## Winter 2024/25

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

**17.12.2024** Jakob Hedicke (Montreal) "tba"

**03.12.2024** Ronen Brilleslijper (Amsterdam) "tba"

**19.11.2024 **Yuan Yao (Nantes) "tba"

**12.11.2024** Olga Bernardi (Padova) "Symplectic billiards: an overview and recent rigidity results"

**Abstract:** Symplectic billiards were introduced by P. Albers and S. Tabachnikov in 2018 as a simple dynamical system where, opposed to Birkhoff billiards, the generating function is the area instead of the length. We first recall the main properties of symplectic billiards in strictly-convex domains. Then we focus on their integrability. In particular, we present this recent result: if the phase-space is fully foliated by continuous invariant curves which are not null-homotopic, then the boundary of the billiard table is an ellipse. Finally, we introduce and discuss the area spectral rigidity for axially symmetric domains. Joint works with L. Baracco and A. Nardi.

**29.10.2024** Murat Saglam (Köln) "On contact structures with Bott-integrable Reeb flows*"*

**Abstract:** We will discuss the characterization of overtwisted contact structures, which allow Reeb flows that are integrable in the sense of admitting an invariant Morse-Bott function.

**22.10.2024** Adi Dickstein (Tel Aviv) *"*Symplectic topology and ideal-valued measures*"*

**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. Thi 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 Yaniv Ganor, Leonid Polterovich and Frol Zapolsky.

## Spring 2024

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

**16.07.2024** Jan Bohr (Bonn) "On transport twistor spaces"

**Abstract**: Transport twistor spaces are complex surfaces that are tailored to the geodesic flow (magnetic flow,...) over a Riemannian surface and capture many interesting features of the flow in complex geometric language. For Zoll geodesic flows a cousin (or rather a quotient) of this construction is at the heart of LeBrun-Masons work on Zoll deformations. More recently, it was discovered that transport twistor spaces provide a unifying picture to several developments in geometric inverse problems, allowing to organise and reinterpret various older results, while at the same time posing new intriguing challenges. In the talk I will sketch the aforementioned picture in more detail and report on some recent work with F. Monard and G.P. Paternain that locally recovers LeBrun-Masons desingularisation of twistor space for arbitrary metrics.

**09.07.2024 **Levin Maier (Heidelberg) "On Mañé's critical value for the two-component Hunter-Saxton system"

Abstract:In this talk, we will introduce Mañé's critical value for a Hamiltonian PDE,

the two-component Hunter-Saxton system. We will introduce the magnetic two-component

Hunter-Saxton system (M2HS), which is a magnetic geodesic equation on an infinite-dimensional

Lie group. We prove that this magnetic system is magnetic isomorphic to a magnetic system

on an infinite-dimensional sphere. Surprisingly each magnetic geodesic is tangent to the 3-sphere

obtained by intersecting the ambient sphere with a complex plane. We use this geometric description

of the (M2HS) to give explicit criteria for blow-ups and prove the existence of global weak solutions.

**04.07.2024** Hans-Bert Rademacher (Leipzig) EXCEPTIONALLY ON THURSDAY IN ROOM IA 1/177 at 16.15

"Closed geodesics and orthogonal geodesic chords without self-intersections"

**Abstract:** We show that for a generic Riemannian metric on a compact manifold of dimension at least three all closed geodesics do not have self-intersections. Similar results are possible for geodesic loops and orthogonal geodesic chords.

**27.06.2024** Monika Yadav (Gandhinagar, India) EXCEPTIONALLY ON THURSDAY IN ROOM IA 1/177 at 14.15 *"*Jones polynomial and its categorification for Legendrian knots in **\((\mathbb{R}^3,\xi_{st})\)***"*

**Abstract:** In this talk, I will talk about the joint work with Dr. Dheeraj Kulkarni. The goal of this talk is to define a new invariant of Legendrian knots in \(\mathbb{R}^3\) with the standard tight contact structure.

We will start by recalling some background on Legendrian knots and the Jones polynomial invariant for smooth knots. I will define two invariants of the Legendrian knot type, namely, the Legendrian Jones polynomial and the Legendrian Khovanov homology. We will see that both of these invariants are natural generalizations of the Jones polynomial and the Khovanov homology for smooth knots to the setting of Legendrian knots. In other words, for a Legendrian knot K, the Legendrian Jones polynomial P_{K}(A, r) reduces to the Jones polynomial of the underlying smooth knot after substituting r = 1. We will also see that the Thurston-Bennequin number which is a classical invariant of the Legendrian knot type, occurs as a grade shift in the Legendrian Khovanov homology.

