# Geometry and topology seminar

Time: Thursdays at 13:15 in room Å64119** **and on Zoom.

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

## NEXT TALK

## Future talks

### 9 June: Martin Bäcke (Uppsala university) *Master's thesis presentation*

Title:* **DG-algebra computations for singular Legendrians*

Abstract:

### 23 June: Yash Deshmukh (Columbia university)

Title:* **TBA*

Abstract: TBA

## Previous talks

### 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.