During the summer semster of 2025 Alberto Abbonandolo, Luca Asselle and Simon Vialaret will host a reading course on ECH.

The course starts Wednesdays at 10.15 in Seminar Room IA 1/181.

Description of the reading course

Embedded contact homology (ECH) is a \topological invariant" which can be de ned on closed
contact three manifolds. It's relevance stems from the fact that, as shown by Taubes in his seminal
work and later by other authors, ECH is isomorphic to the Seiberg-Witten Floer homology de ned
by Kronheimer-Mrowka as well as to Heegard Floer homology. This fact is used for instance in
the proof of the Weinstein conjecture in dimension three. ECH has also some other additional
structures which are used in applications for instance to symplectic embedding problems or to
Reeb dynamics (multiplicity of periodic Reeb orbits, etc.) In the seminar we will focus on the
details (as many as possible!) of the construction of ECH and explain some of the applications.
The reading course takes place on Wednesday 10 am - 12 am in Room IA 1/181. All talks are
broadcasted live on zoom and recorded.

                Zoom Link
 

02.07.2025  Alberto Abbondandolo  ECH capacities, and Weyl law for domains in ℝ⁴

25.06.2025 Simon Vialaret Computation of ECH of 𝕋³, the contact invariant detects the contact structure

04.06.2025 Jan Eyll Definition of the U-map, computation of the ECH for ellipsoids

28.05.2025 Barney Bramham Well-definedness of ∂, ∂² = 0

21.05.2025 Johanna Bimmermann Gromov's compactness for currents, proof of the writhe bound and of the partition condition

14.05.2025 Jonas Fritsch ECH index, index inequality, ECH differential and grading

07.05.2025 Manuel Stange Special properties in dimension four: intersection positivity and adjunction formula

30.04.2025 Jacobus Sander de Pooter Moduli spaces of holomorphic curves, transversality and Fredholm index

23.04.2025 Luca Asselle Introductive session: overview of the construction of ECH, and applications (Weinstein conjecture, embeddings of ellipsoids, C∞-closing lemma, generic density of periodic orbits).