**25.06.2024** David Bechara (Aachen) *"*The asymptotic action of area preserving disk maps and some of its properties*"*

**Abstract:** Given a diffeomorphism of the disk that preserves the standard symplectic form, I will introduce the asymptotic action associated to this map. I will then show a pointwise formula relating the asymptotic action to the asymptotic winding number of pairs of points. As a corollary one obtains a generalization for a well known result by A. Fathi which gives a formula for the Calabi invariant of a disk map in terms of its mean winding numbers. Additionally I will focus on how this formula can be used to study symplectic dynamical information for irrational pseudo-rotations of the disk. This talk includes joint work with Patrice Le Calvez and Abror Pirnapasov.

**11.06.2024** Comlan Edmond Koudjinan (Vienna) *"*On the Birkhoff conjecture for nearly centrally symmetric domains*"*

**Abstract:** In this talk, I will discuss a recent advance on Birkhoff conjecture, namely a proof that : an integrable, nearly centrally symmetric Birkhoff billiard table is necessarily an ellipse. This is done by combining recent breakthroughs by Bialy-Mironov (who prove the conjecture for centrally symmetric Birkhoff billiard tables) and by Kaloshin-Sorrentino (who prove the conjecture for Birkhoff billiard tables close to ellipses). In particular, we shall discuss the use of nonstandard generating functions discovered by Bialy-Mironov. This is a joint work with Ke Zhang and V. Kaloshin.**04.06.2024** Umberto Hryniewicz (Aachen) *"Proof of Hofer-Wysocki-Zehnder’s two or infinity conjecture"* **Abstract:** : In this talk we will describe the main steps in the proof that the Reeb flow of a contact form on a closed connected 3-manifold has exactly two or infinitely many periodic orbits, under the assumption that the first Chern class of the associated contact structure is torsion. This is joint work with Cristofaro-Gardiner, Hutchings and Liu.

**21.05.2024** Robert Cardona (Barcelona) *"*Contact topology and time-dependent hydrodynamics*"*

**Abstract: **More than twenty years ago, Etnyre and Ghrist established a connection between Reeb fields and a class of stationary solutions to the 3D Euler equations for ideal fluids. In this talk, we present a new framework that allows assigning contact/symplectic invariants to large sets of *time-dependent* solutions to the Euler equations on any three-manifold with an arbitrary fixed Riemannian metric, thus broadening the scope of contact topological methods in hydrodynamics. We use it to prove a rather general non-mixing result for the infinite-dimensional dynamical system defined by the equation and to show the existence of new conserved quantities obtained from embedded contact homology spectral invariants. This is joint work with Francisco Torres de Lizaur.

**02.05.2024** Agustin Moreno (Heidelberg) EXCEPTIONALLY ON THURSDAY IN ROOM IA 1/177 at 16.15

*"Symplectic structures from almost symplectic structures"* **Abstract:** In this talk, we will consider a stabilized version of the fundamental existence problem of symplectic structures. Given a formal symplectic manifold, i.e. a closed manifold M with a non-degenerate 2-form and a non-degenerate second cohomology class, we investigate when its natural stabilization to M x T^2 can be realized by a symplectic form. We show that this can be done whenever the formal symplectic manifold admits a positive symplectic divisor. It follows that if a formal symplectic 4-manifold, which either satisfies that its positive/negative second betti numbers are both at least 2, or that is simply connected, then M x T^2 is symplectic. This is joint work with Fabio Gironella, Fran Presas, Lauran Touissant.**16.04.2024** Karla Leipold (Cologne) *"Computing the EHZ-capacity is NP-hard"* **Abstract:** We show that computing the EHZ capacity of polytopes is NP-hard. For this we reduce the feedback arc set problem in bipartite tournaments to computing the EHZ capacity of simplices.

