## WINTER 2016/17

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

**27.6.2017** Milena Pabiniak (Cologne), * The contact version of Arnold Conjecture (by S.Sandon) for lens spaces via a non-linear Maslov index. *

** Abstract:**

Diffeomorphisms in symplectic category posses certain rigidity properties. An important manifestation of rigidity is given by the conjectures posed by V. Arnold describing a lower bound for the number of fixed points of a Hamiltonian diffeomorphism of a compact symplectic manifold, greater than what topological arguments could predict.
Arnold Conjectures present a difficult problem and motivated a lot of important research in symplectic geometry. It has been translated by S. Sandon to the contact geometry setting where one looks for a lower bound for the number of translated points.
Givental's construction of a quasimorphism, called the non-linear Maslov index, allows one to prove the Arnold Conjecture for complex and real projective spaces. Moreover, the properties of this quasimorphism imply that the real projective space is orderable, has a non-displaceable pre-Lagrangian and that its discriminant and oscillation norms are unbounded.
In this talk I will describe my work joint with G. Granja, Y. Karshon and S. Sandon, aimed at constructing a quasimorphism for lens spaces (building on the ideas of Givental) and proving the corresponding statements for these spaces (Contact Arnold Conjecture, orderability, ... ).
I will discuss the difficulties of constructing such a generalization to lens spaces and, if time permits, the possibility of generalizing these ideas even further: to prequantizations of symplectic toric manifolds.

**7.2.2017** Umberto Hryniewicz (Rio de Janeiro), * Linking and global surfaces of section *

** Abstract:**

In this talk I'd like to explain how Schwarzman-Fried-Sullivan theory
can be used to answer the following question: When does a collection
of periodic orbits of a vector field on a 3-manifold bound a global
surface of section? The answer is given in terms of linking assumption
with Schwartzman cycles. Then we explain how Hofer-Wysocki-Zehnder
theory of finite-energy curves can be used to sometimes prove stronger
results for the special case of Reeb vector fields. This is joint work
with Pedro Salomao and Kris Wysocki, and with Joan Licata.

**2.2.2017** Maylis Limouzineau (Universite Paris 6, UPMC), * Constructions of Legendrian cobordism and generating functions.*

** Abstract:**

**31.1.2017** Jian Wang (MPI Leipzig), * The rigidity of toral diffeomorphisms *

** Abstract:**

In this talk, we will give an estimate on the maximal displacement of a
pseudo-rotation homeomorphism of two-torus with respect to its rotation
vector when the homeomorphism satisfies the bounded deviation condition.
As a consequence, we prove that a pseudo-rotation diffeomorphism $f$ of
class $C^k (k\geq 2$) is $C^{k-1}$-rigidity when the rotation vector of
$f$ is totally irrational (resp. irrational but not totally irrational)
non-Brjuno case and $f$ satisfies the bounded mean motion (resp. bounded
deviation condition).

**17.1.2017** Michael Hutchings (Berkeley), * Reeb orbits and contact volume*

** Abstract:**

Consider a closed three-manifold with a contact form and its associated Reeb vector field. There is a relation between the asymptotics of the ECH spectrum (determined by the periods of certain periodic orbits of the Reeb vector field) and the contact volume, established in joint work with Dan Cristofaro-Gardiner and Vinicius Ramos. We review this relation and some of its recent applications to Reeb dynamics.

**17.1.2017** Jo Nelson (Columbia), *An integral lift of contact homology*

** Abstract:**

Contact geometry is the study of certain geometric
structures on odd dimensional smooth manifolds. A contact structure
is a hyperplane field specified by a one form which satisfies a
maximum nondegeneracy condition called complete non-integrability. The
associated one form is called a contact form and uniquely determines a
vector field called the Reeb vector field on the manifold. I will
explain how to make use of J-holomorphic curves to obtain a Floer
theoretic contact invariant whose chain complex is generated by closed
Reeb orbits. In particular, I will explain the pitfalls in defining
contact homology and discuss my work (in part joint with Hutchings)
which gives a rigorous construction of cylindrical contact homology
via geometric methods. This talk will feature numerous graphics to
acclimate people to the realm of contact geometry.

**13.12.2016** Pedro Salomão (São Paulo), * Legendrian contact homology on the complement of Reeb orbits and topological entropy.*

** Abstract:**

A celebrated theorem of Li-Yorke states that a continuous map of the interval possessing a period three periodic point admits a subset where the dynamics is chaotic. I will present a related result for Reeb flows on contact 3-manifolds: the existence of certain periodic orbits implies positivity of topological entropy. This generalizes a result of Denvir and Mackay which asserts that a geodesic flow on the 2-torus with a simple contractible closed geodesic has positive topological entropy. This result relies on a version of Legendrian contact homology on the complement of Reeb orbits. This is joint work with Marcelo Alves (Univ. de Neuchâtel).

