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

## WINTER 2022/23

The talks start at 16.15 .

24.01.2023   Leonardo Garcia Heveling (Radboud) "tba"        in Seminar Room IA 1/53

22.11.2022   Dusan Joksimovic (Paris) "A Hölder-type inequality for the $C^0$ distance and Anosov-Katok pseudo-rotations"                                                                            in Seminar Room IA 1/53

Abstract: In this talk, we will show that sufficiently fast convergence in Hofer/spectral metric forces $C^0$ convergence. We achieve this by proving a H\"older-type inequality for Hamiltonian diffeomorphisms relating the $C^0$ norm, the $C^0$ norm of the derivative, and the Hofer/spectral norm. As an application of our H\"older-type inequality, we prove $C^0$ rigidity for a certain class of pseudo-rotations. In the first part of the talk, we will state the main results and prove the inequality. In the second part, we will introduce the class of Anosov-Katok pseudo-rotations (AKPRs) and prove (using the inequality) that exponentially Liouville AKPRs are $C^0$ rigid. This talk is based on joint work with Sobhan Seyfaddini.

15.11.2022   Sheila Sandon (Strasbourg) "Contact non-squeezing at large scale via generating functions"                                                                                                    in Seminar Room IA 1/53

Abstract: In 2006 Eliashberg, Kim and Polterovich discovered a non-squeezing phenomenon in contact topology: they proved that if \pi r^2 < k < \pi R^2 for some integer k then the prequantization in R^2n x S^1 of the ball of radius R cannot be squeezed by a contact isotopy into the prequantization of the ball or radius r. On the other hand, by a geometric construction based on the existence of a positive contractible loop of contactomorphisms of the sphere S^{2n-1} for n > 1, they also proved that if \pi R^2 < 1 and n > 1 the prequantization of the ball of radius R can be squeezed into the prequantization of the ball of radius r for r arbitrarily small. The case 1 < \pi r^2 < \pi R^2 with no integers between \pi r^2 and \pi R^2 was settled by Chiu in 2017 and Fraser in 2016 using respectively holomorphic curves and microlocal sheaves: they proved that also in this case non-squeezing holds. In 2011 I gave a generating functions proof of the non-squeezing theorem of Eliashberg, Kim and Polterovich, in which an important role was played by translated points of contactomorphisms. In my talk I will present a joint work in progress with Maia Fraser and Bingyu Zhang to obtain a generating functions proof of the general non-squeezing result of Chiu and Fraser, in which a key role is played by translated chains, a generalization of translated points.

08.11.2022   Valerio Assenza (Heidelberg) "Magnetic Curvature and Existence of Closed Magnetic Geodesics"                                                                                                    in Seminar Room IA 1/53

Abstract: A Magnetic System is the toy model for the motion of a charged particle moving on a Riemannian Manifold endowed with a magnetic field. Solutions for such systems are called Magnetic Geodesics and preserve the Kinetic Energy. One of the most relevant investigative interest in the theory is to understand the existence and in case the topological nature of Closed Magnetic Geodesic (periodic solution) in a given level of the energy. I will introduce the Magnetic Curvature, an object which encodes the geometrical properties coming from the Riemannian Curvature structure together with terms of perturbation due to the magnetic interaction. We will see how a positive curved Magnetic System carries a Contractible Closed Magnetic Geodesic for small energies.

25.10.2022   Miguel Paternain (CMAT, Montevideo) "Some applications of loop bundles and algebras of loops"                                                                                                in Seminar Room IA 1/53

Abstract: We introduce a Lie algebra of curves slightly extending Goldman construction. We discuss an algebraic characterization of simple curves in a surface with boundary in terms of the Lie bracket. We shall also show applications of the loop bundle to cohomological equations and Anosov diffeomorphisms.

18.10.2022   Juan Manuel Burgos (Cinvestav, Mexico City) "On the Lagrange-Dirichlet converse in dimension three"                                                                                        in Seminar Room IA 1/53

Abstract: In the context of mechanical systems, the Lagrange-Dirichlet Theorem states that a strict minimum of the potential is a Lyapunov stable equilibrium point. In his treatise of 1892, A. M. Lyapunov recognizes that the converse is non trivial and solves the first partial result. Since then, in the class of real analytic potentials, this problem remains as an open conjecture. After reviewing the state of the art of the problem and possible alternative approaches, It will be shown that for a real analytic potential in dimension three, there is an open and dense subset of the set of non-strict local minimums of the potential whose points are Lyapunov unstable equilibria. This will be achieved by proving a new instability criterion whose main ingredient is the new notion of weakly logarithmic vector field introduced in this work. This is joint work with Miguel Paternain (https://arxiv.org/abs/2208.01139).

