## SPRING 2019

The talks start at 16.15 in **Seminar Room IA 1/181 **.

**30.04.2019** Anna Florio (Université d'Avignon) *"Torsion of C^1 surface diffeomorphisms and homoclinic transverse intersections"
*

** Abstract:** For a C^1 diffeomorphism isotopic to the identity on a Riemannian parallelizable surface f : S → S, the torsion, denoted as Torsion(f), is the limit of the average rotational velocity of tangent vectors under the action of the linearized dynamical system. We will give the definition of torsion as well as some examples. In particular we will discuss what can be said about torsion for fixed hyperbolic points with transverse homoclinic intersections.

**14.05.2019** Michela Egidi (TU Dormtund) *"On eigenvalue estimates of the magnetic Laplacian on a Riemannian manifold"
*

** Abstract:** In this talk we consider the magnetic Laplacian on a compact Riemannian manifold without boundary with a magnetic potential and we discuss eigenvalue estimates similar to the ones obtained in the Lichnerowicz Theorem and the Buser estimate in the classical case. These estimates relates eigenvalues, Ricci curvature, and Cheeger-type constant.

**28.05.2019** Paolo Giulietti (De Giorgi Center, Pisa) *"A transfer operator approach to study statistical properties of parabolic flows"
*

** Abstract:** After a gentle introduction to transfer operator methods to study hyperbolic dynamics, I will show a general strategy to study parabolic flows when a hyperbolic renormalization strategy is available.

**04.06.2019** Albert Fathi (Georgia Tech, Atlanta) *"Recurrence on abelian cover. Application to closed geodesics in manifolds of negative curvature"
*

** Abstract:** If h is a homeomorphism on a compact manifold which is chain-recurrent, we will try to understand when the lift of h to an abelian cover (i.e. the covering whose Galois group is the first homology group of the manifold) is also chain-recurrent.
This is related to the proof by John Franks of the Poincaré-Birkhoff theorem.
It has consequences on density of classes of closed geodesics in a manifold of negative curvature, and more generally on geodesic flows which are Anosov.

**11.06.2019** Egor Shelukhin (Montreal) *"Symplectic cohomology and a conjecture of Viterbo"
*

** Abstract:** We explain how the TQFT operations introduced and studied by Seidel and Solomon in the context of symplectic cohomology and Lagrangian Floer homology yield upper bounds on the Lagrangian spectral norm of exact Lagrangians in cotangent disk bundles. These upper bounds in the case of the n-torus were conjectured by Viterbo.
Time permitting, we discuss different TQFT operations in fixed point Floer cohomology introduced by Seidel, the speaker, and Zhao, and their application to questions in dynamics, such as the Hofer-Zehnder conjecture.

**18.06.2019** Agustin Moreno (Universität Augsburg) *"5-dimensional Bourgeois contact structures are always tight"
*

** Abstract:** Starting from a contact manifold and a supporting open book
decomposition, an explicit construction by Bourgeois provides a contact
structure in the product of the original manifold with the two-torus.
Besides being a simple and elegant construction, this shows e.g. that
the 5-torus is contact, something that had been open since work of Lutz
in the seventies. On the other hand, by work of Eliashberg in dimension
3 and Borman-Eliashberg-Murphy in higher dimensions, we know that
contact manifolds are classified in two flavours: tight, or overtwisted.
In this talk, we will describe how to show that Bourgeois contact
structures are, in dimension 5, always tight, independent on the
classification type of the original contact manifold. This is joint work
with Jonathan Bowden and Fabio Gironella.

**25.06.2019** Nils Waterstraat (Martin-Luther-Universität Halle-Wittenberg) *"On the Fredholm Lagrangian Grassmannian, Spectral Flow and ODEs in Hilbert Spaces"
*

** Abstract:** We consider homoclinic solutions for Hamiltonian systems in symplectic Hilbert spaces and
generalise spectral flow formulas that were proved by Pejsachowicz and us in finite dimensions
some years ago. Roughly speaking, our main theorem relates the spectra of infinite dimensional
Hamiltonian systems under homoclinic boundary conditions to intersections of their stable and
unstable spaces. Our constructions make use of striking results by Abbondandolo and Majer to study
Fredholm properties of infinite dimensional Hamiltonian systems.

