Simon Allais

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Research

Preprints

  • (with Pierre-Alexandre Arlove) $C^1$-Local Flatness and Geodesics of the Legendrian Spectral Distance
    • preprint
    • arXiv link
      • Abstract In this article, we give an explicit computation of the order spectral selectors of a pair of $C^1$-close Legendrian submanifolds belonging to an orderable isotopy class. The $C^1$-local flatness of the spectral distance and the characterisation of its geodesics are deduced. Another consequence is the $C^1$-local coincidence of spectral and Shelukhin-Chekanov-Hofer distances. Similar statements are then deduced for several contactomorphism groups.
  • (with Pierre-Alexandre Arlove et Sheila Sandon) Spectral selectors on lens spaces and applications to the geometry of the group of contactomorphisms
    • preprint
    • arXiv link
      • Abstract Using Givental's non-linear Maslov index we define a sequence of spectral selectors on the universal cover of the identity component of the contactomorphism group of any lens space. As applications, we prove that the standard Reeb flow is a geodesic for the discriminant and oscillation norms, and we define a stably unbounded conjugation invariant spectral pseudonorm.
  • (with Pierre-Alexandre Arlove) Spectral selectors and contact orderability
    • preprint
    • arXiv link
      • Abstract We study the notion of orderability of isotopy classes of Legendrian submanifolds and their universal covers, with some weaker results concerning spaces of contactomorphisms. Our main result is that orderability is equivalent to the existence of spectral selectors analogous to the spectral invariants coming from Lagrangian Floer Homology. A direct application is the existence of Reeb chords between any closed Legendrian submanifolds of a same orderable isotopy class. Other applications concern the Sandon conjecture, the Arnold chord conjecture, Legendrian interlinking, the existence of time-functions and the study of metrics due to Hofer-Chekanov-Shelukhin, Colin-Sandon, Fraser-Polterovich-Rosen and Nakamura.

Papers

  • Morse estimates for translated points on unit tangent bundles
    To appear in Annales l'Institut Fourier.
    • preprint
    • arXiv link
      • Abstract In this article, we study conjectures of Sandon on the minimal number of translated points of contactomorphisms in the special case of the unit tangent bundle of a Riemannian manifold. We restrict ourselves to contactomorphisms of a unit tangent bundle that lift diffeomorphisms of the base homotopic to the identity. We prove that there exist sequences $(p_n,t_n)$ where $p_n$ is a translated point of time-shift $t_n$ with $t_n\to+\infty$ for a large class of manifolds. In the case of Zoll Riemannian manifolds, we also prove estimates relating the number of translated points to either the sum of the Betti numbers of the bundle under a generic assumption or its cuplength under a $C^0$-closedness assumption.
  • (with Marie-Claude Arnaud) The dynamics of conformal Hamiltonian flows: dissipativity and conservativity
    To appear in Revista Matemática Iberoamericana.
    • preprint
    • arXiv link
      • Abstract We study in detail the dynamics of conformal Hamiltonian flows that are defined on a conformal symplectic manifold (this notion was popularized by Vaisman in 1976). We show that they exhibit some conservative and dissipative behaviours. We also build many examples of various dynamics that show simultaneously their difference and resemblance with the contact and symplectic case.
  • On the Hofer-Zehnder conjecture on weighted projective spaces
    Compositio Mathematica, 159 (2023), no. 1, 87–108.
    • published version
    • preprint
    • arXiv link
      • Abstract We prove an extension of the homology version of the Hofer-Zehnder conjecture proved by Shelukhin to the weighted projective spaces which are symplectic orbifolds. In particular, we prove that if the number of fixed points counted with their isotropy order as multiplicity of a non-degenerate Hamiltonian diffeomorphism of such a space is larger than the minimum number possible, then there are infinitely many periodic points.
  • On the Hofer-Zehnder conjecture on $\mathbb{C}\text{P}^d$ via generating functions (with an appendix by Egor Shelukhin)
    International Journal of Mathematics, 33 (2022), no. 10-11.
    • published version
    • preprint
    • arXiv link
      • Abstract We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in $\mathbb{C}\text{P}^d$ of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer-Zehnder conjecture in the non-degenerated case: every Hamiltonian diffeomorphism of $\mathbb{C}\text{P}^d$ that has at least $d+2$ non-degenerated periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of $J$-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type inequality for barcodes of Hamiltonian diffeomorphisms that arise from Floer theory, which lends itself to adaptation to the setting of generating functions.
  • (with Tobias Soethe) Homologically visible closed geodesics on complete surfaces
    To appear in Journal of Topology and Analysis.
    • published version
    • preprint
    • arXiv link
      • Abstract In this article, we give multiple situations when having one or two geometrically distinct closed geodesics on a complete Riemannian cylinder, a complete Möbius band or a complete Riemannian plane leads to having infinitely many geometrically distinct closed geodesics. In particular, we prove that any complete cylinder with isolated closed geodesics has zero, one or infinitely many homologically visible closed geodesics; this answers a question of Alberto Abbondandolo.
  • On the minimal number of translated points in contact lens spaces
    Proceedings of the American Mathematical Society, 150 (2022), no. 6, 2685-2693.
    • published version
    • preprint
    • arXiv link
      • Abstract In this article, we prove that every contactomorphism of any standard contact lens space of dimension $2n-1$ that is contact-isotopic to identity has at least $2n$ translated points. This sharp lower bound refines a result of Granja-Karshon-Pabiniak-Sandon and answers a conjecture of Sandon positively.
  • On periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}\text{P}^d$ via generating functions
    the Journal of Symplectic Geometry, Volume 20 (2022), Number 1, Pages: 1-48.
    • published version
    • preprint
    • arXiv link
      • Abstract Inspired by the techniques of Givental and Théret, we provide a proof with generating functions of a recent result of Ginzburg-Gürel concerning the periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}\text{P}^d$. For instance, we are able to prove that fixed points of pseudo-rotations are isolated as invariant sets or that a Hamiltonian diffeomorphism with a hyperbolic fixed point has infinitely many periodic points.
  • On the growth rate of geodesic chords
    Differential Geometry and its Applications, Volume 73, December 2020, 101668.
    • published version
    • preprint
    • arXiv link
      • Abstract We show that every forward complete Finsler manifold of infinite fundamental group and not homotopy-equivalent to $S^1$ has infinitely many geometrically distinct geodesics joining any given pair of points $p$ and $q$. In the special case in which $\beta_1(M;\mathbb{Z})\geq 1$ and $M$ is closed, the number of geometrically distinct geodesics between two points grows at least logarithmically.
  • A contact camel theorem
    International Mathematics Research Notices, Volume 2021, Issue 17, September 2021, Pages 13153–13181.
    • published version
    • preprint
    • arXiv link
      • Abstract We provide a contact analogue of the symplectic camel theorem that holds in $\mathbb{R}^{2n}\times S^1$ and generalizes the symplectic camel. Our proof is based on the generating function techniques introduced by Viterbo, extended to the contact case by Bhupal and Sandon, and builds on Viterbo's proof of the symplectic camel.
  • Improvement and generalisation of Papasoglu's lemma
    The Graduate Journal of Mathematics, Volume 3, Issue 1 (2018), 31-36.
    • published version
    • preprint
    • arXiv link
      • Abstract We improve an isoperimetric inequality due to Panos Papasoglu. We also generalize this inequality to the Finsler case by proving an optimal Finsler version of the Besicovitch's lemma which holds for any notion of Finsler volume.

Participation in seminars

Past