# 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

## Upcoming seminars

## Previous talks

### September 5: Lukas Nakamura (UU)

**Title:** The worldsheet skein D-module and basic curves on Lagrangian fillings of the Hopf link conormal

**Abstract:** We propose a skein-valued lift of the D-module associated to a link in S^3, which is the quotient of the skein of the cylindrical lift of the unit conormal of the link by an ideal which in the classical limit reduces to the defining ideal of the augmentation variety of the link. The Gromov-Witten partition function of a Lagrangian filling can then be viewed as a morphism from the D-module to the skein of the filling. This imposes constraints on the partition function which for some links can be used to determine the partition function uniquely.

I will then exemplify this general framework in the particular case of the Hopf link. For every point of the augmentation variety, we identify a corresponding Lagrangian filling whose partition function is uniquely determined by the constraints. We obtain explicit expressions for the partition functions in terms of contributions of basic holomorphic curves, which can be viewed as skein-valued versions of the Gopakumar-Vafa formula and the knots-quivers correspondence. This is joint work with T. Ekholm and P. Longhi.

### August 29: Tianyu Yuan (Peking University)

**Title: **Bulk deformed Lagrangian Floer theory on symmetric products

**Abstract: **We discuss some progress on orbifold Lagrangian Floer theory. In the case of symmetric products, it is modeled by the higher-dimensional Heegaard Floer homology (HDHF) and is related to symplectic Khovanov homology. In this talk, we discuss HDHF of cotangent fibers and its relation to various Hecke algebras. This is joint work with Ko Honda, Roman Krutowski and Yin Tian.

### August 15 : Alejandro Vicente (Hebrew University of Jerusalem)

**Title: **On symplectic embedding problems via Lagrangian fibrations**Abstract: **In this talk I will talk about how to study certain symplectic embedding problems in sufficiently symmetric manifolds with the use of Lagrangian fibrations. In particular I will give ideas on how to compute the Gromov width of disk cotangent bundles of ellipsoids of revolution and talk about some interesting open problems that could be worked with similar ideas. This is based on joint work with Brayan Ferreira and Vinicius Ramos, as well as undergoing work with Santiago Achig-Andrango and Renato Vianna.

### May 30: Louis Hainaut (Stockholm University)

**Title**: A classifying space for the handlebody mapping class group

**Abstract: **In recent work join with Dan Petersen, we described a model for the classifying space of the mapping class group of 3-dimensional handlebodies. This model allows us to detect the existence of many classes in the cohomology of this group, in a similar way as in the work of Chan-Galatius-Payne about the surface mapping class group. I will first explain some background about the handlebody mapping class group and Teichmuller space, and then explain the construction of our model. In the remaining time I will explain the connection between our model and the tropical moduli space of curves.

### May 23: Trygve Poppe Oldervoll (NTNU)

**Title:** An A-infinity structure on generating function homology**Abstract:** Generating functions are a tool for producing Lagrangians in cotangent bundles. To a pair of generating functions, one can associate a Morse theoretic invariant known as generating function homology. This is analogous to the Floer homology of the associated Lagrangians. In Floer theory, there are higher operations defined by counting rigid holomorphic polygons. In Morse theory, one can construct higher operations by counting rigid flow trees. This was made precise for standard Morse theory on a compact manifold by Fukaya and Oh. Generalizing the work of Myer, I will explain how to construct such operations for generating function homology.

### April 25: Lukas Woike (University of Burgundy)

**Title: **Applications of factorization homology in quantum algebra

**Abstract: **Factorization homology allows us to integrate algebras over the little n-disks operad over n-dimensional manifolds. After giving a brief introduction to factorization homology, I will explain how factorization homology, in the case n=2, can be used to classify systems of representations of mapping class groups of surfaces called modular functors. The methods can be understood as a far-reaching generalization of skein theory. This is joint work with Adrien Brochier.

### April 18: Tanushree Shah (Chennai Mathematics Institute)

**Title:** Giroux torsion in contact structures on 3-manifolds

**Abstract:** I will start by introducing contact structures. They come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. There infinitely many Tight structures iff the manifold is toroidal. Tight contact structures have been completely classified on very few toroidal 3 manifolds. We will look at what more classification results can we hope to get using the current techniques and what is far-fetched.

### April 18: Marco Golla (Nantes Université)

**Title:** Line arrangements with only triple points

**Abstract: **Line arrangements are unions of lines in a (projective or affine) space over some field and they are studied from a combinatorial or algebro-geometrical standpoint. (Non-trivial) complex line arrangements always have at least a double or a triple point, a point where exactly two or three lines meet. Does there exist 13 lines in the complex projective plane that only meet at triple points? I will discuss some topological ideas around this question and higher-degree variants.

### April 11: Du Pei (University of Southern Denmark)

**Title:** 4d Symplectic Duality

**Abstract:** Symplectic duality can be understood as a non-trivial relation between the Coulomb and Higgs branches of 3d N=4 quantum field theories. In this talk, I will present some emerging evidence for an analogous relation between the Coulomb and Higgs branches for 4d theories.

