- Beyond the Parisi formula. Summer school in high-dimensional probability and its applications, Northwestern University, Chicago (August 2026). [abstract]
The Parisi formula is a landmark result in the theory of mean-field spin glasses. This formula expresses the limit free energy as a variational formula. One of the crucial shortcomings of this formula is that it is invalid for models whose covariance function is not convex. In this talk we explain that, when the spins are assumed to be centered \(\pm 1\), one can derive a formula for the limit free energy that is distinct from the Parisi formula and holds regardless of the convexity of the covariance function. This is based on joint works with Hong-Bin Chen and Jean-Christophe Mourrat.
- On the Hamilton-Jacobi approach to mean-field spin glasses. International Congress of Mathematicians 2026, Philadelphia (July 2026). [abstract]
Spin glasses, though originating in statistical physics as models of disordered magnetic systems, have had a remarkable influence across many scientific disciplines. Their key features—frustration, disorder, complex energy landscapes, and multiple metastable states—make them valuable metaphors and analytical tools in diverse fields. For mean-field spin glass 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 talk, we will discuss the links that exist between this formula and a certain Hamilton-Jacobi equation in Wasserstein space. Using those links, we will derive uniqueness results for maximizers of the Parisi formula for models with scalar and vector spins. This is based on joint work with Hong-Bin Chen and Jean-Christophe Mourrat.
- The convex structure of the Parisi formula for multi-species spin glasses. Séminaire Modélisation et Probabilités, Université Paris Cité, Paris (April 2026). [abstract]
Mean-field spin glass models are statistical physics models where a large number of units interact through a random energy function with many symmetries. Their rich phenomenology makes them ideal for developing tools and ideas aimed at tackling high-dimensional optimization problems. For some of the models relevant to those applications, the units are partitioned into several species.
For many spin glass models, it is known that the free energy can be expressed as the maximal value of an explicit functional; this is the celebrated Parisi formula. Moreover, for single-species models, it is also known that the variational problem appearing in the Parisi formula admits a unique optimizer. This allows one to derive very precise properties of the energy landscape using the Parisi formula.
In this talk, I will explain how those results can be adapted to the setting of multi-species models and the different ideas required to prove them. Surprisingly, this involves studying a Hamilton-Jacobi partial differential equation. This is joint work with Hong-Bin Chen and Jean-Christophe Mourrat.
- Spin glasses, overlap gap and learning. PhD student seminar, École Normale Supérieure de Lyon, Lyon (January 2026). [abstract]
In this talk we will explore the link between problems related to the statistical physics of disordered systems and the procedures used in practice to design and train neural networks in modern machine learning. We will put a special emphasis on the question of uniqueness of Parisi measures for spin glasses and the related overlap gap property which has recently been put forward as a barrier for efficient optimization.
- A Hopf-like formula for mean-field spin glass models. 53rd Saint-Flour probability summer school, Saint-Flour (July 2025). [abstract]
For mean-field spin glass 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 talk, using Fenchel-Moreau duality, we will 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.
- Statistical physics and Hamilton-Jacobi equations in infinite dimensions. ANR project SMOOTH annual meeting, École Normale Supérieure de Lyon, Lyon (June 2025). [abstract]
Disordered systems appear in many branches of physics, from condensed matter to neural networks. In this talk we explain a recently discovered link between those models and a certain class of Hamilton-Jacobi equations in Wasserstein space. We show that this link is useful in several ways. First, it is a source of examples of Hamilton-Jacobi equations with rich structure. For example, it can be shown that the unique solution of those equations is the limit of solutions to finite dimensional approximations, a property that is not satisfied by all infinite dimensional HJ equations. Second, this link also sheds light on some aspects of the Parisi formula and allows one to derive several different versions of this formula which seem better suited for generalization.
- The Parisi formula and beyond: a PDE approach to disordered systems. PhD student seminar, Institut Camille Jordan, Lyon (February 2025). [abstract]
Disordered systems appear in many branches of physics, from condensed matter to neural networks. In this talk, we introduce disordered systems through the dean's problem and the example of the Sherrington-Kirkpatrick model. As initially conjectured by physicists, the free energy of this model can be expressed via a variational formula. We will explain how this formula is related to an infinite dimensional Hamilton-Jacobi equation and how this observation allows one to formulate conjectures for the value of the free energy of more complicated models. If time allows, we will explain how this philosophy can be used to describe the free energy of a simple disorderless model, the Curie-Weiss model.
- The Parisi formula and beyond: a PDE approach to disordered systems. Séminaire Compréhensible, Institut Fourier, Grenoble (February 2025). [abstract]
Disordered systems appear in many branches of physics, from condensed matter to neural networks. In this talk, we introduce disordered systems through the dean's problem and the example of the Sherrington-Kirkpatrick model. As initially conjectured by physicists, the free energy of this model can be expressed via a variational formula. We will explain how this formula is related to an infinite dimensional Hamilton-Jacobi equation and how this observation allows one to formulate conjectures for the value of the free energy of more complicated models. If time allows, we will explain how this philosophy can be used to describe the free energy of a simple disorderless model, the Curie-Weiss model.
- Gaussian isoperimetry and stochastic localization. Working group on Eldan's stochastic localization, Institut Camille Jordan, Lyon (December 2024). [abstract]
The talk follows Section 3 of Eldan's 2022 ICM notes. We describe how one can prove the Gaussian isoperimetric inequality using a measure-valued martingale. Along the way we give a proof of the so-called Borell's noise stability inequality; this result can be understood as the statement that the half-spaces minimize the rate at which the heat escapes under the heat flow on Gaussian space.
- A Hamilton-Jacobi approach to disordered systems. PDE day, École Normale Supérieure de Lyon, Lyon (October 2024). [abstract]
Spin glasses are probabilistic models motivated by statistical physics. We will explain how the limit free energy of one of the simplest spin glass models, 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 one 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.
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