## SPRING 2015

All talks take place at 16.15 in **Seminar Room NA 5/24 **.

**23.06.2015** Sina Türeli (ICTP, Trieste), * A Frobenius Theorem for Continuous Distributions.*

** Abstract:**

We are going to give a new theorem for integrability of rank 2 continuous tangent sub-bundles (distributions) in 3 dimensional manifolds. The theorem depends on approximations and a natural notion of asymptotic involutivity. We will then state some applications to PDE, ODE and Dynamical Systems.

**19.05.2015** Alessandro Gentile (SISSA, Trieste), * Topology of non-holonomic loop spaces.*

** Abstract:**

On a manifold with generic non-holonomic constraints, even though the motion is allowed in the admissible directions only, it is possible to go everywhere along admissible curves (horizontal curves). It is interesting to study the topology of the space of horizontal curves joining two points (the non-holonomic loop space); by applying Morse theory we can relate its topology with the structure of geodesics (critical points of the Energy). If the manifold is R^n, the non-holonomic loop space is contractible, but we can still apply Morse theory and study what happens to the sublevels of the energy as it increases. On one hand, we get more and more geodesics and we find an upper bound for their number; on the other hand even if every Betti number of the loop space must eventually vanish, their sum still grows unbounded. It turns out that Morse inequalities are far from sharp, and we present the "right" order of growth for the topology. The leading coefficient of this growth order is a local invariant of the non-holonomic structure; in the case the non admissible directions are less than 3, this leading coefficient is computed analitically with an integral on the space of skew-symmetric matrices.

## WINTER 2014/15

All talks take place at 16.15 in **Seminar Room NA 5/24 **.

**20.01.2015** Luca Asselle (RUB), * Periodic orbits of magnetic flows for spherical manifolds.*

** Abstract:**

Given a magnetic system (M,g,\sigma), with (M,g) closed Riemannian manifold and
\sigma closed 2-form on M (non necessarily exact or weakly-exact), we define for any k>0
the action 1-form \eta_k, whose zeros correspond to the periodic orbits of the magnetic
flow defined by the pair (g,\sigma) with energy k. We then show a compactness property for
critical sequences of \eta_k with periods bounded and bounded away from zero. Under the
additional assumption that the second fundamental group of M does not vanish, this crucial
property will allow us to prove the existence of a closed contractible magnetic geodesic
with energy k for almost every energy k>0. This is a joint work with Gabriele Benedetti
(Westfälische Wilhelms-Universität Münster).

**13.01.2015** Frederic LeRoux (Paris), * A dynamical construction of Hamiltonian spectral invariants on surfaces.*

** Abstract:**

Inspired by Le Calvez’ theory of transverse foliations for dynamical systems of surfaces, we introduce a new dynamical invariant, denoted by N, for Hamiltonians of the plane and closed surfaces with positive genus. We prove that, on the set of autonomous Hamiltonians, this invariant coincides with the spectral invariants constructed by Viterbo on the plane and Schwarz on closed surfaces of positive genus.
Work in collaboration with Vincent Humilière and Sobhan Seyfaddini.

**16.12.2014** Jean Gutt (Berkeley), * Positive S1-equivariant symplectic homology as an invariant for some contact manifolds.*

** Abstract:**

We will see how positive S1-equivariant symplectic homology allows one to tackle questions about the number of non-diffeomorphic contact structures on the sphere, or the minimal number of periodic Reeb orbits on some contact manifolds. This will be done by establishing properties of positive S1-equivariant symplectic homology, namely a computational property, a functoriality property and an invariance property.

**9.12.2014** Rodolfo Rios-Zertuche (Bonn), * The variational structure of the space of holonomic measures.*

** Abstract:**

We introduce a space of measures that represent submanifolds, and we obtain a set of stability conditions that are strictly more general than the Euler-Lagrange equations. To show this, we give three examples that, respectively, recover those equations, produce higher-dimensional analogues of energy conservation, and give a very general version of the weak KAM theorem. We also discuss how these stability conditions imply regularity properties of the critical points.

**2.12.2014** Stefan Suhr (Paris), * A Counterexample to Guillemin's Zollfrei Conjecture.*

** Abstract:**

Guillemin calls a compact Lorentzian 3-fold "Zollfrei" if the geodesics
flow on the
nonzero lightlike vectors induces a fibration by circles (especially all
lightlike geodesics
are closed). He conjectured that these metric can only exist 3-folds
covered by
$S^2\times S^1$. I will explain a construction of counterexamples on every
circle bundle
over a closed surface. If time permits discuss under what additional
assumptions the
conjecture holds and hint at what is the right conjecture in the general
case.

**25.11.2014** Alessandro Portaluri (Università di Torino), * Index theory in Celestial Mechanics: recent results and new perspectives.*

** Abstract:**

In the last decades a zoo of new symmetric periodic collision-less orbits for the n-body problem appeared in the literature as minimizers of the Lagrangian action functional. Certainly one of the important features of such orbits, for a better understanding of the dynamics, is the knowledge of the Morse index as well as their linear (in)stability properties. A central devices for computing this index is a Morse-type index theorem and a refined computation of the Maslov index. However, a key role in order to penetrate the intricate dynamics of this singular problem is represented by the collision orbits.
In this talk, after a presentation of a new variational regularisation of the Lagrangian action functional, we will show how to define a suitable index theory for a special class of colliding trajectories.
This is a joint work with, V. Barutello, X. Hu and S. Terracini.

**28.10.2014** Silvia Sabatini (Köln), * The role of Chern numbers in the classification of the topological invariants of symplectic manifolds with symmetries .*

** Abstract:**

In this talk I will discuss some problems related to the classificationof some equivariant topological invariants of compact symplectic manifolds endowed
with a symplectic circle action.
In particular I will present some recent results of my work currently in progress,
as well as those obtained with L. Godinho in arXiv:1206.3195 [math.SG],
and A. Pelayo and L. Godinho in arXiv:1404.4541 [math.AT]
involving the Chern numbers of the manifold, and show how these can be used to:
(a) classify the equivariant cohomology ring and Chern classes when
the action is Hamiltonian
and
(b) give a lower bound on the number of fixed points when the action is not Hamiltonian
and the first Chern class of the manifold vanishes.

**21.10.2014** Nils Waterstraat (Humboldt-Universität zu Berlin) * Spectral flow, crossing forms and homoclinics of Hamiltonian systems.*

** Abstract:**

The spectral flow is an integer-valued homotopy invariant for paths of selfadjoint Fredholm operators that was introduced by Atiyah, Patodi and Singer in the Seventies. Later Robbin and Salamon developed an effective method to compute the spectral flow by signatures of certain quadratic forms, which are called crossing forms, under the additional assumption that the operators have compact resolvents.
In the first part of our talk, we recall the definition of the spectral flow, introduce crossing forms, and extend Robbin and Salamon's theory to a broader class of operators. In the second part, we apply our results to one-parameter families of Hamiltonian systems under homoclinic boundary conditions. We obtain a spectral flow formula which involves a relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, if time permits, we discuss in a third part sufficient conditions for bifurcation of homoclinic solutions of one-parameter families of nonlinear Hamiltonian systems.