## Spring 2022

The talks start at 16.15 .

12.07.2022   Jakob Hedicke (RUB) "On the global hyperbolicity of the positively elliptic region"        in Seminar Room IA 1/181

Abstract: It was recently observed by Abbondandolo, Benedetti and Polterovich, that the positive semi-definite symmetric matrices define a conjugation invariant closed cone structure on the linear symplectic group. We show that the open subset of positively elliptic symplectic matrices is globally hyperbolic.

05.07.2022   Erman Cineli (Paris) "Topological entropy and Floer theory"        in Seminar Room IA 1/181

Abstract: In this talk I will introduce barcode entropy and discuss its connections to topological entropy. The barcode entropy is a Floer-theoretic invariant of a compactly supported Hamiltonian diffeomorphism, measuring, roughly speaking, the exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. The topological entropy bounds from above the barcode entropy and, conversely, the barcode entropy is bounded from below by the topological entropy of any hyperbolic locally maximal invariant set. As a consequence, the two quantities are equal for Hamiltonian diffeomorphisms of closed surfaces. The talk is based on a joint work with Viktor Ginzburg and Basak Gurel.

17.05.2022   Alessandro Portaluri (Torino) "An index theory for asymptotic motions in the gravitational N-body problem"                                                                                          in Seminar Room IA 1/181

Abstract: Completely parabolic and total colliding trajectories are the basic representatives of a large class of asymptotic motions. In this talk we sketch the construction of an index theory for such classes of motions. Both problems suffer from a lack of compactness and can be brought in a similar form of a Lagrangian system on the half (time) line by a regularizing change of coordinates which preserves the Lagrangian structure. We introduce a Maslov-type index which is suitable to capture the asymptotic nature of these trajectories as half-clinic orbits and we develop the relative index theory by proving the relation with the Morse index of these trajectories as critical points of the Lagrangian action functional. If time permits, we discuss asymptotic estimates for the growth of the Morse index for such classes of solutions as well as possible applications of non-action minimization methods in the Newtonian N-body problem. This talk is based on a recent joint work with Barutello, Hu and Terracini.

10.05.2022   Yann Chaubert (Paris) "Counting closed geodesics under intersection constraints"                                                                                                                           in Seminar Room IA 1/181

Abstract: On a closed negatively curved surface, Margulis gave the asymptotic growth of the number of closed geodesics of bounded length, when the bound goes to infinity. A natural question is: can we obtain similar counting results for closed geodesics satisfying some (topological or geometrical) constraints? After a short state of the art on this issue, we will discuss some recent results concerning geometric intersection constraints. Namely, we will give the asymptotic growth of closed geodesics for which certain intersection numbers (with a given family of simple closed geodesics) are prescribed. The proof involves a dynamical scattering operator related to the surface (with boundary) obtained by cutting our original surface along the simple curves.

## WINTER 2021/22

The talks start at 16.15 .

18.01.2022   Thibault Lefeuvre (Sorbonne Paris) "On the ergodicity of the frame flow on negatively-curved manifolds"    via Zoom

Abstract: The frame flow on negatively-curved manifolds is one of the first historical examples of partially hyperbolic dynamics. It is known that this flow is ergodic on nearly-hyperbolic manifolds and on odd-dimensional manifolds (dimension not equal to 7). On the contrary, this flow is never ergodic on Kähler manifolds (e.g. complex hyperbolic manifolds). Brin thus naturally conjectured in the 70s that even-dimensional manifolds with 1/4-pinched curvature should have an ergodic frame flow but this question is still widely open today. In this talk, I will explain recent progress achieved on this conjecture: I will show that 4k+2-dimensional (resp. 4k-dimensional) manifolds with ~0.27-pinched curvature (resp. ~0.55-pinched curvature) have an ergodic frame flow. This new approach combines three tools: 1) hyperbolic dynamics (transitivity group, representations of Parry's free monoid), 2) reduction of structure groups on spheres, 3) harmonic analysis on the unit tangent bundle (twisted Pestov/Weitzenböck identities). Joint work with Mihajlo Cekić, Andrei Moroianu, Uwe Semmelmann.

