Probability and Mathematical Physics Seminar

Synopsis:

We show that in the Sherrington-Kirkpatrick model at inverse temperature beta with uniform external field h, replica symmetry holds for the set of pairs (beta, h) predicted by de Almeida and Thouless (1978). The proof proceeds by a direct analysis of the Parisi measure using the characterization provided by Jagannath and Tobasco (2017). This argument was found in the spring of 2026 with the assistance of large language models. If time permits, we will also discuss the landscape of rigorous spin glass theory in the wake of the OpenAI problem release.

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TBA

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Synopsis:

TBA

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TBA

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Synopsis:

A dispersive estimate is a result of dynamical delocalization. It asserts that the supremum norm of wave packets decays like t^{-beta} as time goes on, under Schrodinger evolution. More precisely, || e^{itH} f||_infty \le t^{-beta} || f ||_1. A classical example is when H is the Laplacian on Z, where beta = 1/3. The fastest speed in one dimensional systems prior to our work was beta = 1/2, to the best of our knowledge. 

In this talk, I will discuss a result of very fast dispersion, where for any N no matter how large, we construct a 1d translation invariant graph with long range hopping where the dispersion speed is bounded by t^{-N}. The underlying Floquet function is a Weierstrass nowhere differentiable function W. As a special case we answer a question of Geman and Horowitz from 1980 regarding the existence of the local time of W for sufficiently large lacunarity.

The estimate boils down to controlling oscillatory integrals with highly irregular oscillatory phases, precluding standard van der Corput/stationary phase techniques. I will explain how we overcome this using a Dolgopyat inspired transfer operator method, which first arose in the study of Anosov flows. I will also mention a new van der Coprut estimate as well as some lower bounds on oscillatory integrals showing that our result is essentially sharp. Although the result is deterministic, the model bears analogies with series of random variables and leads to some questions on random Toeplitz matrices.

This is based on a joint work with Gaetan Leclerc and Tuomas Sahlsten.

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We study the open KPZ equation, a prototypical one-dimensional random growth model subject to Neumann boundary conditions. Using stochastic analytic tools, we show that a suitably resampled Brownian motion describes its long-time behavior. This is based on a joint work with Alex Dunlap and Tommaso Rosati.

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Synopsis:

Lattice Yang–Mills theory was introduced by Ken Wilson in 1974, in part to give a mathematical framework for understanding quark confinement—the physical phenomenon that quarks are not observed in isolation. Wilson proposed the area law for Wilson loop expectations as a criterion for confinement: for large rectangular loops, the expectation should decay exponentially in the area enclosed by the loop, rather than merely in its perimeter.

I will start by briefly reviewing joint work with Scott Sheffield and Sky Cao in which we substantially expanded the regime of parameters where an area law is proved. A key ingredient in this work is the center symmetry of the model. The main focus of the talk will be recent and upcoming solo work in the opposite direction: when the relevant Wilson loop is insensitive to the center of the gauge group, we prove a perimeter-law lower bound, ruling out Wilson's confinement criterion. The result holds in dimensions three and higher and, importantly, extends to the weak-coupling regime relevant to the continuum theory. I will also briefly discuss a Lean formalization of the proof and the role that AI tools played in its development.

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Synopsis:

If all goes to plan I will explain the terminology in the title, then describe the analysis of a random Riemann-Hilbert problem, and apply the method to the study of extremal solutions and soliton gases.  Convergence theorems and the overarching quest will be described.  I will focus mainly on joint work with Katerina Gkogkou and Guido Mazzuca.

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