# 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

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## PREVIOUS TALKS

### 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.

**Past seminars from previous years**