Simon Allais

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Research

Preprints

  • (with Marie-Claude Arnaud) The dynamics of conformal Hamiltonian flows: dissipativity and conservativity.
    • preprint
    • arXiv link
      • AbstractWe 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.
  • Morse estimates for translated points on unit tangent bundles.
    • preprint
    • arXiv link
      • AbstractIn this article, we study conjectures of Sandon on the minimal number of translated points in the special case of the unit tangent bundle of a Riemannian manifold. We restrict ourselves to contactomorphisms of $SM$ that lift diffeomorphisms of $M$ homotopic to 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. We also prove Morse estimates on the number of translated points in the case of Zoll Riemannian manifolds.

Papers

  • On the Hofer-Zehnder conjecture on weighted projective spaces.
    Compositio Mathematica, 159 (2023), no. 1, 87–108.
    • published version
    • preprint
    • arXiv link
      • AbstractWe 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
      • AbstractWe 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
      • AbstractIn 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
      • AbstractIn 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
      • AbstractInspired 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
      • AbstractWe 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
      • AbstractWe 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
      • AbstractWe 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