### April 4 : Santiago Achig (UU)

**Title:** Singular Lagrangian Torus Fibrations on the smoothing of algebraic cones.

**Abstract:** In this talk, we explore a new approach to special Lagrangian fibrations on the smoothing of Gorenstein singularities, initially introduced by Gross and further analyzed in the context of the SYZ conjecture of mirror symmetry. Utilizing global coordinates connected to Altmann's characterization of the smoothing, our methodology not only provides alternative proofs for existing theorems about these fibrations but also expands on Symington’s work by creating a convex base diagram in higher dimensions. Additionally, we apply techniques from Pascaleff and Tonkonog to recover Lau's calculation of the potential for certain monotone fibers and investigate the non-displaceability of these fibers. This presentation includes a series of illustrative examples and concludes with an overview of future research directions.

### March 21: Stefan Behrens (Bielefeld University)

**Title:** Generalized Froyshov invariants

**Abstract:** Froyshov invariants are rational valued invariants of 3-manifolds with the same rational homology as the 3-sphere. They can be extracted from monopole Floer theory and contain information about the topology of 4-manifolds that bound a given such 3-manifold. I will discuss a framework to obtain generalized Froyshov invariants using Seiberg-Witten-Floer homotopy types and various tools from equivariant stable homotopy theory. This is work in progress with Tyrone Cutler.

### March 20: Shah Faisal (Humboldt University of Berlin)

**Title:** Extremal Lagrangian tori in convex toric domains

**Abstract:** Cieliebak and Mohnke define the symplectic area of a Lagrangian submanifold of a symplectic manifold as the minimal positive symplectic area of a 2-disk in the symplectic manifold with a boundary on the Lagrangian. I will explain that every Lagrangian torus that maximizes the symplectic area among the Lagrangian tori in the standard symplectic unit ball must lie entirely on the boundary of the ball. This answers a question attributed to L. Lazzarini and settles a conjecture of Cieliebak and Mohnke in the affirmative. I will also explain to which extent this statement is true for general toric domains.

### February 22: Milica Ðukic (UU)

**Title:** A deformation of Chekanov-Eliashberg dg-algebra for Legendrian knots

**Abstract:** Inspired by symplectic field theory and string topology, we

introduce a chain complex for any Legendrian knot, whose homology is an

invariant of the knot up to Legendrian isotopy. The chain complex is

obtained by deforming Chekanov-Eliashberg differential using

pseudoholomorphic annuli in the symplectization. We show how to compute

the invariant combinatorially for any Legendrian knot in R^3.

### February 21: Rémi Leclercq (Université Paris-Saclay)

**Title:** Essential loops of Hamiltonian homeomorphisms

**Abstract:** In 1987, Gromov and Eliashberg showed that if a sequence of diffeomorphisms preserving a symplectic form C⁰ converges to a diffeomorphism, the limit also preserves the symplectic form -- even though this is a C¹ condition. This result gave rise to the notion of symplectic homeomorphisms, i.e. elements of the C⁰-closure of the group of symplectomorphisms in that of homeomorphisms, and started the study of "continuous symplectic geometry".

In this talk, I will present recent progress in understanding the fundamental group of the C⁰-closure of the group of Hamiltonian diffeomorphisms in that of homeomorphisms. More precisely, I will explain a sufficient condition which ensures that certain essential loops of Hamiltonian diffeomorphisms remain essential when seen as "Hamiltonian homeomorphisms". I will illustrate this method (and its limits) on toric manifolds, namely complex projective spaces, rational products of 2-spheres, and rational 1-point blow-ups of CP².

Our condition is based on (explicit) computation of the spectral norm of loops of Hamiltonian diffeomorphisms which is of independent interest. For example, in the case of 1-point blow-ups of CP², I will show that the spectral norm exhibits a surprising behavior which heavily depends on the choice of the symplectic form. This is joint work with Vincent Humilière and Alexandre Jannaud.

### February 8th: Alberto Cavallo (IMPAN)

**Title:** Transverse links in Stein fillable contact 3-manifolds

**Abstract:** We study the behavior of different versions of the Ozsváth-Szabó tau-invariant for holomorphically fillable links in Stein domains. More specifically, we relate the Hedden's version of the invariant, which needs the assumption that our links live in a contact 3-manifold with non-vanishing contact invariant, with the one introduced by Grigsby, Ruberman and Strle, which on the other hand only depends on the pair link-Spin^c 3-manifold and is then a purely topological invariant. This is joint work with Antonio Alfieri.

The main goal of the talk is to describe how our work allows us to recover results about properly embedded holomorphic curves, such as the slice-Bennequin inequality and the relative Thom conjecture, and to find new restrictions on the topology of Stein fillings of certain 3-manifolds. In particular, building on a result of Mark and Tosun, we show that a Brieskorn 3-sphere, with its canonical orientation, never bounds a rational homology 4-ball Stein filling; confirming a conjecture of Gompf.

### February 1: Matt Magill (UU)

**Title:** Functorial QFT and Thom spectra

**Abstract:** Thanks to work of Atiyah and Segal in the late '80s, it has been understood that quantum field theories furnish us with representations of bordism categories. Some particularly interesting QFTs are "anomalous" theories. In this talk, I will spend some time reviewing these ideas, then show how (in certain settings) the language of Thom spectra can be used to classify anomalous theories.

### January 25: Lisa Lokteva (UU)

**Title: **Graph Manifolds with Rational Homology Ball Fillings

**Abstract: **The subfield of low-dimensional topology colloquially called "3.5-dimensional topology" studies closed 3-manifolds through the eyes of the 4-manifolds that they bound. This talk focusses on Casson's question of which rational homology 3-spheres bound rational homology 4-balls. Since rational homology 3-spheres bounding rational homology 4-balls is a rare phenomenon, we will discuss how to construct examples.