**25.06.2019** Jacobo Pejsachowicz (Politecnico di Torino) *"Topology and Bifurcation"
*

** Abstract:** A celebrated Lyapunov-Schmidt method uses the Fredholm property of the
linearization of a nonlinear equation at a point of the trivial branch in order to recast locally
a given bifurcation problem to one with a finite number of equations in a finite number
of indeterminates. Via the above reduction, assuming moreover that singular points of
the linearization are isolated, a number of sufficient conditions for bifurcation of nontrivial
solutions from the trivial branch are obtained.
In this talk I will review a different approach to bifurcation on which I have been
working in the past years. It is based on various homotopy invariants for families of
linear Fredholm operators borrowed from the elliptic topology. These invariants, together
with the J-homomorphism, arise in bifurcation theory as a tool linking the appearance
of nontrivial solutions of nonlinear equations to the nontrivial topology of the parameter
space.

**16.07.2019** Sigurður Freyr Hafstein (Reykjavík)* "Numerical methods for computing Lyapunov functions"
*

** Abstract:** Lyapunov functions correspond to dissipative free energy in
physics and characterize the long term
behavior of dynamical systems. Hence, they are of much
theoretical and practical interest. Unfortunately, their analytical
computation is close to impossible, except in the simplest cases.
We give a short introduction to the theory of (complete) Lyapunov
functions for ODEs and discuss several numerical approaches
to compute them, including methods using collocation, linear programming,
and semidefinte optimization

**16.07.2019** Viktor Ginzburg (Santa Cruz) *"Periodic orbits of Hamiltonian systems: the Conley conjecture and pseudo-rotations"
*

** Abstract:** One distinguishing feature of Hamiltonian dynamical systems is that such systems, with few notable exceptions, tend to have numerous periodic orbits. For instance, for many symplectic manifolds, every Hamiltonian diffeomorphism has infinitely many periodic orbits unconditionally. This fact, usually referred to as the Conley conjecture, has by now been established for a broad class of manifolds. However, the Conley conjecture obviously fails for some, even very simple, manifolds such as the sphere. These spaces admit Hamiltonian diffeomorphisms with few periodic orbits -- the so-called pseudo-rotations -- which are of particular interest and occupy a very special place in dynamics. Symplectic topological methods and, in particular, Floer theory turn out to be the right tools to study pseudo-rotations in all dimensions and recently a connection between the existence of pseudo-rotations and the Gromov-Witten invariants has been discovered.
We will start this talk with background results on the Conley conjecture and then focus on the dynamics of Hamiltonian pseudo-rotations and the connection between pseudo-rotations and quantum homology.

## WINTER 2018/19

The talks start at 16.15 in **Seminar Room IA 1/53 **.

**29.01.2019** Agustin Moreno (Universität Augsburg) *"Planarity in higher-dimensional contact manifolds"
*

** Abstract:** Planar contact 3-manifolds are, in some sense, amongst the simplest contact manifolds. In this talk, we describe a notion of planarity for higher-dimensional contact manifolds, called iterated planarity, which generalizes the situation in dimension 3. We will discuss examples and related notions, and describe how to extend known 3-dimensional results to the higher-dimensional planar setting, using a combination of holomorphic curve techniques and symplectic handle attachments. This is joint work with Bahar Acu.

**29.01.2019** Rodolfo Rios-Zertuche (ENS, Paris) *"Variations of curves: wiggling and beyond"
*

** Abstract:** We study the variations of curves and of slightly more general objects called closed measures on the tangent bundle. We give examples of several different classes of variations, and show that criticality with respect to each class gives different properties, like invariance under the Euler-Lagrange flow, energy conservation, Lipschitz regularity of the momenta, and existence of a global extension of the momenta to an exact form. Along the way, we characterize all minimizable Lagrangian actions as being those that dominate an exact form, which can be interpreted as a critical subsolution of the Hamilton-Jacobi equation.

**22.01.2019 at 17:00** Francisco Javier Torres de Lizaur (MPI, Bonn) *"Invariant measures of divergence free flows and the Seiberg-Witten equations"
*

** Abstract:** In this talk we will present some new results on the invariant measures of 3 dimensional divergence-free flows arising from sequences of solutions to the Seiberg-Witten equations. In the first part we will show that, locally in a flow box, any transverse measure can arise in this way. In the second part we will show how, when $X$ is a Reeb vector field, those invariant measures provide a link between the periodic orbits of $X$ and its contact Riemannian geometry. Along the way, we revisit C. Taubes solution to the Weinstein conjecture through a modified argument that does not rely on the analysis of the 2D vortex equations. This is joint work with A. Enciso and D. Peralta-Salas.

**15.01.2019** Colin Guillarmou (Orsay) *"On the marked length spectrum of Anosov manifolds"
*

** Abstract:** We discuss recent work with T. Lefeuvre on the problem of rigidity of the marked length spectrum for manifolds with Anosov geodesic flow.

