Gérard Ben Arous

About

Gérard Ben Arous is a Professor of Mathematics at the Courant Institute of Mathematical Sciences, New York University, where he arrived in 2002. He also serves as the Director of the Courant Institute and Vice Provost for Science and Engineering Development. A native of France, Professor Ben Arous studied Mathematics at École Normale Supérieure and earned his PhD from the University of Paris VII in 1981. His academic career includes positions at the University of Paris-Sud (Orsay), École Normale Supérieure, and the Swiss Federal Institute of Technology in Lausanne, where he held the Chair of Stochastic Modeling. He has also headed departments of Mathematics and Computer Science and founded the Bernoulli Center, a Mathematics research institute in Lausanne. He is the managing editor of the journal Probability Theory and Related Fields, alongside Amir Dembo of Stanford. His research focuses on probability theory and its applications, including stochastic analysis, large deviations, random media, and random matrices, as well as their connections to partial differential equations, dynamical systems, and physics, particularly statistical mechanics of disordered media. His main interests involve the time evolution of complex systems, the universal aspects of their long-term behavior, and the mechanisms of aging related to complexity and disorder. Recognized for his contributions, he is a Fellow of the Institute of Mathematical Statistics, an elected member of the International…

Research topics

• Computer Science
• Artificial Intelligence
• Mathematics
• Physics
• Statistical physics
• Mathematical analysis
• Geometry
• Algorithm
• Quantum mechanics
• Thermodynamics

Selected publications

• Algorithmic thresholds for tensor PCA

  The Annals of Probability · 2020 · 59 citations

  1st authorCorresponding

  We study the algorithmic thresholds for principal component analysis of Gaussian $k$-tensors with a planted rank-one spike, via Langevin dynamics and gradient descent. In order to efficiently recover the spike from natural initializations, the signal-to-noise ratio must diverge in the dimension. Our proof shows that the mechanism for the success/failure of recovery is the strength of the “curvature” of the spike on the maximum entropy region of the initial data. To demonstrate this, we study the…

  PublisherDOI

• Counting equilibria of large complex systems by instability index

  Proceedings of the National Academy of Sciences · 2021 · 42 citations

  1st authorCorresponding

  We consider a nonlinear autonomous system of [Formula: see text] degrees of freedom randomly coupled by both relaxational (“gradient”) and nonrelaxational (“solenoidal”) random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of “absolute instability” where equilibria are on average exponentially abundant, but typically, all of…

  PublisherDOI

• High‐dimensional limit theorems for SGD: Effective dynamics and critical scaling

  Communications on Pure and Applied Mathematics · 2023-10-04 · 17 citations

  articleOpen access1st author

  Abstract We study the scaling limits of stochastic gradient descent (SGD) with constant step‐size in the high‐dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite‐dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step‐size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former…

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• Landscape complexity beyond invariance and the elastic manifold

  Communications on Pure and Applied Mathematics · 2023-09-14 · 16 citations

  article1st author

  Abstract This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self‐interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as the Larkin mass, confirming formulas of Fyodorov and Le Doussal. One essential, dynamical,…

  PublisherDOI

• Shattering versus metastability in spin glasses

  Communications on Pure and Applied Mathematics · 2023-07-25 · 10 citations

  articleOpen access1st authorCorresponding

  Abstract Our goal in this work is to better understand the relationship between replica symmetry breaking, shattering, and metastability. To this end, we study the static and dynamic behaviour of spherical pure p ‐spin glasses above the replica symmetry breaking temperature . In this regime, we find that there are at least two distinct temperatures related to non‐trivial behaviour. First we prove that there is a regime of temperatures in which the spherical p ‐spin model exhibits a shattering ph…

  PublisherOA PDFDOI

Recent grants

• Slow Dynamics in Random Media

  NSF · $300k · 2008–2012

• Random Matrices, Complexity and Slow Dynamics in Random Media

  NSF · $420k · 2012–2016

Frequent coauthors

• Alice Guionnet

  Unité de Mathématiques Pures et Appliquées

  32 shared

• Alexander Fribergh

  Université de Montréal

  32 shared

• Paul Bourgade

  24 shared

• Jǐŕı Černý

  21 shared

• Alan Hammond

  20 shared

• Aukosh Jagannath

  University of Waterloo

  19 shared

• Reza Gheissari

  18 shared

• Nina Gantert

  17 shared

Labs

• NYU Courant Mathematics DepartmentPI

Education

• Ph.D.

  University of Paris VII

  1981

• Other

  École Normale Supérieure

• Other

  University of Paris-Sud (Orsay)

• Other, Chair of Stochastic Modeling

  Swiss Federal Institute of Technology in Lausanne

Awards & honors

• Senior Lady Davis Fellowship (Israel)
• Rollo Davison Prize (Imperial College, London)
• Montyon Prize (French Academy of Sciences)

Similar researchers at New York University

• Xi Chenh-121
• Kyunghyun Choh-90
• Chen Fengh-46