Webpage of Cyril Falcon

Research interests

I investigate contact rigidity aspects of isotopies through closed Legendrian submanifolds in contact manifolds by studying the algebraic structure of Legendrian submanifolds invariants constructed from generating families. From a general perspective, I am also interested in the categorical correspondence between these invariants and the ones built from pseudo-holomorphic curves and constructible sheaves.

Key words. Legendrian submanifolds, Contact rigidity, Generating families, Moduli spaces analysis.

I am working on a conjecture made by M. B. Henry and D. Rutherford in 2013 regarding how to compute the differential for generating family homology of Legendrian submanifolds from their front projections by counting ”gradient staircases” flow lines. In my thesis, I addressed this question by studying an adiabatic relaxation process for gradient flows of difference functions and developping an appropriate compactness-gluing strategy. I showed that the trajectories of the associated slow-fast dynamical systems "accumulate" on chains of gradient staircases.

Research papers

  1. Gradient staircases and generating family homology of Legendrian submanifolds
    (in preparation).
    Joint work with Frédéric Bourgeois.
  2. Roadblock to non-Euclidean geometry: A planar bias in the identification of straight paths
    (in preparation).
    Joint work with Charlotte Barot and Véronique Izard.
  3. Recursive KalmanNet : Analyse des capacités de généralisation d'un réseau de neurones récurrents guidé par un filtre de Kalman
    In 30e Colloque sur le traitement du signal et des images, GRETSI, (2025).
    Joint work with Mathéo Clavaud, Jean-Philippe Michel and Hassan Mortada.
  4. Recursive KalmanNet: Deep Learning-Augmented Kalman filter for State Estimation and Consistent Uncertainty Quantification
    In 2025 33rd European Signal Processing Conference (EUSIPCO), IEEE, (2025).
    Joint work with Mathéo Clavaud, Yanis Kahil, Jean-Philippe Michel and Hassan Mortada.
    [arXiv version]
  5. Generating family homologies of Legendrian submanifolds and moduli spaces of gradient staircases
    Doctoral thesis, Université Paris-Saclay, defended on January 25th, 2023.
    Supervisor: Professor Frédéric Bourgeois.
    President: Professor Claude Viterbo [Defence report (in French)].
    Referees and examinators: Professor Alexandru Oancea [Report (in French)] and Professor Lisa Traynor [Report].
    Examinators: Baptiste Chantraine, Ph.D. and Professor Urs Frauenfelder.
    [Dissertation and slides]

Research talks

If you are looking for the informal talks I gave, please visit my Teaching and outreach page.

Research events attended

Recurring events

One-time events

Other research-related writings

If you are looking for my other mathematical writings, please visit my Miscellaneous page.