11.01.2022   Shahriar Aslani (ENS Paris) "Mañe generic properties of non-convex Hamiltonians"    in Seminar Room IA 1/53

Abstract: In this talk I will introduce a certain Mañe generic property of non-convex Hamiltonians. A property (g) is called Mañe generic for a given C2 Hamiltonian H : T ∗M → R, if there exists a residual subset of potentials R ⊂ C∞(M) such that for all u ∈ O, H + u satisﬁes (g). Mañe perturbations are closely related to conformal perturbations of Riemannian metrics. If H be a convex Hamiltonian, for a given k ∈ R, there exists a residual subset O ∈ C∞(M) such that (H + u)−1(k), u ∈ O, is a regular energy level and all closed orbits in this energy level are non-degenerate. This result reminds the so-called bumpy metric theorem in the context of Riemannian geometry. The set of Cr (r ≥ 2) bumpy metrics Br(M) on a manifold M, i.e. metrics with no closed degenerate geodesic, is residual in Rr(M), where Rr(M) refers to the set of all the Cr Riemannian metrics on M. However, it is important to note that Mañe perturbations (or conformal perturbations of Riemannian metrics) are much more restrictive than perturbations of Hamiltonians or metrics with respect to Withney topologies. After a quick review of the convex case, we will replace the assumption of convexity with a geometric condition for Hamiltonians, a condition that is weaker than convexity.

14.12.2021      at 14:00   Yael Karshon (University of Toronto) "tba"    via Zoom
at 15:15   Marco Mazzucchelli (ENS Lyon) "tba"    in room ID 04/445
at 17:00   Dan Cristofaro (University of Maryland) "tba"    via Zoom
THESE TALKS TAKES PLACE AS PART OF THE BACH SEMINAR

16.11.2021   Rohil Prasad (Princeton) "Invariant probability measures from pseudoholomorphic curves"    via Zoom

Abstract: We introduce a new method for producing invariant probability measures for a large class of volume-preserving flows on closed, oriented odd-dimensional smooth manifolds; these include all non-singular volume-preserving flows in dimension three. These probability measures arise as "limit sets" of pseudoholomorphic curves with infinite Hofer energy. We will also discuss an application of our method to showing that the characteristic flow on a large class of autonomous Hamiltonian energy levels is not uniquely ergodic.

09.11.2021   Michael Khanevsky (Technion) "Geometry of non-chaotic Hamiltonian diffeomorphisms on surfaces"    via Zoom

Abstract: Traditionally, a lot of research activity is focused at Hamiltonian dynamics with positive entropy, while the non-chaotic part of the field has received less attention. At the same time, it is still far from being well understood, even in the simplest possible setting - when we consider dynamics in dimension two. We will discuss several results that distinguish geometry of autonomous and integrable Hamiltonian diffeomorphisms on surfaces.

12.10.2021   Yaniv Ganor (Tel Aviv) "Big Fiber Theorems and Ideal-Valued Measures in Symplectic Topology"    via Zoom

Abstract: In various areas of mathematics there exist "big fiber theorems", these are theorems of the following type: "For any map in a certain class, there exists a 'big' fiber", where the class of maps and the notion of size changes from case to case. We will discuss three examples of such theorems, coming from combinatorics, topology and symplectic topology from a unified viewpoint provided by Gromov's notion of ideal-valued measures. We adapt the latter notion to the realm of symplectic topology, using an enhancement of Varolgunes’ relative symplectic cohomology to include cohomology of pairs. This allows us to prove symplectic analogues for the first two theorems, yielding new symplectic rigidity results. Necessary preliminaries will be explained. The talk is based on a joint work with Adi Dickstein, Leonid Polterovich and Frol Zapolsky.

## Spring 2021

The talks start at 16.15 .

06.07.2021 at 17:15!   Julian Chaidez (Berkeley) "Curvature And Rotation In Convex Reeb Flows"    via Zoom
THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR

Abstract: The boundary Y of a 4-dimensional, smooth convex domain has the structure of a contact manifold with a natural Reeb flow. Many conjectures exist about the special dynamical properties of Reeb flows arising in this way. In this talk, I will discuss a relationship between the curvature of Y and various invariants of the Reeb flow that measure rotation. In recent work (joint with Oliver Edtmair), we used this relationship to show that convexity is not equivalent to dynamical convexity in dimension 3. I will then give an overview of potential applications of these ideas to questions of Reeb orbit knottedness and generalizations to higher dimensions.

