## SPRING 2016

All talks take place at 16.15 in **Seminar Room NA 4/24 **.

**19.7.2016** Maciej Starostka (Gdansk), * E-cohomological Conley index and the Arnold conjecture.*

** Abstract:**

We quickly review the definition and properties of the classical Conley index. Then we show how, using the E-cohomology theory, the Conley index can be generalized to the case of Hilbert spaces. Finally, we discuss application of our invariant to the Arnold conjecture on 2n-dimensional torus. Both non-degenerate and degenerate case are discussed and the later follows from some additional multiplicative structure of the Conley index.

**22.6.2016** Marco Zambon (KU Leuven), * Deformations of coisotropic submanifolds in symplectic geometry.*

** Abstract:**

Lagrangian submanifolds are a well understood class of submanifolds of symplectic manifolds, and their
deformations (modulo hamiltonian diffeomorphisms) are governed by the De Rham chain complex. Coisotropic
submanifolds include the lagrangian ones as special cases, and their deformation theory turns out to be
governed by L-infinity algebras, by the work of Oh-Park in 2003. After reviewing this notion, I will sketch
some results obtained with Florian Schätz about equivalences of deformations and coisotropic deformations
in symplectic manifolds.

**24.5.2016** Felix Schmäschke (MPI Leipzig), * Symplectic reduction and the magnetic flow.*

** Abstract:**

The magnetic or twisted cotangent bundle arises as a sympletic
quotient. We explain how this viewpoint permits to treat the
Hamiltonian flow associated to a twisted symplectic form as a
Hamiltonian flow associated to a standard symplectic form at the
cost of introducing a symmetry group. We use it to prove that if the
configuration space is compact and not aspherical there exists
closed magnetic geodesics for all positive energies.

**3.5.2016** Christian Lange (Universität zu Köln), * 2-orbifolds all of whose geodesics are closed.*

** Abstract:**

We explain why the geodesic period spectrum of a Riemannian 2-orbifold all of whose geodesics are closed only depends on its orbifold topology in particular. We recover the fact proved by Gromoll, Grove and Pries that in the manifold case all prime geodesics have the same length.

**19.4.2016** Valentine Roos (ENS Paris), * Viscosity and variational solutions of the evolutive Hamilton-Jacobi equation.*

** Abstract:**

Two different notions of weak solutions were introduced for the
evolutive Hamilton-Jacobi equation, they coincide when the Hamiltonian is
convex in the fiber. The talk will point out examples where the two notions
differ even for small time and quadratic integrable Hamiltonians.
Then we will present the construction of an operator giving the variational
solutions for small time, by applying a minmax selector to the explicit
Chaperon's generating family of the geometric solution.

## WINTER 2015/16

**2.2.2015** Sheila Sandon (Strasbourg), * Floer homology for translated points.*

** Abstract:**

Translated points of contactomorphisms are, roughly speaking, fixed points modulo the Reeb flow. If there are no closed contractible Reeb orbits then translated points satisfy an analogue of the Arnold conjecture on fixed points of Hamiltonian symplectomorphisms. In my talk I will decribe a Floer homology theory for translated points that allows to prove this conjecture (in the absence of closed contractible Reeb orbits) and, as I will discuss, has potential to apply also to more general situations. I will then describe some simple applications of this theory.

**8.12.2015** Oliver Fabert (Amsterdam), * Symplectic topology of classical field theories via model theory.*

** Abstract:**

Hamiltonian PDE, arising e.g. in classical field theories and quantum mechanics, can be viewed as infinite-dimensional Hamiltonian systems. In this talk I show that analogues of the classical rigidity results from symplectic topology, such as Gromov's nonsqueezing theorem and the Arnold conjecture, also hold for these Hamiltonian PDE. In order to establish the existence of the relevant holomorphic curves, I use that each separable symplectic Hilbert space is contained in a symplectic vector space which behaves as if it were finite-dimensional. As a concrete result I show (without experiment !) that every Bose-Einstein condensate, which is constrained to a circle and annoyed by a time-periodic exterior potential, has infinitely many time-periodic quantum states.
.

**24.11.2015** Jian Wang (MPI, Leipzig), * A generalization of classical action of Hamiltonian
diffeomorphisms to Hamiltonian homeomorphisms.*

** Abstract:**

In symplectic geometry, a classical object is the notion of
action function, defined on the set of contractible fixed points of the
time-one map of a Hamiltonian isotopy. On closed surfaces, we
generalize the classical function to any Hamiltonian homeomorphism,
provided that the boundedness condition is satisfied. After that, we give
some applications. For example, we generalize the Schwarz's theorem to the
$C^0$-case on surfaces. We also prove that a Hamiltonian homeomorphism is
the identity if the set of contractible fixed points is path connected.

**3.11.2015** Marco Mazzucchelli (ENS, Lyon), * On the multiplicity of isometry-invariant geodesics.*

** Abstract:**

The problem of isometry-invariant geodesics, introduced by K. Grove in the 70s, is a generalization of the closed geodesics one: given an isometry of a closed Riemannian manifold, one looks for geodesics on which the isometry acts as a non-trivian translation. In this talk, after recalling the framework of the problem, we present a few new multiplicity results on certain classes of Riemannian manifolds. We will also discuss a contact-geometric generalization: the existence problem for Reeb orbits that are invariant under a strict contactomorphism. Part of the talk is based on a joint work with Leonardo Macarini.

**27.10.2015** Fabian Ziltener (Utrecht), * Leafwise fixed points for $C^0$-small Hamiltonian flows and local coisotropic Floer homology.*

** Abstract:**

Consider a symplectic manifold $(M,\omega)$, a closed coisotropic submanifold $N$ of $M$, and a Hamiltonian diffeomorphism $\phi$ on $M$. A leafwise fixed point for $\phi$ is a point $x\in N$ that under $\phi$ is mapped to its isotropic leaf. These points generalize fixed points and Lagrangian intersection points. The main result of this talk will be that $\phi$ has a leafwise fixed point, provided that it is the time-1-map of a Hamiltonian flow whose restriction to $N$ stays $C^0$-close to the inclusion $N\to M$. This result is optimal in the sense that the $C^0$-condition cannot be replaced by the assumption that $\phi$ is Hofer-small.
The method of proof of this result leads to a local coisotropic version of Floer homology.