Center for Mathematical Sciences Research
IN-PERSON MATHEMATICAL PHYSICS SEMINARRUTGERS UNIVERSITYHILL 705____________________________________________
SEMINAR WILL BEGIN AT 12:10
COFFEE WILL BE AVAILABLE AT 11:50
THERE WILL BE A BROWN BAG LUNCH AFTER THE SEMINAR AT 1:15 PM.
Alberto Maspero – SISSA Trieste
Wednesday, September 23, 2026
Zoom opens: 10:30AM EDT
Seminar begins: 10:45AM EDT
Transfer of energy for pure-gravity water waves with constant vorticity
We prove growth of Sobolev norms for the two-dimensional periodic gravity water waves with constant nonzero vorticity, in infinite depth and with periodic boundary conditions. Precisely we construct weakly turbulent solutions exhibiting arbitrary large growth of high Sobolev norms, while having lower-order norms of small size, yielding the first rigorous construction of weakly turbulent solutions for the water waves equations. The proof relies on a new mechanism for generating energy cascades in quasilinear dispersive PDEs with sublinear dispersion and a nonlinear transport structure. A central ingredient is to exploit quasi-resonances from 2-wave interactions to produce a resonant transport operator that drives energy to high modes and causes Sobolev norm growth.
Anirvan Sengupta – Rutgers University
Date/Time/Location
Thursday, September 24, 2026, 12:10 pm; Hill Center 705
In-context Denoising, Attention and Diffusion
Modern neural networks can have enough parameters to memorize their training data, yet often generalize remarkably well. I will discuss this tension from a statistical-physics perspective, focusing on attention mechanisms and their relation to associative memory. I will first consider an in-context denoising problem for a one-layer transformer. For several simple data models, the Bayes-optimal denoiser can be derived explicitly and represented by linear or softmax attention, providing a direct connection to modern Hopfield models. In this picture, generalization is associated with a crossover of the Hopfield energy landscape from isolated stored patterns to a representation of the underlying data distribution. I will then discuss multilayer attention-only dynamics, where all samples evolve together. In an appropriate continuum limit, the resulting self-attractive dynamics can be related to anti-diffusion, connecting attention to score-based denoising and empirical Bayes. This suggests a unified view of associative memory, attention, and diffusion-based generative modeling.
Nigel Goldenfeld – University of California San Diego
Wednesday, September 30, 2026
Emergence and Generalization in Machine Learning
The remarkable ability of modern neural networks to generalize improves with increasing network capacity, even when the number of model parameters or effective degrees of freedom exceeds the number of training data points. Here we use dynamical mean field theory to show, in a simple setting of linear regression, that this surprising behavior is the outcome of a phase transition in the stochastic field theory describing the training process. We calculate the critical exponents and scaling function of the double descent phase transition, and show that it is marked by a breakdown of the fluctuation-dissipation theorem associated with broken ergodicity. We describe how this phenomenon is an example of emergent behavior, and that the appropriate response function has the same functional form as the simple London model of the superconducting transition, with the rigidity of the wave function corresponding to the neural network's ability to generalize accurately. Our results are distinct from earlier work, because we calculate the time-dependence specifically, not just the least norm equilibrium solutions. This is what enables us to identify the origin of the emergent behavior. Our work provides a specific framework with which to understand emergent behavior in artificial neural networks, and I outline some future directions.
Work performed in collaboration with Chan Li.
Kasper Larsen – Rutgers University
Thursday, October 1, 2026, 12:10 pm; Hill Center 705
Strict SDE Comparison for Cusp Coefficients and Counterexamples
We provide a tractable sufficient condition for strict comparison for solutions of the one-dimensional stochastic differential equation dX = sigma(X)dB, with initial condition x in an interval I, for sigma positive and continuous. Our proof is based on a two-dimensional Lyapunov argument, which allows us to prove strict comparison for some coefficients in the local Sobolev space W1p, for p between 1 and 2. We illustrate using sigma of x equal to 1 plus the absolute value of x to the power beta, for beta between 0 and 1, and show that strict comparison holds if and only if beta is at least one half. We give examples showing that neither local W1p regularity for p between 1 and 2 nor Hölder regularity of order beta between one half and 1 is sufficient for strict comparison, even when combined with boundedness, uniform ellipticity, global strong existence, and pathwise uniqueness.
Daniel Fisher – Stanford University
Wednesday, October 7, 2026
Eco-Evo Dynamics, Diversification, and Red Queen Phases Near a Snowy Fitness Peak
The caricature of evolution in a fitness landscape with trajectories approaching a fitness peak is very misleading., especially if drawn in two-dimensions. The dimensionality of both organismic phenotype spaces and environmental spaces are very large, and they are inextricably coupled by evolution changing the environment: A better caricature is a fitness snowscape. We show that near a high-dimensional snowy peak, multiple distinct eco-evolutionary phases exist, including a diverse Red Queen phase with continual turnover of strains but no overall individual or community improvement.
M. Cristina Marchetti – UC Santa Barbara
Wednesday, October 14, 2026
Active nematic solids and morphogenesis
Orientational order of elongated cells or muscle fibers is observed ubiquitously in biology and is known to play an important role in the organization of biological tissue. While most previous work has modeled living tissue as active fluids, in many circumstances both cell neighbor exchanges and cell divisions are suppressed and the tissue behaves more like an active solid. A striking example of this is the freshwater polyp Hydra, where topological defects in the muscle fiber orientation have been shown to localize to key features of the body plan. In this talk I will examine the behavior of active nematic solids, where the feedback between alignment, elasticity and biochemical signaling controls shape and structure.
131st Statistical Mechanics Conference Sunday December 13, 2026 - Tuesday December 15 2026 at 0800am - 0500pm nbspHill Center 100 Frelinghuysen Road Room 116 Piscataway NJ nbsp Celebrating the achievements of our guests of honor Eric Carlen, Eduardo Fradkin and Sid Redner
