Colloquia

Yu Deng

The 4 dimensional Anderson model

We establish Gaussian scaling limit, for the (elliptic) Anderson model in 4 dimensions, with small coupling constant. This is a simplest example of a critical model in SPDEs, a theme which has been a major challenge of the field. Our theorem provides the first example where infinitely many renormalization constants are present and controlled. This is joint work with Hao Shen.

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Dallas Albritton

Instability, self-similarity, and non-uniqueness for the Euler equations

We will survey aspects of the (in)stability theory for the Euler equations, to which Susan Friedlander has contributed in a fundamental way. We will conclude with an explanation of how the Vishik non-unique solutions can arise in the inviscid limit.

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Alexis Vasseur

Boundary vorticity estimate for the Navier-Stokes equation and control of layer separation in the inviscid limit

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Helena Nussenzveig Lopes

On local energy conservation

Anomalous dissipation, i.e. the non-vanishing of the dissipation term in the limit of zero viscosity, is a cornerstone of turbulence theory. In the case of periodic 2D incompressible fluid flow there has been a lot of recent work in which anomalous dissipation is ruled out if the vorticity, which is the curl of velocity, is p-th power integrable or even a nonnegative Radon measure. The vanishing viscosity limit in domains with boundary is a classical open problem due mostly to the formation of large gradients near the walls. These may propagate into the bulk of the fluid through boundary layer separation and have the potential to yield anomalous dissipation. In this talk we will discuss recent work on absence of local anomalous dissipation in the vanishing viscosity limit in a bounded domain.

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Nathan Glatt-Holtz

New Approach to Unique Ergodicity for Degenerate SPDEs with the 2D Stochastic Navier–Stokes Equations as a Paradigm

We consider the two-dimensional stochastic Navier–Stokes equations on a bounded domain with no-slip boundary conditions, driven by a white-in-time noise acting through a small finite set of smooth, divergence-free profiles fixed independently of the Grashof number. In this setting we prove that

(i) the invariant probability measure is unique under a spectral condition (M) on the Malliavin covariance operator;
(ii) condition (M) follows from a Hörmander-type Lie bracket condition (H) on the forcing profiles; and
(iii) on the unit disk six explicit clamped polynomial profiles satisfy (H), giving a concrete instance in which uniqueness holds for every value of the viscosity.

Our approach is built on the asymptotic coupling criterion. Within this framework we develop a noninfinitesimal approach to infinite-dimensional hypoellipticity, which replaces the moment bounds on the tangent flow required by the standard approach, and unavailable on a domain, by a suitable localization in probability space. At the core of our analysis is an optimal coupling for a carefully tuned multiscale Wasserstein metric, which is employed to match the low modes of two solutions. Our new approach opens the possibility of extending unique ergodicity and mixing under hypoelliptic conditions to a variety of models inaccessible to the original Hairer–Mattingly theory. This is joint work with Vincent Martinez (CUNY), Geordie Richards (Guelph), and Alexandra Stavrianidi (Reed College).

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Gigliola Staffilani

New and Old Mathematical Results on Wave Turbulence Theory

In this lecture we give an overview of some recent mathematical advances in wave turbulence theory. We start with the original perspective of Bourgain on the study of the energy spectrum for a periodic nonlinear Schrödinger equation. Then we introduce wave kinetic equations as effective equations for the energy spectrum derived from weakly nonlinear dispersive equations and we end with examples of rigorous proofs energy transfer.

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Peter Constantin

Ideal Magnetic Reconnection

Magnetic fields do not change topology during smooth dynamics of ideal MHD. But topology change does occur in nature. Magnetic resistivity and near singularities have been suggested as a possible explanation. In this talk I will focus on a different explanation, also suggested in the physics literature: reconnection due to magneto-hydrodynamic inertia. I will describe 2D models that exhibit rigorous ideal topology change. The models are Hamiltonian, have global smooth solutions and also global unique Yudovich class solutions. The reconnection is obtained from merger of a pair of active scalars transported by incompressible velocities they create. Finite time merger without singularities and without resistivity or viscosity is proved rigorously. Limited smoothness of patch boundaries of one class of models persists through merger. Numerics suggest a very rich phase space. This is based on joint work with Lydia Boubendir, Hezekiah Grayer II and Zhongtian (Kevin) Hu.

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Tarek Elgindi

Aspects of Steady States to the 2D Euler equation

I will discuss various results about the flexibility and rigidity of steady states to the 2D Euler equations.

