Oberseminar 2013/2014
Faculties » Faculty of Mathematics » Chairs » Mathematics VII - Analysis


15.07.2014   Michela Egidi (Durham University), Principal bundle, Grassmannian and parallel transport.

The oriented k-th Grassmannian of a manifold is the space of all k-planes of the tangent bundle of the manifold taken with an intrinsic orientation. Parallel transporting a k-plane along a curve, we can either do it along an intrinsic direction or along a general one, where intrinsic means that the starting vector of the curve belongs to the k-plane and general means otherwise. Using the language of principal bundles and frame flows I will prove that any smooth function on the oriented k-th Grassmannian invariant under all intrinsic parallel transports is invariant under all parallel transports.

08.07.2014   Richard Sharp (University of Warwick), Growth and amenability on regular covers.

Let $M$ be a compact Riemannian manifold. A beautiful theorem of Brooks says that the bottom of the spectrum of the Laplacian on a regular cover of $M$ is equal to zero if and only if the covering group is amenable. In the case where $M$ has negative sectional curvatures, we will give an analogous result for the growth of closed geodesics. This is joint work with Rhiannon Dougall.

03.06.2014   Felix Schmäschke (RUB), Pearl homology for pairs.

I construct a spectral sequence converging to the Floer homology of a pair of two monotone Lagrangian submanifolds intersecting cleanly. The notion of cleanly intersecting Lagrangians is a useful generalization of transversely intersecting Lagrangians and naturally arises for instance in situations with present symmetries. The spectral sequence is obtained via an invariant, which I call pearl homology for pairs, because it is a direct generalization of the pearl homology associated to a single monotone Lagrangian as introduced by Biran and Cornea. As an application, I deduce from the spectral sequence new obtructions on the topology of the fixed point set of an Hamiltonian diffeomorphism.

06.05.2014   Jan Philipp Schröder (RUB), Minimal rays on surfaces of genus greater than one.

For Finsler metrics on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove that for almost all asymptotic directions (i.e. points at infinity) the minimal rays with these directions laminate the universal cover and that the Busemann functions with these directions are unique up to adding constants. Moreover, we show that for almost all pairs of points at infinity, there is a unique minimal geodesic connecting these two points. If time permits, we indicate future research in the area.

29.04.2014   Urs Fuchs (RUB), Gromov compactness revisited.

I will outline a proof of Gromov compactness for closed pseudoholomorphic curves by following Gromov's original article and emphasizing the study of the metrics induced on pseudoholomorphic curves. Then I will indicate, how doubling constructions allow to carry over this proof to situations, where the pseudoholomorphic curves have totally real or otherwise controlled boundary. This talk relies on joint work with Lizhen Qin.

08.04.2014   Gabriele Benedetti (Cambridge University), The contact property for non-exact magnetic flows on surfaces.

The aim of this talk is to give a qualitative description of the dynamics of a charged particle on a surface moving under the effect of a non-exact magnetic force. Our approach is based on the study of the contact property of the level sets of the kinetic energy of the particle sitting in the tangent bundle of the surface. In particular, for the case of a nowhere vanishing magnetic field on the two-sphere, we prove that there exist either two or infinitely many periodic motions on every low kinetic energy level.

WINTER 2013/14

28.01.2014   Leonardo Macarini (Rio de Janeiro), Dynamical convexity and elliptic orbits for Reeb flows.

A classical conjecture in Hamiltonian Dynamics states that the Reeb flow of any convex hypersurface in even-dimensional euclidean space carries an elliptic closed orbit. Dell'Antonio-D'Onofrio-Ekeland proved it in 1995 for antipodal invariant convex hypersurfaces. In this talk I will present a generalization of this result using contact homology and a notion of dynamical convexity first introduced by Hofer-Wysocki-Zehnder for tight contact forms on the 3-sphere. Applications include geodesic flows under pinching conditions, magnetic flows and toric contact manifolds. This is joint work with Miguel Abreu.

14.01.2014   Fabian Schwarzenberger (Chemnitz), On the Existence of the Integrated Density of States.

Ich habe vor kurzem zum Thema "die integrierte Zustandsdichte (IDS) für Operatoren auf Cayley Graphen" promoviert. Dabei habe ich mich insbesondere mit der Existenz der IDS für verschiedene Operatoren (deterministisch/zufällig) auf unterschiedlichen Geometrien (gegeben durch endlich erzeugte Gruppen) beschäftigt. Die benötigten Methoden kommen zum Beispiel aus der Large Deviations-Theorie und Ergodentheorie.