## Winter 2023/24

The talks start at 16.15 in Seminar Room IA 1/53 .**20.02.2024** Karim Mosani (Tübingen) *"Geometry of trapped photon region in a class of stationary spacetimes"* **Abstract:** In Einstein’s general relativity, extremely strong gravity can trap light. In a spacetime admitting a singularity, we say that light (or a “photon”) is trapped if it neither escapes to spatial infinity nor falls into the singularity. Null geodesics govern the trajectories of light. In the Schwarzschild spacetime with posi- tive mass M, there exist (unstable) circular orbits of trapped photons at the Schwarzschild radius r = 3M, outside the black hole horizon at r = 2M. These orbits fill a three-dimensional submanifold of topology S2 ×R called the photon sphere of the Schwarzschild spacetime. In general, a region in spacetime that is a union of all trapped null geodesics is called the Trapped Photon Region (TPR) of the spacetime. In this talk, we will consider a particular stationary space-time class constructed by the Newman-Jenis algorithm. We will see that, unlike the TPR of Schwarzschild spacetime, the TPR in such spacetimes is not a submanifold of the spacetime in general. However, the lift of TPR in the phase space is a five-dimensional submanifold. This result has applications in various problems in mathematical relativity (This work is an extension of the similar result but in Kerr spacetime, by Cederbaum and Jahns- 2019). This is a joint work with Carla Cederbaum.**13.02.2024** James Farre (Leipzig) *"Horospheres, Lipschitz maps, and laminations"* **Abstract:** Every horocycle in a closed hyperbolic surface is dense, and this has been known since the 1940's. We study the behavior of horocycle orbit closures in Z-covers of closed surfaces, and obtain a fairly complete classification of their topology and geometry. The main tool is a solution of a surprisingly delicate geometric optimization problem: finding an optimal Lipschitz map to the circle and the associated lamination of maximal stretch. This is joint work with Yair Minsky and Or Landesberg.**05.12.2023** Álvaro del Pino (Utrecht) *"Convex integration with avoidance"* **Abstract:** Convex integration is one of the most important tools in the construction of solutions of partial differential relations. It was first introduced by J. Nash in his work on C^1 isometric embeddings and later generalised by M. Gromov to deal with a large class of differential relations satisfying a geometric condition called ampleness. Gromov developed various flavours of ampleness to which convex integration applies. Roughly speaking, there is an "easy" to check version (ampleness in all principal directions) that is limited in its applications, and an "impossible" to check version (ampleness via convex hull extensions) that is extremely general. This will motivate me to discuss a new version of convex integration and a corresponding notion of ampleness, called ampleness up to avoidance. This notion is checkable in practice and applies in more generality than ampleness in all principal directions. This is joint work with F.J. Martínez Aguinaga.**28.11.2023** Valerio Assenza (Heidelberg) *"Geometrical aspects of magnetic flows"* **Abstract:** To a Riemannian manifold endowed with a magnetic form we associate an operator called Magnetic Curvature. Such an operator encodes the geometrical properties of the Riemannian structure together with terms of perturbation due to the magnetic interaction and carries relevant information about the magnetic dynamics. In the first part of the talk we will see how a level of the energy positively curved in a magnetic sense carries a contractible periodic orbit. The second part is devoted to the generalization of the Hopf’s rigidity to the magnetic case and to the notion of magnetic flatness.**14.11.2023** Michael Rothgang (Berlin-Humboldt) *"Equivariant transversality for holomorphic curves"* **Abstract:** We study closed holomorphic curves in symplectic G-manifolds, with respect to a G-equivariant almost complex structure. We should not expect the moduli space of such curves to be a manifold (after all, transversality and symmetry are famously incompatible). However, we can hope for a clean intersection condition: the moduli space decomposes into disjoint strata which are smooth manifolds; the dimensions of the strata are explicitly computable. I'll present this decomposition for simple curves and indicate how to extend this to multiple covers. These are the first steps towards a well-behaved theory of equivariant holomorphic curves.**07.11.2023** Christian Ketterer (Freiburg) *"Characterization of the null energy via displacement convexity of entropy"* **Abstract:** I will present a characterization of the null energy condition for an (n+1)-dimensional, time-oriented Lorentzian manifold in terms of convexity of the relative (n-1)-Renyi entropy along null displacement interpolations on null hypersurfaces. More generally, I consider a Lorentzian manifold with a weight function and I introduce a synthetic Bakry-Emery N-null energy condition that we characterize in terms of null displacement convexity of the relative N-Renyi entropy. Here the relative N-Renyi entropy is given w.r.t. the co-dimension 2 reference measure induced by the Lorentz metric and the weight. As applications we prove Hawking’s area monotonicity theorem for the area of a black hole horizon and a Penrose singularity theorem in the context of weighted Lorentzian manifolds.