22.06.2021 at 17:15!   Dušan Joksimović (Jussieu) "No symplectic-Lipschitz structures on S^{2n \geq 4} "    via Zoom
THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR

Abstract: One of the central questions in C^0-symplectic geometry is whether spheres (of dimension at least 4) admit symplectic topological atlas (i.e. atlas whose transition functions are symplectic homeomorphisms). In this talk, we will prove that the answer is „no“ if we replace the word „topological“ with „Lipschitz“. More precisely, we will prove that every closed symplectic-Lipschitz manifold has non-vanishing even degree cohomology groups with real coefficients. The proof is based on the fact that one can define analogs of differential forms and de Rham complex on Lipschitz manifolds which share similar properties as in the smooth setting.

15.06.2021   Georgios Dimitroglou Rizell (Uppsala) " Lagrangian Unlinking and Recurrence"    via Zoom

Abstract: We give an introduction to two related phenomena: Lagrangian unlinking and Lagrangian Poincaré recurrence. More precisely, we discuss recent work with L. Côté where we show that rational weakly exact Lagrangian tori in the cotangent bundle are Hamiltonian unlinked by using pseudoholomorphic foliation techniques. In an ongoing project with E. Opshtein we then use these techniques to study the problem of Lagrangian reccurrence in the product of a surface and an annulus.

25.05.2021   Comlan E. Koudjinan (IST, Austria) " Some applications of KAM Theory to billiards"    via Zoom

Abstract: In this talk, I will illustrate the power of KAM Theory (viewed as a tool) by discussing some of its applications to the study of billiards.

04.05.2021   Tobias Weich (Paderborn) "SRB measures for Anosov actions"    via Zoom

Abstract: SRB (Sinai-Ruelle-Bowen) measures for hyperbolic flows (Anosov flows) are known to be among the most significant invariant measures. Using recently established spectral techniques (Ruelle-Taylor resonances) we construct SRB-measures for general higher rank Anosov actions. We examine their properties and provide a Bowen formula which allows to express these measures in terms of closed orbits. This is based on joint work with Y. Guedes-Bonthonneu, C. Guillarmou and J. Hilgert.

27.04.2021   Alexandre Jannaud (Neuchâtel) "Dehn-Seidel twist, C^0 symplectic geometry and barcodes"    via Zoom

Abstract: In this talk I will present my work initiating the study of the $C^0$ symplectic mapping class group, i.e. the group of isotopy classes of symplectic homeomorphisms, and present the proofs of the first results regarding the topology of the group of symplectic homeomorphisms. For that purpose, we will introduce a method coming from Floer theory and barcodes theory. Applying this strategy to the Dehn-Seidel twist, a symplectomorphism of particular interest when studying the symplectic mapping class group, we will generalize to $C^0$ settings a result of Seidel concerning the non-triviality of the mapping class of this symplectomorphism. We will indeed prove that no power of the square of the generalized Dehn twist is not in the connected component of the identity in the group of symplectic homeomorphisms. Doing so, we prove the non-triviality of the $C^0$ symplectic mapping class group of some Liouville domains.

## WINTER 2020/21

The talks start at 16.15 .

26.01.2021   Klaus Niederkrüger (Lyon 1) "About symplectic fillings of real projective spaces "    via Zoom
THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR

Abstract: (Joint work with Paolo Ghiggini.) The standard contact structure of a real projective space is always strongly fillable, but it was shown by Eliashberg-Kim-Poltervich, that from dimension 5 on, they are not Weinstein fillable. I will explain a geometric argument that shows that every real projective space of dimension 4k+1 is not Liouville fillable.

19.01.2021   Lucas Dahinden (Heidelberg) "Introducing sub-Riemannian Billiards"    via Zoom

Abstract: In sub-Riemannian geometry we study Riemannian manifolds with movement restricted to a bracket generating sub-bundle of the tangent bundle. We introduce a billiard reflection law in two natural ways: on the one hand by a variational principle and on the other hand by a symplectic description as characteristic flow lines at the boundary of a domain in the cotangent bundle. The two definitions coincide. Finally, we examine the highly symmetric example of the Heisenberg group, which reveals strong connections to magnetic billiards.

12.01.2021   Johanna Bimmermann (Heidelberg) "On the Hofer–Zehnder capacity for twisted tangent bundles over closed surfaces"    via Zoom

Abstract: In this talk I will present the computation of the Hofer–Zehnder capacity for magnetic systems on closed surfaces with constant (weak) magnetic field. While finding a lower bound for the Hofer–Zehnder capacity is relatively easy, as any admissible Hamiltonian function provides one, finding an upper bound is much harder. By a theorem of G. Lu for closed symplectic manifolds an upper bound is given by the symplectic area of a homology class that has a non-vanishing Gromov–Witten invariant. Our strategy is therefore, to find an embedding of the magnetic system into a closed symplectic manifold. We will then use the theorem to find an upper bound of the Hofer–Zehnder capacity. Finally we will see that upper and lower bound agree and therefore determine the Hofer-Zehnder capacity.

