Program

Lecture courses:

Jean Gutt (Cologne)

Title: Introduction to positive equivariant symplectic homology and some applications

Abstract: We shall define first (positive) symplectic homology then (positive) equivariant symplectic homology. We shall study some properties of this latter homology and apply them to some dynamical and embeddings questions.

Lecture notes

Myeonggi Kwon (Gießen)

Title: Brieskorn manifolds and positive equivariant symplectic homology

Abstract: Brieskorn manifolds and Milnor fibers have played an important role in contact and symplectic topology. In particular, they provide computable examples for Floer theoretical homology which are in general very hard to compute explicitly. In the lectures, I will give computations of (positive equivariant) symplectic homology of Brieskorn Milnor fibers using Morse-Bott spectral sequences. For this, we first study topological and contact topological properties of Brieskorn manifolds. The main application will be about exotic contact structures on spheres.

Additional lectures:

Jungsoo Kang (Seoul)

Title: Introduction to Morse and Floer homology

Abstract: In the first half of the lectures, I will give a gentle introduction to Morse homology. The second half will be devoted to explaining the Floer homology for closed symplectic manifolds in the simplest setting. No prerequisite in symplectic geometry is needed.

Lara Simone Suárez López (Bochum)

Title: Whitehead torsion and Floer complex

Abstract: In this talk I will introduce a classical algebraic invariant, the Whitehead torsion of an acyclic bounded chain complex over a group ring. Following work of M. Sullivan, Lee-Hutchings and Abouzaid-Kragh we will introduce the Whitehed torsion of the Floer complex and study its invariance. If time allows we will study some applications of this construction to symplectic geometry.

Frol Zapolsky (Haifa)

Title: Orientations in Hamiltonian Floer theory

Abstract: The use of coherent orientations is ubiquitous in symplectic geometry. However there is no need to use them. Using the canonical chain complex for a cellular filtration of a topological space as a motivation, I'll describe an approach to orientations in Floer theory which, unlike coherent orientations, does not involve any choices (or at least, the choices are of an entirely different nature) and leads to a canonical chain complex defined over the integers in the case of Hamiltonian Floer theory on monotone symplectic manifolds, and hopefully I'll have time to prove d^2 = 0.

Research talks:

Takahiro Oba (Kyoto)

Title: Lefschetz-Bott fibrations on disk bundles over symplectic manifolds

Abstract: A Lefschetz-Bott fibration on a symplectic manifold is a smooth map to a surface with only Lefschetz-Bott critical points, which model complexifications of Morse-Bott critical points. As Lefschetz fibrations play an important role in the study of Stein fillings of contact manifolds, we expect Lefscehtz-Bott fibrations to help us understand symplectic fillings. However, little is known about symplectic manifolds admitting such fibrations. In this talk, we show that a good class of disk bundles over symplectic manifolds admits Lefschetz-Bott fibrations over the unit disk. As an application, we give a geometric interpretation of symplectic mapping class group relations of Milnor fibers given by Acu and Avdek.

Alexander Fauck (Berlin)

Title: Distinguishing contact structures via Symplectic homology

Abstract: In my talk, I will introduce the notion of asymptotically finitely generated (a.f.g.) contact structures. Such contact structures posses a sequence of Reeb vector fields R_l such that the number of its relevant closed orbits is uniformly bounded in a fixed degree. I will discuss that a large class of fillable contact manifolds is a.f.g. and I will show how this property can be used to distinguish infinitely many fillable contact structures on the same differentiable manifold.

Joontae Kim (Seoul)

Title: Equivariant wrapped Floer homology and real Lagrangians

Abstract: We explore an open string analogue of symplectic homology, called wrapped Floer homology, and its applications. Firstly, we introduce a nice class of real Lagrangians in a Brieskorn variety and compute its wrapped Floer homology using Morse-Bott argument. This will show a positivity of slow volume entropy of certain symplectomorphisms. Secondly, we introduce equivariant wrapped Floer homology using (anti-)symplectic involutions. With careful analysis on index iterations, we obtain the minimal number of geometrically distinct symmetric Reeb orbits. This is joint work with Seongchan Kim and Myeonggi Kwon.

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