higgs moduli space etc. 

16 February: Paolo Ghiggini (Institut Fourier Grenoble)

Title: Heegaard Floer homology as a Floer field theory and genus two mutations

Abstract: I will show that the hat version of Heegaard Floer homology can be cast into a modified version of Wehrheim and Wodward's Floer field theory: roughly speaking, a 2+1 TQFT with values in compact Fukaya categories. The main difference from previous unpublished work by Lekili and Perutz is that we use a "cylindrical reformulation" of the Fukaya category of a symmetric product allowing us to avoid quilted Floer homology. As an application, I will show that the total rank of Heegaard Floer homology is not changed by a surgery operation called "genus two mutation". This is a work in progress in collaboration with Ina Petkova.

9 February: Tasos Moulinos (Université Paris 13)

Title: A tour through the topological K-theory of dg categories

Abstract: I will discuss some of my past work on the topological K-theory of dg categories. This is an invariant which shares some formal properties with algebraic K-theory but which lands in the infinity category of KU-module spectra.

In particular it is a “localizing invariant of complex-linear differential graded categories” which outputs a KU-module spectrum. I will begin the talk with motivations coming by way of Hodge theoretic mirror symmetry, and will proceed to describe the basic construction, originally due to Blanc. I will then describe a variant of this construction from previous work, relative to any base complex scheme, together with applications of such a construction towards computations in twisted K-theory, and towards the theory of variations of Hodge structures.  Time permitting, I will describe some open questions.

2 February: Tobias Ekholm (Uppsala university)

Title: Counting bare curves IV

Abstract: This is a follow up of the talk held on November 24th.

12 January: Noah Porcelli (Imperial College London)

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.

Past seminars from previous years