Faculties » Faculty of Mathematics » Chairs » Mathematics VII - Analysis

WINTER 2019/20

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

15.10.2019   Simon Allais (ENS, Lyon) "Application of generating functions to symplectic and contact rigidity"

Abstract: In 1992, Viterbo introduced new tools to study the Hamiltonian dynamics of ${\mathbb R}^{2n}$ by applying Morse-theoretical methods to generating functions. Among his results, he gave a new proof of Gromov's non-squeezing theorem (1985) and sketched a proof of the more subtle symplectic camel theorem. A part of this work was generalized to the contact case by Sandon (2011) who provided a new way to derive the contact non-squeezing theorem of Eliashberg, Kim and Polterovich (2006). We will recall the main points of this theory and show how it allows us to derive a contact analogue of the symplectic camel theorem.

22.10.2019   Christian Lange (Köln) "Rigidity of Zoll magnetic systems on surfaces"

Abstract: I report on recent work joint with Luca Asselle on rigidity phenomena of Zoll magnetic systems on surfaces. In particular, we show that on a surface of positive genus constant curvature metrics and constant magnetic functions provide the only examples of magnetic systems whose Hamiltonian flow has all orbits closed, on every energy level.

05.11.2019   Beatrice Pozzetti (Heidelberg) "Orbit growth rate in higher rank Teichmüller theories"

Abstract: Higher rank Teichmüller theories are unexpected connected components of the variety of homomorphisms of the fundamental group of a hyperbolic surface in a semisimple Lie group, that only consist of injective homomorphisms with discrete image. They thus generalize the Teichmüller space, and can be thought of as parametrizing certain locally symmetric spaces of infinite volume. After motivating the study of higher rank Teichmüller theories, I will discuss joint work with Andres Sambarino and Anna Wienhard in which we prove a sharp upper bound for the exponential orbit growth rate of the associated actions on the symmetric space.

19.11.2019   Nikhil Savale (Köln) “Sub-leading asymptotics of ECH capacities“

Abstract: On a closed, contact three manifold the asymptotics of its ECH (Embedded contact homology) spectrum are known to recover the contact volume. This has applications to the existence of at least two, and in some cases two or infinitely many, Reeb orbits as well as the density of the union of periodic Reeb orbits for generic contact forms. In this talk, we improve the asymptotic formula for the ECH spectrum with a subleading estimate. This has applications to a Weyl law for the ECH spectrum and the region of analyticity of the ECH zeta function.

10.12.2019   Ettore Minguzzi (Florenz) "A gravitational collapse singularity theorem consistent with black hole evaporation and chronology violation"

Abstract: The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work I show that the causality conditions in Penrose's theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in absence of predictability and hence compatibly with quantum field theory and black hole evaporation.

10.12.2019   Dr. Wolfgang Schmaltz (Gießen) "The Steenrod problem for orbifolds and polyfold Gromov-Witten invariants"

Abstract: Following the approach of Thom, we solve the Steenrod problem for closed orientable orbifolds, proving that the rational homology groups of a closed orientable orbifold have a basis consisting of classes represented by suborbifolds whose normal bundles have fiberwise trivial isotropy action. Using this, we demonstrate that the polyfold Gromov–Witten invariants, originally defined via integration of differential forms, may equivalently be defined as intersection numbers against a basis of representing suborbifolds. Thus, the traditional geometric interpretation of the Gromov–Witten invariants as a "count of curves'' is made literal.

21.01.2020   Annegret Burtscher (RU, Nijmegen) "On the metric structure and convergence of spacetimes"

Abstract: Riemannian manifolds naturally carry the structure of metric spaces, and standard notions of metric convergence interact with the Riemannian structure and (weak) curvature bounds. No such theory is yet available for Lorentzian manifolds because the distance function does not give rise to an obvious metric structure. For spacetimes with suitable time functions, however, the recently introduced null distance of Sormani and Vega provides an alternative metric that naturally interacts with the causal structure and yields an integral current space. We compare different notions of convergence for the null distance of warped product spacetimes, in particular, we show that uniform, Gromov-Hausdorff and Sormani-Wenger intrinsic flat convergence agree if the sequence of (continuous) warping functions converges uniformly. This is joint work with Brian Allen.

21.01.2020   Dario Corona (Università di Camerino) "Orthogonal Geodesics Trajectories in Manifolds with Boundary"

Abstract: Let us consider an autonomous Hamiltonian function even with respect to the momenta. Are there multiplicity results for periodic solutions of Hamilton's equations having fixed energy? For a special class of periodic solutions, called brake orbits, this problem is equivalent to a variational one for curves in a compact manifold $\Omega$ with boundary. In fact, in the case of a natural Hamiltonian function, the brake orbits are linked with orthogonal geodesic chords in $\Omega$, endowed with a Riemannian structure. In the more general case of a Hamiltonian convex function which is even with respect to the momenta, the problem is related to orthogonal geodesic chords in $\Omega$, with respect to a Finsler metric, thus a metric which generalizes the Riemannian one. In this seminar, I will present a preliminary study for multiplicity results in the case of a general Lagrangian integral and some results to study the Finsler case.

28.01.2020   Gleb Smirnov (ETH, Zürich) “Symplectic triangle inequality“

Abstract: We describe necessary and sufficient conditions for the existence of a Lagrangian projective plane in a three-fold blow-up of the symplectic ball.

28.01.2020   Stephan Mescher (Leipzig) "Spherical complexities and closed geodesics"

Abstract: In the 1930s, L. Lusternik and L. Schnirelmann discovered a fundamental relation between a homotopy invariant of a manifold and the minimal number of critical function on the manifold. This relation has subsequently been generalized and extended to various settings. In my talk, I will introduce new integer-values homotopy invariants of spaces that are well-suited to study critical points of functions on loop and sphere spaces, so called spherical complexities. After defining these invariants and outlining a corresponding Lusternik-Schnirelmann-type theorem, I will apply this result to energy functionals on free loop spaces of closed Riemannian/Finsler manifolds and discuss how additional topological arguments lead to a new existence result for closed geodesics (e.g. on complex projective spaces) of pinched Riemannian/Finsler metrics that requires a cohomology assumption, but no nondegeneracy assumption on the metric.