## Spring 2023

The talks start at 16.15 in Seminar Room IA 1/181 .**20.06.2023** Filip Broćić (Montreal) *"Riemannian distance and symplectic embeddings in cotangent bundle"* **Abstract:** In the talk, I will define a distance-like function d_W on the zero section N of the cotangent bundle T*N. The function d_W is defined using certain symplectic embeddings from the standard ball to the open neighborhood W of the zero section. Using such a function, one can define a length structure on the zero section. The main result of the talk is that in the case when W is equal to the unit disc-cotangent bundle with respect to some Riemannian metric g, the length structure is equal to the Riemannian length. In the process of explaining the proof I will present some results related to the relative type of Gromov width in T*N, and I will give the proof of the strong Viterbo conjecture for the product of two Lagrangian discs in R^{2n}. In the joint work with Dylan Cant, we were able to give a sharper bound on the relative Gromov width, under some constraints, using bordism classes in the free loop space. We also prove the existence of periodic orbits for a large class of Hamiltonians using the same technic. Time permitting, I will present how to use bordism classes to prove these results.**13.06.2023**

16:15: Roman Golovko (Prague) *"On non-geometric augmentations of Chekanov-Eliashberg algebras"* **Abstract:** Legendrian contact homology is a modern invariant of Legendrian submanifolds of contact manifolds defined by Eliashberg–Givental–Hofer and Chekanov, and developed by Ekholm–Etnyre–Sullivan for the case of the standard contact vector space. It is defined to be the homology of the Chekanov-Eliashberg algebra of a given Legendrian submanifold. This invariant is difficult to compute, and, in order to make it computable, one needs to use augmentations. Some augmentations come from certain geometric objects called exact Lagrangian fillings, some do not. We will discuss non-geometric augmentations for high dimensional Legendrian submanifolds. Along the way, we prove a Künneth formula for (linearized) Legendrian contact homology for high spuns of Legendrian submanifolds. If time permits, we will also discuss whether algebraic torsion appears in Legendrian contact homology.

17:30: Sayani Bera (IACS, Calcutta) *"On non-autonomous attracting basins"* **Abstract:** The goal of this talk is to discuss briefly the idea of the proof of the Bedford's conjecture (formulated by Fornæss-Stensønes in 2004), on uniform non-autonomous attracting basins of automorphisms of C^k, k \ge 2 and Fatou-Bieberbach domains. Thus we also affirmatively answer Bedford's question (2000) on uniformizations of the stable manifolds, corresponding to a hyperbolic compact invariant subset of a complex manifold. This is a joint work with Dr. Kaushal Verma.**11.05.2023** Stefano Baranzini (Turin) *"Morse Index Theorems for Graphs"* in Seminar Room IA 1/177 at 14:15**Abstract:** In this talk I will discuss some Morse Index Theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Given a graph G and a di erentiable functional A de ned on a suitable subspace of continuous function on G, one could ask: "How does the index of a critical point change when we change the topology of the graph?". The general formula I will present tries to answer this question. It expresses the di erence of Morse Indices of two Hessians, related to two di erent graphs or two di erent sets of boundary conditions, in terms of a suitable symplectic invariant: the Maslov Index. If time permits application of the formula will be given. For instance, it can be used to produce a certain type of discretization formulae to reduce the complexity of the computation of Morse Index to a nite dimensional problem or it can be specialized to the case of periodic extremals to get iteration for- mulae. From a more hands-on perspective this formula can be used to compute numerically the Morse Index of some speci c problems such as the non-linear Schrödinger equation on symmetric trees. This is a joint work with A. Agrachev and I. Beschastnyi.**09.05.2023** Gian Marco Canneori (Turin) *"The N-centre problem on Riemannian surfaces: a variational approach"* **Abstract:** The classical N-centre problem of Celestial Mechanics describes the behaviour of a point particle under the attraction of a finite number of motionless bodies. Considered as a limit case of a (N+1)-body problem, it has been the object of several results concerning integrability, investigation of chaos and existence of periodic orbits, mostly when the motion is constrained to the Euclidean plane. In particular, variational approaches are convincing in this situation and have produced classes of collision-less periodic solutions, after imposing topological constraints of different natures. Looking for genuine solutions of second order differential equations, the most delicate step resides in avoiding collisions with the centres. Picturing a more realistic situation, a natural extension of these results could be the one in which the motion is constrained to a prescribed Riemannian surface. In this talk we state the N-centre problem on orientable surfaces and we show how it is possible to use variational arguments in order to produce collision-less periodic solutions. Such trajectories will be found among homotopy classes of loops, and their variational and topological properties will be described. This is a joint work with Stefano Baranzini.**02.05.2023**

