Center for Mathematical Sciences Research
Jed Pixley – Rutgers University
Date/Time/Location
Thursday, October 22, 2026, 2:00 pm; Hill Center 313
PLEASE NOTE SPECIAL TIME AND LOCATION OF THIS SEMINAR
A “critical” take on the magic-angle in twisted bilayer graphene
Twisted bilayer graphene near the magic-angle has attracted a considerable amount of attention following the discovery of correlated insulating phases and superconductivity. Usually incommensurate effects are assumed to be weak or ignored entirely. We demonstrate that at the magic-angle, the system undergoes a single-particle, incommensurate transition where the wavefunctions delocalize in momentum space, concomitant with the onset of a finite density of states at the Dirac node energy, which has an observable effect on the screening of local vacancies. To make these effects pronounced enough to observe in experiment requires aligning twisted bilayer graphene with the hexagonal Boron Nitride sample, which we show both theoretically and experimentally reveals multifractal eigenstates.
Lai-Sang Young - NYU
Wednesday, October 21, 2026
Zoom opens: 10:30AM EDT
Seminar begins: 10:45AM EDT
"Neuroscience meets dynamical systems: building a model of primary visual cortex"
In this talk, I would like to discuss some modeling work in which simple mathematical ideas were used to shed light on neural mechanisms. The model in question simulates a small region of the primary visual cortex (V1) of the Macaque monkey, whose visual system is very similar to that of humans. Our model is a dynamical network of spiking neurons capable of reproducing dozens of V1 function and is (to a degree) built to "compute like the real brain". I will attempt to clarify/propose answers to some basic questions in visual neuroscience, such as how exactly orientation selectivity (OS) works, and the origin of direction selectivity (DS). OS and DS are visual functions enjoyed by neurons in V1 but not by their LGN afferents.
Yi-Zhi Huang – Rutgers University
Thursday, October 15 , 2026, 12:10 pm; Hill Center 705
Mathematical construction and study of conformal-field-theoretic tensor categories
Tensor categories now play an important role in the study of the representation theory of infinite-dimensional Lie algebras and other algebras related to two-dimensional conformal field theories. They have also been viewed as a type of generalized symmetries in physics (symmetry topological field
theories) and are used to describe and study fractional quantum Hall systems, anyons, and other phenomena in condensed matter physics. Recently there are a lot of progresses in the general construction and study of tensor categories arising from the mathematical approaches to conformal field theories, especially in the representation theory of vertex operator algebras. I will discuss some of these main progresses and will also discuss the connection with the probability construction of the Liouville conformal field theory.
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.
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.
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.
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.
Anirvan Sengupta – Rutgers University
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.
Alberto Maspero – SISSA Trieste
Wednesday, September 23, 2026
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.
Arup K Chakraborty – MIT
Wednesday, September 9, 2026
How the Immune System Learns
The humoral immune system, comprised of B cells and their antibody and memory B cell products, plays an important role in protecting us from infection. This system is also a learning algorithm. Antibodies and memory B cells are produced by a Darwinian evolutionary process. This is a non-equilibrium stochastic dynamic process that allows the immune system to learn about a new antigen (pathogen or vaccine component). I will first describe results obtained from statistical physics-based models of these processes and complementary data from animals and humans. These studies show that the human immune system has a remarkable ability to learn to develop responses that can respond to previously unseen variant antigens upon “training” with only a few exposures to the same antigen. The mechanism underlying this ability to generalize will be discussed. I will then discuss how exploring and exploiting analogies between how the immune system learns and how machines learn is now enabling us to address basic scientific questions that can potentially guide better strategies to cure and prevent disease. Finally, I will comment on similarities and differences between learning in the immune system and deep learning networks.
Eduardo Fradkin – University of Illinois at Urbana Champaign
Wednesday, September 2, 2026
The Quantum Roughening Problem revisited
In 1978 Andreev and Parshin conjectured that the surface of a quantum crystal would become rough due to effects of quantum fluctuations. Two papers published in 1983, one by Daniel Fisher and John Weeks and another one by me, showed that quantum fluctuations make these surfaces smooth and are not rough. In my talk I will revisit these arguments and sinus some interesting connections between this problem and other problems of interest such as quantum dimer models and Josephson junction arrays. In particular I will discuss the case of a model with a global conservation law in its quantum dynamics and the existence of two types of smooth phases in this case.
Karin Dahmen – University of Illinois at Urbana Champaign
Wednesday, August 26, 2026
Universal Avalanches and Prediction: from Nanoindentation and the Brain to Earthquakes and Stars?
