Certain simple symplectic manifolds (symplectic vector space, Milnor fibres of certain complex surface singularities,...) contain sets of symplectically distinct Lagrangian tori which have the following remarkable property: they remain symplectically distinct under embeddings into any reasonable (i.e. geometrically bounded) symplectic manifold. This leads to a vast extension of the class of spaces in which the existence of exotic tori is known, especially in dimensions six and above. In this talk we mainly focus on recent joint work with Johannes Hauber and Joel Schmitz which treats the more intricate case of dimension four.

I will explain recent work on relationships among geometric quantization, deformation quantization, Berezin-Toeplitz quantization and brane quantization.

Given a degenerating family of Kähler-Einstein metrics it is natural to study from a differential geometric perspective the collection of all metric limits at all possible scales, a typical example being the emergence of Kronheimer’s ALE spaces near the formation of orbifold singularities for Einstein 4-manifolds. In this talk, I will describe, focusing on the discussion of some concrete and elementary examples, how it should be possible to use algebro geometric tools to investigate such problem for algebraic families, leading in the non-collapsing case to an inductive argument identifying the so-called metric bubble tree at a singularity (made of a collection of asymptotically conical Calabi-Yau varieties) with a subset of the non-Archimedean Berkovich analytification of the family. Based on joint work with M. de Borbon.

Given a flow on a manifold, how to perturb it in order to create a periodic orbit passing through a given region? This question was originally asked by Poincaré and was initially studied in the 60s. However, various facets of it remain largely open. Recently, several advances were made in the context of Hamiltonian and contact flows. I will discuss a connection between this problem and Gromov-Witten invariants, which are "counts" of holomorphic curves. This is based on a joint work with Julian Chaidez.

The symplectic non-squeezing theorem, discovered by Gromov in 1985, has been the first result showing a fundamental difference between symplectic transformations and volume preserving ones. A similar but more subtle phenomenon in contact topology was found by Eliashberg, Kim and Polterovich in 2006, and refined by Fraser in 2016 and Chiu in 2017: in this case non-squeezing depends on the size of the domains, and only appears above a certain quantum scale.

In my talk I will outline the geometric ideas behind a proof of this general contact non-squeezing theorem that uses generating functions, a classical method based on finite dimensional Morse theory. This is a joint work with Maia Fraser and Bingyu Zhang.