# 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

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## PREVIOUS TALKS

### 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*

**Abstract: **Follow-up of the talk held on April 13.

**27 April: Lukas Nakamura (Uppsala university)**

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

**Abstract: **We define a pseudo-metric on the contactomorphism group of a cooriented contact manifold M which is bounded from above by the Shelukhin-Hofer metric and non-degenerate if, in addition, M is orderable. If M is closed, we show that its metric topology agrees with the interval topology of Chernov and Nemirovsky, thereby positively answering their question of whether the interval topology of an orderable contact manifold is Hausdorff. We discuss similar results on the universal cover of the contactomorphism group and on spaces of Legendrians. If time permits, we explain connections to Hedicke's recently defined Lorentzian distance function.

### 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*

**Abstract: **Intuitively, a collection of points on a variety is in "general position" if there is no "unexpected" subvariety passing through them. In this talk, I will discuss configurations of points in general position in projective spaces and various constructions related to them. In particular, I will discuss their relation to moduli spaces and related structures.

**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 *

**Abstract:**** **In the paper "Floer homology and algebraic geometry", Donaldson raised the question of constructing an algebraic theory for moduli spaces of vector bundles on algebraic surfaces which are singular along a nodal divisor. A succesful candidate for such a theory would enable one to compute sheaf-theoretic invariants of a smoothing of the singular surface in terms of relative invariants of its normalization, similar to the degeneration formulas in Gromov-Witten or Donaldson-Thomas theory.

I will present work in progress on a good theory of sheaves on degenerations that works when the surface degenerates together with a fibration to a curve. This opens the door for many interesting computations - for example, one can recover and strengthen results about the geometry of moduli spaces on elliptic 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)

*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.

**Past seminars from previous years**