# Geometri- och topologiseminariet

Wednesdays at 14.15 on ZOOM.

24 June (new date!): Baptiste Chantraine (Université de Nantes)

Title: Invariant of Legendrian submanifolds in non co-orientable contact structure

Abstract: I want to report on some work in progress on invariants of Legendrian submanifolds in contact manifolds where the contact structure is not coorientable. We will see two structures arising when considering the Z_2 action on the lift of the Legendrian to its coorientation cover: one is a structure of involution on the augmentation category and the other is a Z_2 equivariant version of linearised Legendrian contact homology. I will begin by considering natural examples of such Legendrian submanifolds and end by explaining how these invariants relate to other equivariant theories and how this could be used to compute it in these natural situations.

Meeting ID: 640 3867 0355

Past seminars 2020:

10 June: Russell Avdek

Title: A closed, tight contact 3-manifold with vanishing contact homology

Abstract: Contact homology (CH) is an invariant which assigns a differential graded algebra to each closed contact manifold. While this invariant shares some formal properties with Heegaard-Floer homology, CH is comparatively not well-understood due to a current lack of computational techniques. In this talk, we will describe the first example of a closed, tight contact 3-manifold whose CH is the zero algebra. In proving that CH=0 for this contact manifold, we will summarize some computational tools which relate surgeries and cobordisms to dynamical systems and holomorphic curves in a combinatorial fashion.

3 June: Lionel Lang (Stockholm University).

Title: Coamoebas, dimers and vanishing cycles

Abstract: In this joint work in progress with J. Forsgård, we study the topology of maps P:(C*)^2 \to C given by Laurent polynomials P(z,w). For specific P, we observed that the topology of the corresponding map can be described in terms of the coamoeba of a generic fiber. Under these circumstances, the coamoeba contains a dimer (a particularly nice graph) together with a distinguished collection of cycles (the fundamental cycles) that turn out to be the vanishing cycles of the map P. For a given degree, the existence of such P is widely open: it relies on the existence of maximal coamoebas. In the meantime, we can bypass coamoebas by going directly to dimers using a construction of Goncharov-Kenyon. We obtain a virtual correspondence between the fundamental cycles of the dimer and the vanishing cycles. In this talk, we will discuss how this (virtual) correspondence can be used to compute the monodromy of the map P.

27 May: Tobias Ekholm (Uppsala Univerisity).

Title: Alexander polynomial, Z-hat, and curve counts

Abstract: We describe how to express the Alexander polynomial of a knot K via its augmentation polynomial. The formula allows for a natural deformation that connects to the physics inspired Z-hat invariant for 3-manifolds of the knot complement, F_K. In fact it leads to an enumerative geometry interpretation and an a-deformation of F_K that is annihilated by the quantum augmentation polynomial. We discuss possible geometric origins of non-uniqueness of solutions to the quantum augmentation polynomial which is related to branching in the augmentation variety.

Meeting ID: 665 0488 6595

May 20: Agnès Gadbled (Université Paris-Saclay)

Title: Weinstein handlebodies for complements of smoothed toric divisors.

May 13: Yang Huang (University of Southern Denmark)

Title: IKEA in contact 5-manifolds.

April 29: Jian Qiu (Uppsala Univerisity).

Title: Toric Hyper Kahler varieties and special functions

Abstract: The talk will be about some work done together with Andreas, Nikos, Maxim. But I will be a bit more broad and try to give some context of why are we doing this. The plan of the talk is
1. some elements of equivariant cohomology algebra
2. special functions constructed out of H_{eq} and their properties, with simple examples.
3. HyperKahler varieties and 3-Sasaki structures
4. some combinatorial results on special functions constructed from 3.
5. why are we doing this, and outlook.

April 22: Paolo Ghiggini (Université de Nantes).

Title: The standard contact structure on RP5 is not Liouville fillable.

April 15: Tobias Ekholm (Uppsala University).

April 8: Noémie Legout (Uppsala University).