16:15: Michael Jung (Amsterdam) *"A geometric computation of cohomotopy sets in codegree one"* **Abstract:** It is a classical fact that for closed manifolds X the homotopy classes of maps X^n\rightarrow S^n are classified by their degree. The Pontryagin-Thom construction provides a similar construction when X and the sphere have different dimensions, and thus generalizes the notion of degree. In particular, the homotopy classes of maps X^{n+1}\rightarrow S^n are in one-to-one correspondence with framed circles up to framed cobordism in X, and the corresponding set comes equipped with a group structure. In this talk, we introduce the Pontryagin-Thom construction and the concept of framed cobordism classes, and we compute the group of homotopy classes X^{n+1}\rightarrow S^n in terms of topological information of X.

17:30: Lauran Toussaint (Amsterdam) *"Classifying proper Fredholm maps"* **Abstract:** Many partial differential equations are encoded by proper Fredholm maps between (infinite dimensional) Hilbert spaces. By the Pontryagin-Thom construction these maps correspond to finite dimensional framed submanifolds. This gives a connection between finite and infinite dimensional topology. In this talk, I will use this relation to classify proper Fredholm maps (up to proper homotopy) between Hilbert spaces in terms of the stable homotopy groups of spheres. This is based on work in progress with Thomas Rot.**25.04.2023** Francesco Morabito (Paris) *"Hofer Pseudonorms on Braid Groups and Quantitative Heegaard-Floer Homology"* **Abstract:** Given a lagrangian link with k components it is possible to define an associated Hofer pseudonorm on the braid group with k strands. In this talk we are going to detail this definition, and explain how it is possible to prove non degeneracy if k=2 and certain area conditions on the lagrangian link are met. The proof is based on the construction, using Quantitative Heegaard-Floer Homology, of a family of quasimorphisms which detect linking numbers of braids on the disc.

## Winter 2022/23

The talks start at 16.15 .**21.03.2023** Khadim War (IMPA) *"Cross section for Codimension one Anosov flows"* in Seminar Room IA 1/53**Abstract:** In this talk we will prove that every codimension one Anosov flow on a manifold of dimension at least four admits a global cross section. The proof is done by constructing a time change of the flow along periodic orbits. This gives a proof of the Verjovsky Conjecture.**24.01.2023** Leonardo Garcia Heveling (Radboud) *"When is topology change physically reasonable?"* in Seminar Room IA 1/53**Abstract:** General Relativity is famous for merging the concepts of space and time into a single entity, spacetime, represented by a 4-dimensional manifold. In practice, however, most interesting spacetime manifolds can still be equipped with a function that plays the role of time. Topology change refers to the situation when, say, the {time=1} set has a different topology than the {time=0} set. There is an ongoing debate about whether the laws of physics should allow topology change or not. This is partly because some examples are known where the change in topology leads to unphysical behaviour, such as infinite energy bursts of a quantum field propagating on spacetime. In this talk I will review previous ideas and work related to topology change and present recent progress on a conjecture of Borde and Sorkin which aims to distinguish between "good" and "bad" topology change.**10.01.2023** Sergi Burniol Clotet (Paris) *"Unique ergodicity of the horocyclic flow on surfaces without conjugate points"* in Seminar Room IA 1/53**Abstract:** There are strong connections between the dynamics of the geodesic flow and the horocyclic flow defined on the unit tangent bundle of certain Riemannian surfaces. Furstenberg and Marcus proved in the 70s that the horocyclic flow of a negatively curved compact surface is uniquely ergodic, i.e. it admits a unique invariant probability measure. I will explain why this result still holds for a compact surface without conjugate points, genus greater than 1 and continuous Green bundles. The proof uses the construction of the measure of maximal entropy for the geodesic flow in a recent paper of Climenhaga-Knieper-War, and the semiconjugation of the geodesic flow with a an expansive continuous flow with local product structure, established by Gelfert-Ruggiero.**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).