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Misha Vishik

Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid

We discuss non-uniqueness for the initial value problem for the Euler equations in dimension two with the corresponding vorticity in Lebesgue function spaces. Study of the linear (in)stability spectrum and the rescaling argument are necessary ingredients for the proof of the non-uniqueness theorem.

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Roman Shvydkoy

Large data relaxation for solutions of kinetic models arising in collective dynamics

In this talk we will present a new estimate on the spectral gap of a general alignment force defined by a symmetric communication kernel. Such forces arise in various models of collective dynamics including the kinetic Cucker-Smale and symmetrized Motsch-Tadmor models. The novelty of the estimate is that it demonstrates dependence of the spectral gap only on the thickness of the flock, and not on the upper bound on the density as in previously known attempts. As an application, we will prove relaxation of solutions to the Fokker-Planck-alignment models starting from large data.

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Alexander Kiselev

Singularity suppression by fluid flow

Transport by fluid flow can provide one of the less understood regularization mechanisms in PDE. In this talk, I will focus on the 2D Keller-Segel equation for chemotaxis set on a general domain and coupled via buoyancy with the fluid obeying Darcy’s law – a much studied model of the incompressible fluid flow in porous media. It is well known that solutions to the 2D Keller-Segel equation can form singularities in finite time if the mass of the initial data is larger than critical. It turns out that if the equation is coupled with fluid flow obeying Darcy’s law via buoyancy, this completely regularizes the system, leading to globally regular solutions for arbitrarily large initial data. One of the key ingredients in the proof is a new generalized Nash inequality, which employs anisotropic norm that is natural in the context of the incompressible porous media flow. This talk is based on works joint with Kevin Hu, Naji Sarsam, and Yao Yao.

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Anna Mazzucato

Global existence for the 2D Kuramoto-Sivashinsky equation

I will present recent results concerning global existence for the Kuramoto-Sivashinsky equation in 2 space dimensions with and without advection by an incompressible flow. The KSE is a model of flame-front propagation in combustion and more generally a model of long-wave instability in dissipative systems.

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Tom Hou

Nonuniqueness of Leray–Hopf Solutions for the Unforced 3D Incompressible Navier–Stokes Equations

The nonuniqueness of Leray–Hopf solutions to the unforced 3D incompressible Navier–Stokes equations is a central open problem in mathematical fluid dynamics. In joint work with Yixuan Wang and Changhe Yang, we develop the first rigorous computer-assisted proof of such nonuniqueness. Inspired by earlier work in this area, we construct a Leray–Hopf solution in a self-similar setting and then prove the existence of a second solution by studying the linearized operator around this profile and showing that it admits an unstable perturbation. Our approach combines a novel high-precision numerical method for computing candidate solutions with a rigorous framework for establishing exact solutions in their neighborhood. A key ingredient is the decomposition of the linearized operator into a coercive part and a compact perturbation, followed by a finite-rank approximation of the compact part up to a small error. We then rigorously verify, by computer-assisted proof, the invertibility of the linearized operator restricted to the image of this finite-rank approximation. This yields a certified unstable eigenpair and, consequently, a second solution—indeed, infinitely many Leray–Hopf solutions.

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Alexey Cheskidov

Energy cascades for the Navier-Stokes equations: from instantaneous blow-up to global existence of smooth solutions for distributional initial data

In this talk we will discuss the role of the forward and inverse energy cascades for non-uniqueness, blow-up, or global existence of smooth solutions of the unforced 3D Navier-Stokes equations. For smooth initial data, inverse energy cascade can result in an instantaneous blow-up and produce smooth for positive time solutions that enjoy the same regularity properties as the Koch-Tataru solutions. For distributional initial data, forward energy cascade can lead to enhanced dissipation and produce global in time very weak solutions. We characterize the class of distributional initial data for which classical weak solutions exist. In particular, any divergence-free initial datum in W^{s,p}, gives rise to a global weak solution, smooth for positive time, if and only if s>=-1. For instance, take any divergence-free initial datum that belongs to W^{-1,p} for some finite p>1 but not H^{-1}. Then the solution to the heat equation is not in L^2_{x,t}. On the other hand, the constructed weak solution is in L^2_{x,t}, as well as C([0,infinity);W^{-1,p}), and smooth for positive time. Hence, the NSE solutions can enjoy better integrability than the solutions of the heat equation. The nonlinear term regularizes the solution via a forward energy cascade, which also results in enhanced dissipation and hence better integrability than the heat flow.

