Tag - Symplectic geometry

Yakov Eliashberg: Contractibility of the space of tight contact structures on ℝ3

18th October, 2021

30 years ago I proved that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. I also optimistically claimed at the same paper that similar methods could be used to prove a multi-parametric version: the space of tight contact structures on the 3-sphere, fixed at a point, is contractible. In our recent joint with N. Mishachev paper we proved this result. While the proof indeed roughly follows the strategy of my 1991 paper, it is much more involved. In particular, it uses a new criterion for tightness of a characteristic foliation on the 2-sphere, which is valid without any contact convexity assumptions.

Umberto Hryniewicz: Results on abundance of global surfaces of section

15th October, 2021

One might ask if global surfaces of section (GSS) for Reeb flows in dimension 3 are abundant in two different senses. One might ask if GSS are abundant for a given Reeb flow, or if Reeb flows carrying some GSS are abundant in the set of all Reeb flows. In this talk, answers to these two questions in specific contexts will be presented. First, I would like to discuss a result, obtained in collaboration with Florio, stating that there are explicit sets of Reeb flows on S3 which are right-handed in the sense of Ghys; in particular, for such a flow all finite (non-empty) collections of periodic orbits spans a GSS. Then, I would like to discuss genericity results, obtained in collaboration with Colin, Dehornoy and Rechtman, for Reeb flows carrying a GSS; as a particular case of such results, we prove that a C∞-generic Reeb flow on the tight 3-sphere carries a GSS.

Andrew Hanlon: Tropical Lagrangian sections and Looijenga pairs

11th October, 2021

We will discuss the first steps in an approach to proving homological mirror symmetry for Looijenga pairs through tropical Lagrangian sections. Namely, we will see how to construct these Lagrangian sections from tropical data corresponding to line bundles on the mirror and include them in a version on the Fukaya-Seidel category. Moreover, the Lagrangian Floer coholomogy of certain sections corresponds with integral points of polytopes that encode theta functions on the mirror.

Rima Chatterjee: Cabling of knots in overtwisted contact manifolds

8th October, 2021

Knots associated to overtwisted manifolds are less explored. There are two types of knots in an overtwisted manifold – loose and non-loose. Non-loose knots are knots with tight complements whereas loose knots have overtwisted complements. While we understand loose knots, non-loose knots remain a mystery. The classification and structure problems of these knots vary greatly compared to the knots in tight manifolds. Especially we are interested in how satellite operations on a knot in overtwisted manifold changes the geometric property of the knot. In this talk, I will discuss under what conditions cabling operation on a non-loose knot preserves non-looseness.

Jean-Philippe Chassé: Convergence and Riemannian bounds on Lagrangian submanifolds

8th October, 2021

Recent years have seen the appearance of a plethora of possible metrics on spaces of Lagrangian submanifolds. Indeed, on top of the better-known Lagrangian Hofer metric and spectral norm, Biran, Cornea, and Shelukhin have constructed families of so-called weighted fragmentation metrics on these spaces. I will explain how — under the presence of bounds coming from Riemannian geometry — all these metrics behave well with respect to the set-theoretic Hausdorff metric.

Leo Digiosia: Cylindrical contact homology of links of simple singularities

8th October, 2021

In this talk we consider the links of simple singularities, which are contactomoprhic to S3/G for finite subgroups G of SU2(ℂ). We explain how to compute the cylindrical contact homology of S3/G by means of perturbing the canonical contact form by a Morse function that is invariant under the corresponding rotation subgroup. We prove that the ranks are given in terms of the number of conjugacy classes of G, demonstrating a form of the McKay correspondence. We also explain how our computation realizes the Seifert fiber structure of these links.

Helmut Hofer: The Floer Jungle: 35 years of Floer Theory

16th July, 2021

An exceptionally gifted mathematician and an extremely complex person, Floer exhibited, as one friend put it, a 'radical individuality'. He viewed the world around him with a singularly critical way of thinking and a quintessential disregard for convention. Indeed, his revolutionary mathematical ideas, contradicting conventional wisdom, could only be inspired by such impetus, and can only be understood in this context.

Poincaré's research on the Three Body Problem laid the foundations for the fields of dynamical systems and symplectic geometry. From whence the ancestral trail follows Marston Morse and Morse theory, Vladimir Arnold and the Arnold conjectures, through to breakthroughs by Yasha Eliashberg. Likewise, Charles Conley and Eduard Zehnder on the Arnold conjectures, Mikhail Gromov's theory of pseudoholomorphic curves, providing a new and powerful tool to study symplectic geometry, and Edward Witten's fresh perspective on Morse theory. And finally, Andreas Floer, who counter-intuitively combined all of this, hitting the "jackpot" with what is now called Floer theory.

Laurent Côté: Action filtrations associated to smooth categorical compactifications

9th July, 2021

There is notion of a smooth categorical compactification of dg/A∞ categories: for example, a smooth compactification of algebraic varieties induces a smooth categorical compactification of the associated bounded dg categories of coherent sheaves. In symplectic topology, wrapped Fukaya categories of Weinstein manifolds admit smooth compactifications by partially wrapped Fukaya categories. The goal of this talk is to explain how to associate an "action filtration" to a smooth categorical compactifications, which is invariant (up to appropriate equivalence) under zig-zags of smooth compactifications. I will then discuss applications to symplectic topology and categorical dynamics.

Felix Schlenk: Symplectically knotted cubes

2nd July, 2021

While by a result of McDuff the space of symplectic embeddings of a closed 4-ball into an open 4-ball is connected, the situation for embeddings of cubes C4 = D2×D2 is very different. For instance, for the open ball B4 of capacity 1, there exists an explicit decreasing sequence c1,c2,… → 1/3 such that for c<ck there are at least k symplectic embeddings of the closed cube C4(c) of capacity c into B4 that are not isotopic. Furthermore, there are infinitely many non-isotopic symplectic embeddings of C4(1/3) into B4.

A similar result holds for several other targets, like the open 4-cube, the complex projective plane, the product of two equal 2-spheres, or a monotone product of such manifolds and any closed monotone toric symplectic manifold.

The proof uses exotic Lagrangian tori.

Ben Wormleighton: Lattice formulas for rational SFT capacities of toric domains

25th June, 2021

Siegel has recently defined 'higher' symplectic capacities using rational SFT that obstruct symplectic embeddings and behave well with respect to stabilisation. I will report on joint work with Julian Chaidez that relates these capacities to algebro-geometric invariants, which leads to computable, combinatorial formulas for many convex toric domains.