**29.11.2016** Iskander Taimanov (Novosibirsk), * The spaces of non-contractible closed curves in compact space forms*

** Abstract:**

We calculate the rational equivariant cohomology of the spaces
of non-contractible loops in compact space forms and show how to apply
these calculations for proving the existence of closed geodesics.

**29.11.2016** Marcelo Alves (Neuchâtel), * Symplectic rings and modules, their algebraic growth and applications to dynamics*

** Abstract:**

Inspired by ideas from Geometric Group Theory, Matthias Meiwes recently introduced the notion of algebraic growth of Floer theoretical invariants of symplectic/contact manifolds. These notions turn out to be powerful tools to study the growth rate of symplectic invariants of Liouville domains, because they are stable under several geometric modifications of Liouville domains. As one application of these ideas we obtain several new examples of high-dimensional contact manifolds with positive topological entropy. These examples include contact structures with positive topological entropy on "almost every" simply-connected 5-manifold, and on spheres of sufficiently high dimension. This is joint work with Matthias Meiwes.

**22.11.2016** Martin Schmoll (Clemson University, USA), * Billiards and the Teichmueller flow.*

** Abstract: **

This is an overview talk where we briefly provide a background on the Teichmueller flow and its connection to polygonal billiards.
A gainful transformation to Teichmueller dynamics is possible for a billiard in rational angled polygons. In fact a rational billiard table unfolds to a compact Riemann surface with a linear flow for which billiard trajectories appear as straight lines. Stretching the Riemann surface in the trajectory direction ad shrinking in the perpendicular one while keeping the area constant defines the Teichmueller flow in modul space. Studying billiards via properties of its associated Teichmueller dynamics has triggered a large program that came to live through the last two decades. At the foundation of the method are orbit closure results, such as the one for genus 2 by Curtis T. McMullen and a recent one by Eskin and Mirzakhani, roughly saying that $SL(2,R)$-orbit closures in moduli space are manifolds.
For our talk we will pick a selection of questions (on billiards) that can be answered by using the translation into Teichmueller dynamics.

**15.11.2016** Murat Saglam (MPI Leipzig), * A search for finer topological information via holomorphic curves in the symplectizations: the case of lens
spaces and their unit cotangent bundle.*

** Abstract:**

It is classically known that two lens spaces L(p,q) and L(p,q')
and their products with S^2 are diffeomorphic if and only if q'\equiv \pm
q^{-\pm 1} in mod p. The non-trivial part of this statement is established
via the Reidemeister torsion.
In this talk, we explore the possibility of capturing the above relation
using holomorphic curves in the symplectizations and symplectic
cobordisms. We endow
lens spaces with the contact form that is the quotient of the standard
contact form on S^3 and in the case of unit cotangent bunndles, the
contact form is the quotient of the contact form on the unit cotangent
bundle of S^3 induced by the round metric. In both cases, the moduli
spaces of curves with at most two non-contractible ends are easy to be
described. Using these moduli spaces in a neck-stretching procedure, we
aim to show that given a contactomorphism between two lens spaces or two
unit cotangent bundles, the above relation is fulfilled. It turns out
that, this is not possible using the standard
methods, which is expected because of the theoretical background of the
problem.
After pointing out what goes wrong along the procedure, we show that
imposing conditions on the contactatomorphism solves both the technical
and essential issues of the procedure. In the case of unit cotangent
bundles, the conditions we impose are global bounds on the
contactomorphism, while in the case of lens spaces, the condition is the
strictness along a single contractible orbit.

**8.11.2016** Joanna Janczewska (Gdansk) and Marek Izydorek (Gdansk), * Strong force Hamiltonian systems.*

** Abstract:**

First talk:

We will consider a planar Newtonian system $\ddot{q}+\nabla V(q)=0$
with a potential $V\colon\mathbb{R}^2\setminus\{\xi\}\to\mathbb{R}$
possessing a singularity at a point $\xi$: $V(x)\to -\infty$ as $x\to\xi$,
and a strict global maximum $0$ that is achieved at two distinct points
$a$ and $b$ in $\mathbb{R}^2\setminus\{\xi\}$.
Applying a variational approach we will establish the existence of
homoclinic and heteroclinic solutions winding around $\xi$ provided that
nearby the singularity the potential $V$ satisfies a strong force condition.

Second talk:

We study the existence of homoclinic solutions for a system
of the type $(\nabla\Phi(\dot{u}(t)))'+ V_u(t,u(t))=0$,
where $t\in\R$, $\Phi\colon\R^2\to[0,\infty)$ is a $G$-function
in the sense of Trudinger,
$V\colon\R\times\left(\R^2\setminus\{\xi\}\right)\to\R$
is a $C^1$-smooth potential with a single well of infinite depth
at a point $\xi\in\R^2\setminus\{0\}$
and a unique strict global maximum $0$ at the origin.
Under a strong force condition around the singular point $\xi$,
via minimization of an action integral, we will prove the existence
of at least two geometrically distinct homoclinic solutions
$u^{\pm}\colon\R\to\R^2\setminus\{\xi\}$.