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Mimi Dai

Blowup phenomena for fluid equations

We will discuss some constructions of blowup for fluid equations. In the case of the Naiver-Stokes equations, instantaneous blowup is constructed by exploiting backward energy cascade; for its dyadic models, we explore both instantaneous and finite time blowup. For the 2D unforced incompressible porous media equation, we present a construction of finite time blowup through an angular amplification mechanism.

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Jerry Bona

Bore propagation with dynamic boundary conditions

The lecture will sketch the derivation and analysis of a model for bore propagation that includes cross-river variation and dynamical boundary conditions.

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Anthony Suen

Existence, asymptotic behaviour and convergence of a generalised 3D Muskat problem in stable regime L∞

We address a generalised three-dimensional α-Muskat model that comes from the fluid interface problem given by two incompressible fluids with different densities in the stable regime. We establish local-in-time wellposedness when α∈[0,1) and also prove global-in-time existence for strong solutions when α∈[0,1/2) with initial data controlled by explicit constants. We obtain maximum principles for the L^∞-norms of both the solutions and their gradients, and we further acquire the corresponding decay rates of these L^∞-norms. Finally, we give some convergence results for strong solutions as α→ 0^+. This is a joint work with Qasim Khan and Bao Quoc Tang.

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Hongqiu Chen

Theory on Higher-order Water Wave Model

Considered here is a class of higher-order models for the unidirectional propagation of small amplitude long waves on the surface of an ideal fluid, namely,

(1) η_t + η_x − γ_1 β η_{xxt} + γ_2 β η_{xxx} + δ_1 β^2 η_{xxxxt} + δ_2 β^2 η_{xxxxx}
+(3/4) α (η^2)_x + α β (γ(η^2)_{xx} − (7/48) η^2_x )_x − (1/8) α^2 (η^3)_x = 0.

It was derived by Bona, Carvajal, Panthee and Scialom, where η = η(x, t) is the deviation of the free surface from its rest position at the point corresponding to x at time t. The two physical parameters α, β > 0 are small compared to one and the Stokes’ number S = α/β is of order one. The five parameters γ1, γ2, δ1, δ2 and γ are not arbitrary. The restrictions are spelled out in the details of the work by Bona and his collaborators just mentioned.

The pure initial-value problem is the problem (1) with initial data

(2) η(x, 0) = η_0 (x), x ∈ R

viewed as known. Assuming that δ_1 appearing in front of the η_{xxxxt}-term is positive and γ_1 is not too negative, if the initial data is selected from the L_2-based Sobolev space H^s (R) for s ≥ 1, then the initial-value problem (1)-(2) is locally well-posed in H^s (R). If, in addition, γ=(7/48), the well-posedness is global. If time allows, I will also discuss the initial-value problem with bore-like initial data.

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Showing of the video of  Terence Tao’s Lecture

“The Potential for AI in Science and Mathematics”

Monday, January 26th, 3:30 – 4:30 pm, KAP 414

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Harold Williams, USC

Monday, February 2nd, 3:30 – 4:30 pm, KAP 414

Title: An Introduction to Automated Theorem Proving in Lean

Abstract: In this talk we will give an introduction to Lean, a programming language adapted to the formal verification of mathematical proofs. Lean and its flagship library, mathlib, have been the focus of a dedicated user community for about a decade, but their visibility has grown significantly in light of recent advances in AI.
Lean-based automated theorem provers are now improving at a steady pace, and we will compare and contrast the training of such systems with the training of large language models such as GPT, Claude, or Gemini. We will also give a brief introduction to Monte Carlo Tree Search, an algorithm whose presence or absence provides one of the main high-level distinctions among different state-of-the-art automated provers. Its use in particular distinguishes Seed-Prover and Aristotle, two frontier models developed respectively by ByteDance and Harmonic, the latter in collaboration with an academic team including myself, Sergei Gukov, Dan Halpern-Leistner, and Alex Meiburg.

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Sergei Gukov, Caltech

Monday, February 9th, 3:30 – 4:30 pm, KAP 414

Title: The role of AI in mathematical (re)search

Abstract: At its core, scientific research is a search, a search for new ideas, new patterns, and new ways to explain or prove things. In this talk, I invite you to explore how AI is reshaping different stages of this process. We will see that while AI excels at many tasks, it still hesitates on others, such as long-horizon reasoning or far-out-of-distribution generalization. I view this as good news: it highlights how much meaningful AI research remains to be done. In fact, the goal of expanding AI’s role in mathematical research has become a motivation for advancing AI itself. I am genuinely excited that these two fields have come into such close contact over the past few years.

