It has been noticed a while ago that several fundamental transformation groups of symplectic and contact geometry carry natural bi-invariant causal structures, i.e., bi-invariant fields of tangent convex cones. Quite often, the latter come together with bi-invariant Lorentz–Finsler metrics, a notion originated in relativity theory, which enable one to do geometric measurements with timelike curves. This includes finite-dimensional linear symplectic groups, where these metrics can be seen as Finsler generalizations of the classical anti-de Sitter spacetime, and infinite-dimensional groups of contact transformations, with the simplest example being the group of circle diffeomorphisms. I will discuss some of the properties of these bi-invariant Lorentz-Finsler metrics, both on the finite and on the infinite dimensional groups. This talk is based on a joint work with Gabriele Benedetti and Leonid Polterovich.

In this second talk, we will give example of Lee flows (which are the analogue of Reeb flows in locally conformally symplectic manifolds) with "unexpected" behaviors compared to the contact or symplectic setting: flows without periodic orbits or exhibiting dense orbits. These examples rely on a twisted version of the locally conformal symplectization of contact spaces. This is a joint work with Marie-Claude Arnaud.

The \(C^0\)-distance on the space of contact forms on a contact manifold has been studied recently by different authors. It can be thought of as an analogue for Reeb flows of the Hofer metric on the space of Hamiltonian diffeomorphisms. In this talk, I will explain some recent progress on the stability properties of the topological entropy with respect to this distance. This is joint work with Lucas Dahinden, Matthias Meiwes and Abror Pirnapasov.

We will introduce the notion of locally symplectic manifolds: they are manifolds endowed with a symplectic form that is only locally defined, and we will study their autonomous Hamiltonian systems. We will explain that they may have dissipative and conservative behaviours in a robust way. This is a joint work with Simon Allais, Université de Paris Cité.
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Locally maximizing billiard configurations (orbits) are those which give local maxima for the Length functional between any two impact points. For example, in two dimensions, rotational invariant curves and Aubry-Mather sets are filled by locally maximizing orbits. In this talk I discuss the Twist maps and Birkhoff billiards in higher dimensions. In particular, I shall give an effective criteria of local maximality and prove that the class of locally-maximizing orbits does not depend on the choice of generating function, similar to the result by P. Bernard and M. Mazzucchelli-A. Sorrentino on Tonelli Hamiltonians. Based on the joint works with Robert MacKay, Andrey E. Mironov, Sergei Tabachnikov and Daniel Tsodikovich.
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It is possible to measure oscillations of a function by means of the theory of persistence modules and barcodes. I will explain how Sobolev norms can control such measurements, and describe some applications motivated by attempts to generalize the Courant nodal domain theorem. The talk is based on a joint work with Jordan Payette, Iosif Polterovich, Leonid Polterovich, Egor Shelukhin, and Vukašin Stojisavljević, and on a joint work with Aleksandr Logunov and Mikhail Sodin.
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We will discuss lower semi-continuity properties of the surface area of a closed Lagrangian submanifold with respect to the gamma-distance. The talk is based on a joint work with Viktor Ginzburg and Basak Gurel.

Spectral invariants defined via Embedded Contact Homology (ECH) or the closely related Periodic Floer Homology (PFH) satisfy a Weyl law: Asymptotically, they recover symplectic volume. This Weyl law has led to striking applications in dynamics and \(C^0\) symplectic geometry. For example, it plays a key role in the proof of the smooth closing lemma for three-dimensional Reeb flows and area preserving surface diffeomorphisms, and in the proof of the simplicity conjecture. In this talk I will report on work in progress concerning the subleading asymptotics of these Weyl laws and the connection to symplectic packing problems.
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Weak KAM theory originally connected Mather theory of Lagrangian Systems with Viscosity Theory of the solutions of the corresponding Hamilton-Jacobi Equation, at least when the Hamiltonian is obtained from a Lagrangian. In such a case the Mañé potential is the minimal action necessary to join two points in arbitrary time. We will show that we can recover just from the Mañé potential concepts like Peierls barrier, Aubry sets, viscosity subsolutions and solutions. This allows the theory to apply in the more general framework of compact metric spaces, opening a way to define solutions of the Hamilton-Jacobi equation on general metric spaces.
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The Birkhoff attractor was defined by G.D. Birkhoff in 1932 for dissipative maps having an attracting bounded annulus, and it was further studied by P. Le Calvez in the framework of dissipative twist maps. In a work in progress with Olga Bernardi and Martin Leguil, we start from some concrete examples in order to understand how the Birkhoff attractor of a dissipative billiard map could look like.

Bolotin's Conjecture, now a joint theorem of Bialy, Mironov and the speaker, is a polynomial version of the Birkhoff Conjecture. It deals with a billiard whose flow admits a non-trivial first integral polynomial in the velocity. It states that 1) if its boundary is \(C^2\)-smooth, nonlinear and connected, then is a conic; 2) if it is piecewise \(C^2\)-smooth, then it consists of arcs of conics from a confocal pencil and segments of "admissible lines" for the pencil; 3) the minimal degree of integral may be only 1, 2, or 4. In the talk we extend this result to the planar projective billiards introduced by Sergei Tabachnikov. Each of them is a planar curve equipped with a transversal line field. It defines a reflection acting on oriented lines and a billiard flow. We classify those piecewise \(C^4\)-smooth non-polygonal projective billiards that are rationally integrable, i.e., whose flow admits a non-constant first integral that is a rational 0-homogeneous function of the velocity. (Case of \(C^4\)-smooth connected projective billiards was previously treated by the speaker.) We show that: 1) the minimal degree of integral may be arbitrary even number; 2) a rationally integrable projective billiard associated to a dual pencil of conics may have integral of minimal degree 2, 4, or 12.