Title: Lagrangian cobordisms not coming from immersed ones after surgery

Abstract: I will give examples of Legendrian knots admitting Lagrangian fillings of genus g with p transverse double points, but no Lagrangian fillings of genus g-1 with p+1 double points (with some additional conditions). This follows from a count of equivalence classes of augmentations: for Legendrian links L^- and L^+, if there is  a Lagrangian cobordism from L^- to L^+ we show that the number of equivalence classes of augmentations of L^- must be smaller than that of L^+. This is a joint work with Orsola Capovilla-Searle, Maÿlis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.

April 1: Georgios Dimitroglou Rizell (Uppsala University).

Title: Refined Floer homology for singular Lagrangians

Abstract: We explain a construction in joint work with T. Ekholm and D. Tonkonog where we use Symplectic Field Theory techniques to define Floer homology groups for pairs of singular exact Lagrangians. The examples that we consider are self-plumbings of Lagrangian surfaces, where these Floer homology groups become derived hom's of representations of "derived (higher genus) multiplicative preprojective algebras".

March 11: Thomas Kragh (Uppsala University).

Title: Using co-algebras, co-modules, injective resolutions and Serre fibrations to compute SWF-bar.

March 4: Olof Bergvall (Uppsala University).

Title: Del Pezzo surfaces and rational curves

Abstract: In this talk we will consider Del Pezzo surfaces, i.e. smooth algebraic surfaces with ample anticanonical bundles, and rational curves on such surfaces. More precisely, I will give very concrete descriptions of certain moduli spaces of Del Pezzo surfaces marked with rational curves and explain how to compute their cohomology (as representations of certain Weyl groups).

February 26: Jérémy Guéré (Université Grenoble Alpes).

Title: Hodge-Gromov-Witten theory

Abstract: Hodge-Gromov-Witten theory of a smooth projective variety X deals with the cap product of the virtual fundamental cycle on the moduli space of stable maps to X with the Euler class of the Hodge vector bundle. I recently studied its deformation invariance to singular varieties, allowing explicit computations in many cases. An important application of my theorem is a calculation of genus-zero GW invariants for some hypersurfaces in weighted projective spaces which do not satisfy the so-called convexity property. It is a first step towards a mirror symmetry statement for these hypersurfaces.
In a second part of the talk, I will describe my plan towards a calculation of GW invariants of the quintic hypersurface in P^4. It is based on a theorem Costello proved in 2003 expressing genus-g GW invariants of a projective variety X in terms of genus-0 GW invariants of the (g+1)-st symmetric power of X.

February 19: Mark Lawrence (Nazarbayev University).

Title: Polynomial hulls, knots, and holomorphic motions

​Abstract: The study of polynomial hulls of compact sets in Cn is too challenging to have a useful general answer. Even for smooth manifolds, there is little known. One restriction is to look at tori or unions of tori in S1 × C which fiber over the circle. The knot class of the torus plays a crucial role, both in a positive and negative direction. The main positive result of the author and J. Duval is that a torus modeled on a square root has a polynomial hull which is fibered by varieties. On the negative side, it appears that the knot type of the torus can exclude any hull from appearing over the unit disc, but only preliminary results exist in this direction. Using the theorem of Lawrence and Duval, new types of holomorphic motions
can be constructed, which have a limited amount of branching.

January 22: Wanmin Liu (Uppsala University).

Title: Contractibility of space of stability conditions on projective plane via global dimension function.

Abstract: The global dimension function $\gldim$ is a continuous function defined on Bridgeland stability manifold, and it maps a stability condition to a non-negative real number. We compute the global dimension function $\gldim$ on the space of stability conditions on projective plane. It has the minimal value $2$, and $\gldim^{-1}(2)$ is contractible. Moreover, $\gldim^{-1}(2)$ is contained in the closure of the subspace consisting of geometric stability conditions. We show that $\gldim^{-1}([2,x)$ contracts to $\gldim^{-1}(2)$ for any real number $x>=2$. The $\gldim$ sheds some lights on why a space of Bridgeland stability conditions should be contractible. This is a joint work with Yu-Wei Fan, Chunyi Li and Yu Qiu.