15.12.2020   Otto van Koert (Seoul) "A generalization of the Poincare-Birkhoff fixed point theorem
and the restricted three-body problem"
via Zoom
THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR

Abstract: In joint work with Agustin Moreno, we propose a generalization of the Poincare-Birkhoff fixed point theorem. We start with a construction of global hypersurfaces of section in the spatial three-body problem, describe some return maps and suggest some generalizations of the Poincare-Birkhoff fixed point theorem. We use symplectic homology in the proof of our theorem.

08.12.2020   Benoit Joly (Paris) "Dynamical construction of barcodes of Hamiltonian homeomorphisms of surfaces"    via Zoom

Abstract: The notion of barcodes appears to be a useful tool in C^0 symplectic topology and can be seen as a « path » of all spectral invariants. Nevertheless, the construction relies on Floer Homology which needs a C^2 setting. I will present a new construction of barcodes of Hamiltonian homeomorphisms of oriented compact surfaces which relies on Le Calvez's transverse foliation theory. For a homeomorphism of a surface Le Calvez proved that there exists C^0 foliations associated to some « maximal » isotopies. Moreover, in the case of Hamiltonian homeomorphisms, the foliations are « gradient-like ». This property allows us to construct new barcodes without Floer Homology. I will present the dynamical tools and then I will give the ideas to construct theses barcodes.

01.12.2020   Samuel Lisi (Oxford, USA) "Spinal Open Book Decompositions"    via Zoom
THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR

Abstract: In work with Jeremy Van Horn-Morris and Chris Wendl, we have introduced the notion of a spinal open book decomposition for contact manifolds in dimension 3. This can be thought of as a generalization of an open book decomposition. The main purpose is to describe a large class of contact manifolds as supported by planar (but spinal) open books. By use of J-holomorphic curve techniques, we are then able to classify symplectic fillings.

17.11.2020   Matthias Meiwes (Aachen) "Hofer's geometry and entropy"    via Zoom

Abstract: A central object in the study of Hamiltonian diffeomorphisms on a symplectic manifold is Hofer's metric, a bi-invariant metric on the group of Hamiltonian diffeomorphisms that displays rigidity features that are special for those diffeomorphisms. The geometry of this metric has been thoroughly studied since its discovery by Hofer and his work in the early 90's and a central theme is to link Hofer's geometry to dynamical properties of the underlying maps. In my talk I will discuss some conditions under which dynamical complexity persists under bounded perturbations in Hofer's geometry. This leads to stable lower bounds on topological entropy and on orbit growth in various situations. This talk is partly based on joint work with Arnon Chor.

10.11.2020   Bernd Stratmann (Bochum) "Nowhere vanishing primitive of a symplectic form and removing parametrized rays symplectically"    via Zoom

Abstract: In the talk, two new results will be explained. The first one is that an exact symplectic form always admits a nowhere vanishing primitive. The second result says that one can extract a ray from an (open) symplectic manifold and the resulting manifold is always symplectomorphic to the inital manifold. The latter can be generalized under a condition to parametrized rays. Both results are proven with classical methods.

03.11.2020   Will Merry (ETH Zürich) "Symplectic cohomology of magnetic cotangent bundles"    via Zoom
THIS TALK TAKES PLACE AS PART OF THE BACH SEMINAR

Abstract: Joint work with Y. Groman (and, time-permitting, Seongchan Kim). We construct a family version of symplectic Floer cohomology for magnetic cotangent bundles.

27.10.2020   Shira Tanny (Tel Aviv) "The Poisson bracket invariant: soft and hard approaches"    via Zoom

Abstract: In 2006 Entov and Polterovich proved that functions forming a partition of unity with displaceable supports cannot commute with respect to the Poisson bracket. In 2012 Polterovich conjectured a quantitative version of this theorem. I will discuss three interconnected topics: a solution of this conjecture in dimension two (with Lev Buhovsky and Alexander Logunov), a link between this problem and Grothendieck's theorem from functional analysis (with Efim Gluskin), and new results related to the Floer-theoretical approach to this conjecture (with Yaniv Ganor).