**13.12.2018** Keon Choi (São Paulo) *"Embedded contact homology and symplectic embedding problems"
*

** Abstract:** Since Gromov first demonstrated rigidity of symplectic mappings in 1985, determining when a symplectic manifold embeds into another has remained a challenging problem. Little progress has been made until 2008 when McDuff showed numerical criteria for an ellipsoid embedding into another. In recent years, we have learned more about embeddings of so-called toric symplectic manifolds. These results (including Gromov's original theorem) rely on finding pseudoholomorphic curves to provide an obstruction to embedding. In this talk, we will talk about this recent progress and how embedded contact homology has been useful in providing many relevant pseudoholomorphic curves.

**04.12.2018** Naiara de Paulo (São Paulo) *"Systems of Transverse Sections for Hamiltonian Flows"
*

** Abstract:** Due to conservation of energy, one usually restricts the study of a Hamiltonian system in R4 to a fixed 3-dimensional energy level. The existence of a global surface of section in such a level provides additional reduction of the flow to an area preserving surface map. In case global sections do not exist or they are unlikely to be found, one may still search for the so called systems of transverse sections. These systems are singular foliations of the energy level so that the singular set is formed by finitely many periodic orbits and the regular leaves are transverse to the Hamiltonian vector field. The Hamiltonian flow determine transition maps between some regular leaves of a system of transverse sections and such maps may provide valuable infor- mation about the Hamiltonian dynamics, such as the multiplicity of periodic orbits and homoclinics. In this talk I will discuss some results obtained with Pedro Salomão (University of São Paulo) on that regard. I will also discuss results on the existence of systems of transverse sections applied to classical Hamiltonian problems, namely the Euler problem of two fixed centers and the Hénon-Heiles Hamiltonian, which are works in progress with Pedro Salomão and Umberto Hryniewicz (Federal University of Rio de Janeiro), and with Alexsandro Schneider (Unicentro) and André Vanderlinde (Federal University of Santa Catarina), respectively.

**20.11.2018** Jaime Bustillo (ENS, Paris) *"A coisotropic non-squeezing theorem in symplectic geometry"*

** Abstract:** I will explain how generating functions and Viterbo's capacities
can be used to prove a coisotropic non-squeezing theorem for Hamiltonian
diffeomorphisms of R^{2n} generated by sub-quadratic Hamiltonians H. We
will then see the relation of this theorem with the middle dimensional
symplectic rigidity problem.

**13.11.2018** Yaniv Ganor (Tel Aviv) * “A homotopical viewpoint at the Poisson bracket invariants for tuples of sets"
*

** Abstract:** Poisson bracket invariants for triples and quadruples of sets were introduced by Buhovski, Entov and Polterovich (2012) as a means to study the C^0 symplectic topology of closed subsets of a symplectic manifold. They were found to have applications to the study of Hamiltonian chords (Entov-Polterovich, 2016) and were also applied to the study of a symplectic topological invariant of Lagrangian submanifolds (Entov-G-Membrez, 2016). Interestingly, they manifest various aspects of both rigid and flexible phenomena.
We suggest a homotopically flavoured description of the Poisson bracket invariants for tuples of closed sets in symplectic manifolds. It implies a flexibility themed result; namely, that in fact these invariants depend only on the union of the sets, along with some topological data which encodes the manner of decomposition into a tuple of sets.

**23.10.2018** Luca Asselle (Giessen) “On Zoll magnetic flows on the two-torus”

** Abstract:** Magnetic systems for which the flow induces a free circle action
on the unit tangent bundle are a natural generalization of Zoll
geodesic flows and are relevant for instance in low-dimensional
systolic geometry, as they satisfy a local systolic inequality
analogous to the one satisfied by Zoll metrics on the two-sphere.
In this talk, after recalling definitions and basic properties,
I will present some recent results about the structure of the
set of magnetic pairs defining Zoll systems on the two-torus.
If time permits I will also discuss some of the open problems
in this topic. This is joint work with Gabriele Benedetti.

**16.10.2018** Egor Shelukhin (Montreal)* “On the Lagrangian spectral norm and barcodes”*

** Abstract:** We discuss upper and lower bounds on the spectral norm in Lagrangian Floer theory, their implications for the associated barcodes, and additional applications. This talk is partially based on joint work with Asaf Kislev.

**25.9.2018** Leonid Polterovich (Tel Aviv) *“Quantum manifold learning”
*

** Abstract:** Does a semiclassical particle remember the phase space topology? We discuss this question in the context of the Berezin-Toeplitz quantization and quantum measurement theory by using tools of topological data analysis.