Victor Issa

Preprints

• A Hopf-like formula for mean-field spin glass models. Preprint (2024). [abstract] [pdf] [arXiv] [tex]

  We study mean-field spin glass models with general vector spins and convex covariance function. For those models, it is known that the limit of the free energy can be written as the supremum of a functional, this is the celebrated Parisi formula. In this paper, we observe that the Parisi functional extends into a concave and Lipschitz functional on the set of signed measures. We use this fact and Fenchel-Moreau duality to derive an un-inverted version of the Parisi formula. Namely, we show that the limit of the free energy can be written as the infimum of a functional related to the Parisi functional. This un-inverted formula can be interpreted as a Hopf-like formula for some Hamilton-Jacobi equation in Wasserstein space.

• Weak-strong uniqueness principle for Hamilton-Jacobi equations. Preprint (2024). [abstract] [pdf] [arXiv] [tex]

  We show that if a Hamilton-Jacobi equation admits a differentiable solution whose gradient is Lipschitz, then this solution is the unique semi-concave weak solution. Our result does not rely on any convexity (nor concavity) assumptions on the initial condition or the nonlinearity, and can therefore be utilized in contexts where the viscosity solution admits no standard variational representation.

• Existence and uniqueness of permutation-invariant optimizers for Parisi formula. Preprint (2024). [abstract] [pdf] [arXiv] [tex]

  It has recently been shown that, upon constraining the system to stay in a balanced state, the Parisi formula for the mean-field Potts model can be written as an optimization problem over permutation-invariant functional order parameters.
  
  In this paper, we focus on permutation-invariant mean-field spin glass models. After introducing a correction term in the definition of the free energy and without constraining the system, we show that the limit free energy can be written as an optimization problem over permutation-invariant functional order parameters. We also show that for some models this optimization problem admits a unique optimizer. In the case of Ising spins, the correction term can be easily removed, and those results transfer to the uncorrected limit free energy.
  
  We also derive an upper bound for the limit free energy of some nonconvex permutation-invariant models. This upper bound is expressed as a variational formula and is related to the solution of some Hamilton-Jacobi equation. We show that if no first-order phase transition occurs, then this upper bound is equal to an unconditional lower bound previously derived in the literature. We expect that this hypothesis holds at least in the high-temperature regime.
  
  Our method relies on the fact that the free energy of any convex mean-field spin glass model can be interpreted as the strong solution of some Hamilton-Jacobi equation.

Publications

• Uniqueness of semi-concave weak solutions for Hamilton-Jacobi equations. Accepted by Annales de la Faculté des Sciences de Toulouse. [abstract] [pdf] [arXiv] [tex]

  It is well known that when the nonlinearity is convex, the Hamilton-Jacobi PDE admits a unique semi-convex weak solution, which is the viscosity solution. In this paper, motivated by problems arising from spin glasses, we show that if the Hamilton-Jacobi PDE with strictly convex nonlinearity and regular enough initial condition admits a semi-concave weak solution, then this solution is the viscosity solution.

Invited Talks

• A Hamilton-Jacobi approach to disordered systems. PDE day, École Normale Supérieure de Lyon, Lyon (October 2024). [abstract] [slides]

  Spin glasses are probabilistic models motivated by statistical physics. We will explain how the limit free energy of one of the simplest spin glass model, the SK model, can be expressed using the unique viscosity solution of a specific infinite dimensional Hamilton-Jacobi equation. The limit free energy of more complicated spin glass models, such as the bipartite model, is not well understood. This observation about the limit free energy of the SK model allows to formulate a conjectural expression for the limit free energy of the bipartite model in terms of the viscosity solution of another specific infinite dimensional Hamilton-Jacobi equation. The justification of this conjecture is a long-term project that requires a combination of advanced probability and analysis techniques. We will give a list of analysis and PDE problems related to this conjecture. We will present partial resolutions of two of those problems.

Teachings

• (Fall 2024) Exercise sessions for the Licence 3 course "Integration and Measure" at ENS de Lyon in collaboration with Thomas Buc-d'Alché and Mathieu Mourichoux, lectures are given by Laurent Laflèche.
  • TD01 : Ensembles et dénombrabilité. [pdf]
  • TD02 : Ensembles mesurables et mesures. [pdf]
  • TD03 : Applications mesurables & Intégration de fonctions étagées. [pdf]
  • TD04 : Intégration I. [pdf]
  • TD05 : Intégration II. [pdf]
  • TD06 : Cantor & Stieljes. [pdf]
• (Spring 2024) Exercise sessions for the Master 1 course "Stochastic Processes" at ENS de Lyon, lectures were given by Emmanuel Jacob.
  • TD01 : Continuous Time Stochastic Processes. [pdf]
  • TD02 : Poisson Processes. [pdf]
  • TD03 : Continuous Time Markov Chains. pdf]
  • TD04 : Stopping Times. [pdf]
  • TD05 : Kolmogorov Equations. [pdf]
  • TD06 : Invariant Measures. [pdf]
  • TD07 : Brownian Motion. [pdf]
  • TD08 : Construction of the Brownian Motion. [pdf]
  • TD09 : Regularity of Brownian Trajectories. [pdf]
  • TD10 : Continuous Time Martringales. [pdf]
  • TD11 : Donsker's Invariance Principle. [pdf]
  • TD12 : Harmonic Functions. [pdf]

Mathematics Outreach

• (with William Sarem) À la recherche des triplets pythagoriciens. Images des Mathématiques (2021). [link]
• La formule de Riemann pour compter les nombres premiers. Journal de Mathématiques de l'École. Hors série n°4, 25-32 (2019). [link]

Conferences Attended

• Statistical and probabilistic analysis of random networks and processes. Laboratoire Jean Alexandre Dieudonné, Nice (September 2024).
• 52th Probability summer school. Saint-Flour (July 2024).
• Conceptual Challenges in AI: from ML to Average-Case Computation and Cryptography. Bocconi University, Milano (May 2024).
• PDE & Probability in interaction: functional inequalities, optimal transport and particle systems. Centre International de Rencontres Mathématiques, Marseille (January 2024).
• High dimensional statistics and random matrices. Porquerolles (June 2023).
• Midwest probability colloquium. Northwestern University, Chicago (October 2022).
• Fields-CFI workshop on impacts of climate change on economics, finance, and insurance. Fields Institute, Toronto (September 2022).

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