A Stochastic Differential Equation (SDE) which describes the evolution of a stochastic process is intimately related to the evolution equation that governs the law of the random variable, $X_t \sim f(t,x)$, called the Fokker-Planck equation.  Taking existence results for both of these problems for granted, we will show that the uniqueness of martingale solutions to the SDE and uniqueness of weak solutions to the Fokker-Planck equation are equivalent. 

The Cauchy-Lipschitz theorem for ODEs provides a well-posedness theory for transport equations, provided the advecting field is smooth. When the field is rough, it is well known that solutions may cease to be unique. In this talk I will present the DiPerna-Lions theory, which shows the well-posedness for Sobolev vector fields. Time permitting, I will describe an application of these techniques to weak sequential compactness of solutions to the compressible Navier Stokes equations. 

When training parametrized models (such as neural networks), the objective function can be written in terms of the output or the parameters. I will show how the dynamics of gradient flow can change depending on the metric chosen in parameter space, and why this might be useful. Joint work with Thomas Chen. 

We will talk about the global existence of weak solutions for various models of fluid motion which are subsequently more delicate.

Most real analysis classes cover two “interpolation theorems”, that of Marcinkiewicz and that of Riesz–Thorin. Both are used throughout analysis, for example to provide streamlined proofs of the Hausdorff–Young theorem or the boundedness of singular integral operators. However, these theorems are just the tip of the iceberg of what is now called interpolation theory, the study of vector spaces which lie between two others. In this talk, we will discuss the two main types of “interpolation spaces”, their properties, and their applications to analysis.

We define Kakeya sets and their properties. These sets have a surprising application in the Ball Multiplier theorem: a result about the Fourier transform on $L^p$. We finish with a discussion of the Kakeya conjectures and their relationship between other conjectures in analysis.

I will be talking about posterior sampling, which is where you sample from an unknown distribution given certain information about your sample. This turns out to be a very difficult problem, even when the prior is nice. As such, our friends in the ECE department said we should abandon this goal entirely and instead sample from a slightly noisy version of the true posterior and that this objective is much more achievable. I will explain why this seems to be true.

In the first part of the talk, we review some fundamental results in elliptic regularity theory, motivated by Hilbert’s 19th problem. This problem concerns the regularity of minimizers of variational problems whose Euler–Lagrange equations are elliptic. In the second part, we discuss the Monge–Ampère equation, which arises in optimal transport. This equation is elliptic but not uniformly so, and therefore falls outside the classical framework described in the first part of the talk. If time permits, I will explain how ideas from elliptic regularity can nevertheless be used to obtain nontrivial stability estimates for optimal transport. This last part on stability is based on current joint work with Matias Delgadino and Jun Kitagawa.

Logarithmic Sobolev inequalities (LSI) play an important role in the analysis of diffusion processes, high dimensional probability, and optimal mass transport. We’ll introduce log-Sobolev inequalities in finite-dimensional Euclidean space, examine some basic consequences, and make a natural connection to linear diffusions via the Bakry-Émery criterion. Time permitting, we’ll sketch the Euclidean proof of Bakry-Émery (Gamma calculus), see a relaxed criterion, and perhaps review some issues in the “high-dimensional” setting. 

For hyperbolic conservation law PDEs, we can prove the existence of solutions for all time when the initial data has small total variation using front tracking approximations. I will review some details of the Riemann problem and then describe the basic front tracking algorithm. I will also prove that the Glimm potential of our approximation is decreasing in time, which will imply that the number of wave fronts in our algorithm stays finite.

Generative diffusion models have emerged as a favored framework with which to generate image and video data. These models work by simulating a “forward” process that applies noise to sample data, then inferring a “reverse” process to recover an approximation of the sample data from noise. I will discuss a PDE framework through which we can study diffusion models. In particular, I will show how to view the reverse diffusion dynamics as a transport equation, then use this to uncover how the distribution of the generated sample compares to that of the original sample.

In this talk, we motivate the notion of invariant measures through finite-dimensional Hamiltonian systems and the Fermi–Pasta–Ulam experiment, with the Poincaré recurrence theorem as a first illustration. We then turn to the complex-valued modified KdV equation on the one-dimensional torus, viewed as the second equation in the nonlinear Schrödinger hierarchy and as a completely integrable Hamiltonian PDE. After discussing the construction of Gibbs and generalized Gibbs measures associated to suitable conserved quantities in the hierarchy, we revisit Bourgain’s invariant measure argument in the context of the cubic NLS on the torus and explain how this strategy can be adapted to the generalized Gibbs setting. We conclude by describing recent works which, together, construct an infinite sequence of measures invariant under the flow of the complex-valued mKdV on the torus, one corresponding to each coercive conservation law other than the mass. This talk is based on joint work with Nataša Pavlović, Gigliola Staffilani, and Nicola Visciglia.

We study viscosity solutions for a degenerate one-phase free boundary problem of the form $\Delta w = \frac{h(\nabla w)}{w}.$ We assume the existence of a star-shaped domain $D$ such that h < 0 in $D$, $h = 0$ on $\partial D$, and $h > 0$ in $\bar{D}^c$. This type of degenerate one-phase free boundary problem arises when a canonical transformation is performed to a semilinear equation $\Delta u = f(u)$, and $f$ morally behaves like $u^{-(\gamma + 1)}$ for some $\gamma > 0$. In this case, known as the Alt-Phillips equation for negative powers, $h(\nabla u) = c(|\nabla u|^2 – 1)$. We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as $\gamma$ degenerates to 2, the free boundary converges to a minimal surface.