I used my NWO Veni to bring young researchers to Utrecht University for a couple of weeks to share their expertise with us using a minicourse format. The topics covered dealt with the theory of tangent distributions (contact or otherwise) and related topics (symplectic geometry, subriemannian geometry, and others).

After two sessions, the event was superseded by the Dutch Differential Geometry and Topology seminar. See here.

Former speakers:

### November 27th – December 6th 2019

Lucas Dahinden is a postdoc based at the Ruprecht-Karls-Universität Heidelberg. His research deals with Contact Topology and Dynamics. During his visit he will give a minicourse on some recent extremely interesting developments regarding topological entropy.

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**Title:**Rigidity of topological entropy for positive contactomorphisms.**Location:**Uithof campus in Utrecht, Minnaert gebouw, room 0.15.**Time:**29th November 2019. 11:00-13:00 and then 15:00-17:00.**Abstract:**In this series of talks I aim to present a result in Symplectic Geometry. Historically, Symplectic Geometry developed as the geometry behind Physics because it is the right framework to define the Hamiltonian equations. Nowadays, Symplectic Geometry is a research field on its own, with many subbranches and connections to other fields. We take a journey beginning with the definition of “symplectic”, to more and more specialized theories. We will follow the following structure:

1. Symplectic Geometry in general

2. Floer homology

3. Rabinowitz-Floer homology

4. The main result.

The proof of (4) uses the machinery introduced in (1-3), but I hope to show you that this machinery is also of independent interest by explaining ideas and mentioning other applications. The main result identifies a class of spaces on which a natural class of maps has positive topological entropy (i.e. they are chaotic in some sense).

For the non-experts: A generalization of geodesic flows on a generalization of rationally hyperbolic manifolds is always chaotic.

For the experts: In Liouville fillable contact manifolds with exponentially growing wrapped Floer homology, every positive contactomorphism has positive topological entropy.

### April 2nd-16th 2019

Daniel Álvarez-Gavela is a postdoc at the IAS. His work deals with flexibility phenomena in contact and Symplectic Topology, Singularity Theory, and parametrised Morse theory. During his stay in Utrecht he gave two beautiful minicourses.

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**Title:**The flexibility of caustics and its applications to the arborealization of Lagrangian skeleta**Location:**Uithof campus in Utrecht, Buys Ballot gebouw, room 077.**Time:**5th April 2019. 11:00-13:00 and then 14:30-16:30.**Abstract:**Starting with the results of Smale-Hirsch on immersion theory, there has been extensive work in the mathematics literature devoted to the problem of simplifying the singularities of smooth maps. The over-arching principle is that such a problem, a priori geometric in nature, often reduces to the underlying homotopy theoretic problem, to which the tools of algebraic topology can be applied. Recently it has been shown that such a reduction also exists for the problem of simplifying the singularities of Lagrangian and Legendrian fronts, also known as caustics. This latter problem is intimately linked to the problem of simplifying the singularities of the skeleta of Weinstein manifolds, which are the symplectic analogues of Stein manifolds (smooth affine complex varieties). In the best case scenario the simplification yields a skeleton with only arboreal singularities, which are a particularly natural and simple class of singularities for which the pseudo-holomorphic invariants of Weinstein manifolds become combinatorial.

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**Title:**K-theoretic invariants in symplectic and contact topology via parametrized Morse theory**Location:**Uithof campus in Utrecht, Buys Ballot gebouw, room 077.**Time:**12th April 2019. 11:00-13:00 and then 14:30-16:30.**Abstract:**The generating family construction provides a link between the study of Lagrangian (resp. Legendrian) submanifolds of cotangent bundles (resp. 1-jet spaces) and the study of parametrized Morse theory on fibre bundles. By the work of Smale, Hatcher, Wagoner, Igusa, Waldhausen and others, the homotopy type of the space of (stable) h-cobordisms of a manifold can be computed in terms of K-theory using the tools of parametrized Morse theory. Therefore it is not surprising that K-theoretic invariants can be used to exhibit rigidity phenomena in symplectic (resp. contact topology). It is currently unclear the precise way in which the K-theoretic invariants are related to the invariants provided by pseudo-holomorphic curve theory. However, several new examples of the connection between K-theory and symplectic (resp. contact) topology have recently appeared in the literature, making the study of this connection a fertile topic for research.