Slow nanoindentation, or slow compression of nano-crystals, bulk metallic glasses, rocks, granular materials, and the earth all show intermittent slips or “quakes”. We find that although these systems span 12 decades in length scale, they all show the same scaling behavior for their slip size distributions and other statistical properties. Remarkably, the size distributions follow the same power law multiplied with a stress-dependent cutoff, indicating an underlying nonequilibrium phase transition. A simple model for avalanches of slipping weak spots explains the agreement across scales. It predicts the observed slip-size distributions and the observed stress-dependent cutoff function. The analysis draws on tools from statistical physics and the renormalization group. The results enable extrapolations from one scale to another, and from one force to another, across different materials and structures, from nanocrystals to earthquakes. Connections to neuron avalanches in the brain and recent observations on stars will also be discussed, extending the range of scales to 16 decades in length.
Cris Moore– Santa Fe Institute
Wednesday, August 19, 2026
Which links matter most? Sparsifying network dynamics with effective resistance
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“Sparsification” is the act of reducing a network to a subset of its edges while approximately preserving its properties: either to reduce the computational cost of solving problems about it, or to identify which edges are the most important in some sense. Computer scientists have developed beautiful techniques for sparsifying a graph using physics-related ideas like the effective resistance. However, while these methods preserve the spectral properties of the Laplacian, it is not obvious to what extent they preserve the behavior of nonlinear dynamical systems. Using a mobility network from the United States as an example, I’ll show that they do very well for the SIR epidemic model, including the probability each node becomes infected and its distribution of arrival times, even when the sparse network includes less than 10% of the original edges. Choosing edges using purely topological methods, or by thresholding edge weights, does not perform nearly as well. I will end by discussing the possibility of using sparsification to “denoise” networks from bioinformatics, and present some preliminary results on the Kuramoto model of coupled oscillators.
This is joint work with Alexander Mercier (Harvard School of Public Health), Emmie Fitz-Gibbons (Brown), and Sam Scarpino (Northeastern).
Christopher Jarzynski – University of Maryland
Wednesday, July 29, 2026
Estimating free energy differences with virtually escorted trajectories
The convergence of numerical free energy estimation methods can be accelerated using artificial fields that “escort” simulated trajectories along near-equilibrium paths. Unfortunately, designing such fields is not easy. Taking a cue from the mathematics behind diffusion models – a class of generative models in machine learning – we introduce a method based on virtual escorting. This method adopts a post-processing approach. Given a fixed set of nonequilibrium trajectories, a parameter-dependent virtual escorting field is constructed, possibly using a neural network. This field is used in combination with the trajectories to produce an estimate of the desired free energy difference. The parameters are then adjusted to optimize the convergence of the estimate. I will describe the method, and will discuss conditions under which it produces a zero-variance estimator of the free energy difference.
Uri Alon - Weizmann Institute
Wednesday, July 22, 2026
Why is the upper tail of human lifespan so rigid? Insights from Langevin threshold-crossing models of aging
Human life expectancy has doubled over the last two centuries, yet the upper tail of human lifespan has barely moved. Why is median lifespan so plastic while extreme longevity remains so rigid? I will present a stochastic threshold-crossing view of aging, in which physiological damage follows time-dependent Langevin dynamics and death occurs as a first-passage event across a critical threshold. In this framework, the exponential increase in mortality with age corresponds to escape from a linearly declining barrier. The model separates two classes of parameters: **robustness parameters**, such as noise amplitude and threshold height, which affect the probability of crossing the barrier, and **senogenic parameters**, which govern the deterministic damage production and removal. Changes in robustness can improve median survival while preserving extreme-lifespan distributions, whereas changes in senogenic parameters shift extreme longevity. I will discuss how this distinction helps interpret historical mortality improvements, lifestyle and socioeconomic effects, which all seem to only affect robustness parameters, as well as familial longevity, and progeroid diseases.
Larry Abbott - Columbia University
Wednesday, July 15, 2026
Biological Systems for Spatial Orientation
I will discuss several topics related to how animals represent and update their sense of orientation in the world. In flies, this is done with a set of neurons that collectively form a ring-attractor network. Activity in this system indicates the azimuthal orientation of the fly. I will discuss how the polarization pattern of the sky is used as keep this encoded angle aligned with the physical world. Vertebrates have an analogous system for head orientation, but it is more complex. I will discuss various ways that a ring-attractor could be embedded in a large population of neurons, with results relevant for studies of the synaptic connections in these circuits. Finally, in both flies and vertebrates, these systems have been studied primarily in two dimension (that is, considering only yaw rotations). I will discuss different ideas about how they might respond to three-dimensional rotations.
Itamar Procaccia - Weizmann Insitute Wednesday, July 8, 2026 Zoom opens: 10:30AM EDT Seminar begins: 10:45AM EDT Amorphous Solids: More than Meets the Eye Understanding the mechanical response of amorphous solids, like glasses and granular media, requires considerations that go
Michael Shelley - Flatiron Institute & NYU Wednesday, July 1, 2026 Zoom opens: 10:30AM EDT Seminar begins: 10:45AM EDT Modeling chromatin as an active polymer melt In this talk I'll discuss a line of work wherein chromatin in the nucleus