17.12.2013   Patrice Le Calvez (Paris), Homological discrete Conley index of isolated invariant acyclic continua

The non existence of minimal homeomorphisms in a punctured two-dimensional sphere can be explained by using the discrete Conley index. In a joint work with Luis Hernandez-Corbato and Francisco Ruiz del Portal, we state some general properties of the discrete homological Conley index of an acyclic continuum (e.g. a cellular set or a fixed point) that is invariant by a homeomorphism of R^d and locally maximal. In case where d=3 and f is orientation reversing, we deduce that its Lefschtez index is smaller than 2. As a corollary, we prove that there are no minimal orientation-reversing homeomorphisms in \R^3.

10.12.2013   Sobhan Seyfaddini (ENS Paris), The displaced disks problem via symplectic topology

We will show that a C^0-small area preserving homeomorphism of S^2 can not displace a disk of large area. This resolves the so-called displaced disks problem posed by F. Béguin, S. Crovisier, and F. Le Roux.

3.12.2013   Pedro Salomão (São Paulo), 3-Dimensional Reeb flows and finite energy foliations

In this talk I will discuss some results on nite energy foliations in symplectizations of 3-manifolds equipped with contact forms. The first result is a criterium for a given Reeb orbit to be the binding orbit of a ra- tional open book decomposition with disk-like pages for lens spaces equipped with universally tight contact forms. This is joint work with J. Licata (Aus- tralian National University) and U. Hryniewicz (Federal University of Rio de Janeiro). The second result, which is a work in progress with J. Fish (IAS), U. Hryniewicz and R. Siefring (Max-Planck Institut), provides more general foliations on connected sums of lens spaces, where the set of binding orbits include hyperbolic Reeb orbits with Conley-Zehnder index 2. If time permits, I will also discuss on a recent result with Naiara de Paulo (University of São Paulo) on systems of transversal sections near certain critical energy levels of Hamiltonian systems in R^4.

3.12.2013   Umberto Hryniewicz (Rio dei Janeiro), Multiplicity results for closed Reeb orbits in three dimensions

In this talk I will discuss how certain simple versions of contact homology can be used to rigorously obtain closed Reeb orbits in various speci fic situations of dynamical interest, with no issues of transversality. I will present applications of these techniques to geodesic flows, focusing on a Poincare-Birkhoff type theorem for Reeb flows on the tight three- sphere. Some of the results that I intend to discuss are joint work with Al Momin and Pedro A. S. Salomao.

26.11.2013   Enrico Valdinoci (WIAS Berlin), Phase separation and minimal surfaces, from local to nonlocal

I would like to discuss some questions related to some semilinear equations driven by a nonlocal elliptic operator (for example, the Allen-Cahn equation, in which the classical Laplace operator is replaced by a fractional Laplacian). In particular, I would like to study the qualitative properties of the solutions, such as symmetry, density estimates of the level set, asymptotic behaviors, etc. The limit interfaces of these problems are related to both the local and the nonlocal perimeter functionals on this topic, I would like to discuss some recent rigidity and regularity results and to present some open problems.

5.11.2013   Richard Siefring (MPI Leipzig), Tba

22.10.2013   Ana Rechtman (Université de Strasbourg), On the minimal set of Kuperberg's plug

In 1993 K. Kuperberg constructed examples of smooth and real analytic ows without periodic orbits on any closed 3-manifold. These examples continue to be the only known examples with such properties and are constructed using plugs. After reviewing K. Kuperberg's construction, I will present part of a study of the minimal set in these plugs. Under some generic assumptions, the minimal set has topological dimension 2 and is strati ed: the 1-dimensional strata has two connected components, both dense in the 2-dimensional strata. I will explain the main properties of this set.

08.10.2013   Will Merry (ETH Zürich), Orderability and the Weinstein Conjecture

In 2000 Eliashberg-Polterovich introduced the natural notion of orderability of contact manifolds; that is, the (non)existence of positive loops of contactomorphisms. I will explain how one can study orderability questions using the machinery of Rabinowitz Floer homology. Our main result is that the Weinstein Conjecture holds (i.e. there exists a closed Reeb orbit) whenever there exists a positive (not necessarily contractible) loop of contactomorphisms. This applies for example to all (non-trivial) prequantization spaces of integral symplectic manifolds. This is a joint work with Peter Albers and Urs Fuchs.