It is well known that spectral invariants coming from Floer theory are a very useful tool to study the dynamics of symplectic maps. In this talk we will see that it can also be used to study the dynamics of conformally symplectic maps, i.e. maps that multitply the symplectic form by a non-trivial factor (this includes for instance Hamiltonian systems with a friction term). More specifically, we will see how to generalize to higher dimension the classical Birkhoff attractor (1932) which was so far only defined in the 2-dimensional annulus. This is based on joint work with Marie-Claude Arnaud and Claude Viterbo.

A periodic orbit (or finite set of periodic orbits) of a surface diffeomorphism gives rise to a braid in the mapping torus. We show that if the diffeomorphism is area-preserving and nondegenerate, then each such braid is stable under Hamiltonian perturbations that are small with respect to the Hofer metric. This generalizes results of Alves-Meiwes. The proof uses foundational results about holomorphic curves that enter into the construction of embedded contact homology and periodic Floer homology, in particular the ECH partition conditions.

A non-degenerate contact form is lacunary if the indexes of every contractible periodic Reeb orbit have the same parity. To the best of my knowledge, every contact form with finitely many periodic orbits known so far is non-degenerate and lacunary. I will show that every non-degenerate lacunary contact form on a suitable prequantization of a closed symplectic manifold \(B\) has precisely \(r_B\) contractible closed orbits, where \(r_B=\dim H_*(B;{\mathbb Q})\). Examples of such prequantizations include the standard contact sphere and the unit cosphere bundle of a compact rank one symmetric space (CROSS). This is ongoing joint work with Miguel Abreu.

Escape orbits in celestial mechanics can often be seen as semilocal "singular" periodic orbits of Reeb vector fields. We prove that generic Reeb vector fields associated with singular contact forms in dimension 3 have (at least) \(2N\) or an infinite number of escape orbits, where \(N\) is the number of compact connected components of the critical set. This is reminiscent of the "two or infinitely many periodic orbits conjecture" for Reeb vector fields. On the other side of the spectrum, we can construct singular contact structures to customize the number of singular periodic orbits by introducing "singular bubbles". This construction shall lead us to a counterexample of the singular Weinstein conjecture. This talk is based on several joint works with Josep Fontana-McNally, Cédric Oms, and Daniel Peralta-Salas (some of them in progress).
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In this talk, we explain how the Toda lattice model can be used to show that certain Lagrangian product configurations in the classical phase space are symplectomorphic to toric domains. In particular, we will discuss the relationship between symplectic balls and (generalized) rhombic dodecahedrons. The talk is based on a joint work with Vinicius Ramos and Daniele Sepe.
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We discuss some extensions of Bezout's theorem in the context of spectral geometry and multi-dimensional complex analysis. Our approach involves methods of topological persistence. Joint work (in parts, in progress) with Lev Buhovsky, Jordan Payette, Iosif Polterovich, Egor Shelukhin and Vukasin Stojisavljevic.
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From Reeb vector fields of contact forms, we extend the existence of Birkhoff sections and broken book decompositions to the case of Reeb vector fields of Stable Hamiltonian Structures (SHS) on closed 3-manifolds and under a non-degeneracy hypothesis. There are a Reeb vector fields of SHS that have no periodic orbits, part of the results presented study the cases in which these flows have none or finitely many periodic orbits. This is joint work with R. Cardona.
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We find in every ball \(B^4(a)\) a finite union of planar Lagrangian discs \(\Delta\) such that \(B^4(a) \setminus \Delta\) symplectically embeds into the symplectic cylinder \(D^2(1) \times \mathbb R^2\). This extends a result of Sackel-Song-Varolgunes-Zhu and Brendel from \(a< 3\) to all \(a\). Among the applications are: capacity killing; non-displaceability of the Clifford torus \(T(1/d,1/d)\) from \(\Delta\) in \(B^4(d)\); and the existence of very short Reeb chords from a Legendrian knot back to itself or to \(\Delta\). This is a joint work with Emmanuel Opshtein.
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The strong closing property for Hamiltonian/Reeb flows concerns the ability to create periodic orbits passing through any given open set, via local \(C^\infty\) perturbations. This property is known to hold in dimensions 2-3 and is generally open in higher dimensions. I will discuss an approach to this problem that is inspired by the works of McDuff-Siegel and Hutchings. This approach uses direct measurements of pseudoholomorphic curves without alluding to any algebraic structure. It allows establishing the strong closing property for a class of flows we call "Hofer nearly periodic". This is based on a work in progress joint with Julian Chaidez.

We explain how one can deal with Homogenization for the stochastic Hamilton-Jacobi equation
\(\qquad\left\{ \begin{aligned} & \partial_t u^\varepsilon(t,x;\omega)+H(\tfrac{x}{ \varepsilon }, \partial_x u^\varepsilon (t,x;\omega))=0,\\ &u^\varepsilon (0,x;\omega)=f(x). \end{aligned} \right. \)
Under some coercivity assumptions on \(p\) - but without any convexity assumption - we prove that for a.e. \(\omega \in \Omega\) we have
\(\qquad\displaystyle C^0\mbox{-}\lim_{\epsilon\to0} u^\varepsilon (t,x;\omega)=v(t,x),\)
where \(v\) is the variational solution of the homogenized equation
\(\qquad \left\{ \begin{aligned} &\partial_t v (t,x)+\overline{H}(x, \partial_x v (t,x))=0,\\ & v (0,x)=f(x). \end{aligned} \right .\)