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Gunnar Carlsson, Stanford University and BluelightAI Inc.

Monday, February 23rd, 3:30 – 4:30 pm, KAP 414

Title: Topology, Data Science, and Deep Learning

Abstract: Approximating data sets by graphs and simplicial complexes has been shown to be a very useful way to obtain qualitative information about data, and more recently has been shown to similarly contribute to artificial intelligence. I will discuss the mathematics around this, with examples from various domains.

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Javier Gomez Serrano, Brown University

Monday, March 2nd, 3:30 – 4:30 pm, KAP 414

Title: Mathematical Exploration and Discovery at Scale

Abstract: Machine learning is transforming mathematical discovery, enabling advances on longstanding open problems. In this talk, I will discuss AlphaEvolve, a general-purpose evolutionary coding agent that uses large language models to autonomously discover old and new mathematical constructions and potentially go beyond them. AlphaEvolve tackles a wide variety of problems across analysis, geometry, combinatorics, and number theory. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. This illustrates how general-purpose AI systems can systematically successfully explore broad mathematical landscapes at an unprecedented speed, leading us to do mathematics at scale.

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Shanghua Teng, USC

Monday, March 9th, 3:30 – 4:30 pm, KAP 414

Title: Understanding and Characterizing Regularization: Learnability and Physics-Based Energy Guidance

Abstract: The quintessential learning algorithm of empirical risk minimization (ERM) is known to fail in various settings for which uniform convergence does not characterize learning. Relatedly, the practice of machine learning is rife with considerably richer algorithmic techniques, perhaps the most notable of which is regularization. Nevertheless, no such technique or principle has broken away from the pack to characterize optimal learning in these more general settings. The purpose of this research direction is to understand the role of regularization in data-driven machine learning.

First, we focus on an abstract statistical learning framework, present our work on characterizing the power of regularization in perhaps the simplest setting for which ERM fails: multiclass learning with arbitrary label sets. Using one-inclusion graphs (OIGs), we exhibit a local-regularization approach to obtain optimal learning algorithms that dovetail with tried-and-true algorithmic principles: Occam’s Razor as embodied by structural risk minimization (SRM), the principle of maximum entropy, and Bayesian inference.

Second, we share some of our on-going progress on designing physics-guided energy regularization for data-driven learning of the weak solution to parabolic PDEs. The goal here is to extend semi-supervised learning by exploiting auxiliary data and the underlying physical model to construct stronger regularization, enabling more efficient learning with optimal estimation and faster generalization.

Joint work (COLT 2024) with Julian Asilis, Siddartha Devic, Shaddin Dughmi, and Vatsal Sharan

Joint work with Xiaohui Chen and Zixiang Zhou.

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Alexey Cheskidov, Westlake University

Monday, March 23rd, 3:30 – 4:30 pm, KAP 414

Title: Instantaneous Type I blow-up and non-uniqueness of smooth solutions to the Navier-Stokes equations

Abstract: For any smooth, divergence-free initial data, we construct a solution of the Navier–Stokes equations that exhibits Type I blow-up of the L^\infty norm, while remaining smooth. An instantaneous injection of energy from infinite wavenumber initiates a bifurcation from the classical solution, producing an infinite family of spatially smooth solutions with the same data and thereby violating uniqueness of the Cauchy problem. A key ingredient is the first known construction of a complete inverse energy cascade realized by a classical Navier-Stokes flow, which transfers energy from infinitely high to low frequencies. This is a joint work with Mimi Dai and Stan Palasek.

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Scott Aaronson, University of Texas at Austin

Monday, April 13th, 3:30 – 4:30 pm, Pacific Time (US and Canada), Join Zoom Meeting: https://usc.zoom.us/j/98591406199

Title: Theoretical Computer Science and AI Alignment

Abstract: I’ll survey some areas where I think theoretical computer science, math, and statistics can potentially contribute to the urgent quest to align powerful AI with humane values. These areas include the watermarking of AI outputs, mechanistic interpretability (including Christiano’s “No-Coincidence Principle”), and guarantees for out-of-distribution generalization.

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