The informal seminar is open to all visitors at IHES, researchers, and students in the Paris area. We aim at keeping the informality, which implies, many (friendly) interruptions, many (naive and basic, occasionally smart) questions, spontaneous (blackboard only, sometimes unfinished) talks, and a welcoming atmosphere. We expect the seminars to be in English only.

## Organizers: Thierry Bodineau, Piet Lammers, Yilin Wang

If you want to be added to the mailing list, please write to Yilin ([email protected])

## Past Sessions

## Time and place:

IHES,

Talks are 90 minutes long.

**Amphi Motchane**,**Fridays**at**2 pm**(unless indicated otherwise)Talks are 90 minutes long.

## Upcoming seminars

**November 27, 2024 (Unusual time)**

**Hong-Bin Chen (IHES)**

**On free energy in non-convex mean-field spin glass models**

We start by reviewing the classical Sherrington-Kirkpatrick (SK) model. In this model, +1/-1-valued spins interact with each other subject to random coupling constants. The covariance of the random interaction is quadratic in terms of spin overlaps. Parisi proposed the celebrated variational formula for the limit of free energy of the SK model in the 80s, which was later rigorously verified in the works by Guerra and Talagrand. This formula has been generalized in various settings, for instance, to vector-valued spins, by Panchenko. However, in these cases, the convexity of the interaction is crucial. In general, the limit of free energy in non-convex models is not known and we do not have variational formulas as valid candidates. Here, we report recent progress through the lens of the Hamilton-Jacobi equation. Under the assumption that the limit of free energy exists, we show that the value of the limit is prescribed by a characteristic line; and the limit (as a function) satisfies an infinite-dimensional Hamilton-Jacobi equation "almost everywhere". This talk is based on a joint work with Jean-Christophe Mourrat.

**December 6, 2024**

Ilya Losev (Cambridge)

Ilya Losev (Cambridge)

**December 13, 2024**

Justin Salez (Paris-Dauphine & PSL)

Justin Salez (Paris-Dauphine & PSL)

**A new approach to the cutoff phenomenon**

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergoned by certain Markov processes in the limit where the number of states tends to infinity. Discovered forty years ago in the context of card shuffling, it has since then been established in a variety of contexts, including random walks on graphs and groups, high-temperature spin systems, or interacting particles. Nevertheless, a general theory is still missing, and identifying the general mechanisms underlying this mysterious phenomenon remains one of the most fundamental problems in the area of mixing times. In this talk, I will give a self-contained introduction to this fascinating question, and then describe a new approach based on entropy and curvature.

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## Past seminars

**September 13, 2024**

Slava Rychkov

Slava Rychkov

**(IHES)**

**Real-space renormalization of 2D lattice models with tensor networks**

Tensor networks are a recent cool addition to physicist’s toolkit used to study renormalization of lattice models. However the mathematical theory of tensor network renormalization group (TNRG) is still in its infancy. I will aim to transmit my excitement about the tensor networks. Rough plan:

1. Wilson’s conjecture about renormalization group fixed points describing criticality - can we prove it?

2. Why are tensor networks better than other approaches to renormalization (e.g. spin blocking).

3. Numerical algorithms for TNRG - what do people see numerically?

4. Discrete scaling operator

5. A few mathematically rigorous results about TNRG

6. Open problems

**October 11, 2024**

Renan Gross (Cambridge)

Renan Gross (Cambridge)

**A sharp lower bound on the small eigenvalues of surfaces**

The Laplacian is a central operator in the analysis of surfaces (and life in general). In this talk, we investigate how small its small eigenvalues can be, giving a sharp, quadratic bound on the k-th eigenvalue of a surface in terms of k, the surface's genus g, and its global geometry via the injectivity radius. The techniques involve extremal length, spectral embedding, and volume arguments.

Joint work with Guy Lachman and Asaf Nachmias, based on the paper: https://arxiv.org/abs/2407.21780

For an exposition and overview of the paper, see here: https://sarcasticresonance.wordpress.com/2024/08/02/new-paper-on-arxiv-a-sharp-lower-bound-on-the-eigenvalues-of-surfaces/

@ IHES, Bures-sur-Yvette, France