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G2 Manifolds Program Seminar: Sonja Hohloch, EPFL Lausanne

Title: Hyperk\”ahler Floer theory as an explicit Hamiltonian system
Speaker: Sonja Hohloch, EPFL Lausanne
Date: Wednesday, September 24, 2014
Time: 02:00pm – 03:00pm
Place: Lecture Hall 102, Simons Center

 
 

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Abstract: Floer homology was developed by Andreas Floer at the end of the 1980’s in order to approach Arnold’s conjecture about the number of fixed points of so-called Hamiltonian diffeomorphisms. Nowadays Floer theory has turned into a powerful tool with applications in various areas of geometry.

In a joint work with Dietmar Salamon and Gregor Noetzel, we had generalized Floer homology to hyperk\”ahler geometry. More precisely, we had defined and computed it on flat, compact hyperk\”ahler manifolds. In contrast to the classical Floer setting where the critical points of the symplectic action functional are periodic Hamiltonian orbits, the critical points of the hypersymplectic action functional are 3-dimensional tori or spheres solving a certain `triholomorphic’ equation, also called `Fueter equation’.

Since the whole setting still seems somewhat mysterious (at least to me), the idea is to come up with another way to describe the solutions.
In this talk, we will show, using an explicit construction, that the `triholomorphic tori’ can be seen as periodic solutions of a Hamiltonian system on the iterated loop space.