The purpose of this talk is to give an introduction to the Hartree equation. This is an example of a well-known semi-linear Schrödinger equation, which describes the mean-field dynamics of many-body quantum-mechanical gases. We will discuss both its mathematical properties as a PDE, and its emergence from physical systems.

While the “flavor” of real and complex analysis are often quite different, the two are linked in a variety of ways. In this talk, we will examine theorems in complex analysis that can be proved using ideas from real analysis, and conversely theorems in real analysis that can be proved by applying the theory of complex functions.

We will present multiple proofs of classical Poincaré inequalities and prove a weighted one. Then, we will see an application in fluid mechanics.

We work through several proofs of the Strong and Weak Laws of Large Numbers, and discuss the different techniques they use. We end with Etemadi’s elementary 1981 proof of the Strong Law. No background in probability is required, only some basic real analysis.

 I will discuss shock and rarefaction solutions to conservation laws, solve the Riemann problem, and (hopefully) start the existence theory via front tracking for initial value problems with small BV initial data.

This is part two of last week’s talk.

The derivation of the quantum Boltzmann equation (qBe) from first principles is a longstanding open problem in mathematical physics. On the other hand, in recent years, the field has seen tremendous progress in the development of mathematical tools for the study of mean-field theories. This has led to the rigorous development of key physical ideas describing the dynamics of cold gases of fermions, including the “bosonization” of their statistics. In this talk, we present the main results of https://arxiv.org/abs/2306.03300. We revisit the emergence of the qBe, taking into account the effect of this phenomenon.

The classical isoperimetric inequality exemplifies a beautiful relationship between the perimeter and volume of a set. It arises from the goal to find perimeter-minimizing sets with fixed volume, a problem which dates back to antiquity, and characterizes these minimizers as the equality case (balls). Quantitative stability concerns the natural follow-up question: If your set almost minimizes the perimeter, is it close to a ball? This area of study has flourished over the past two decades, leading to many generalizations. One such example is the anisotropic perimeter, where certain directions are preferred. We prove a strong sharp quantitative stability result for so-called crystalline surface tensions, whose anisotropic perimeter minimizers are polytopes.

We will discuss how to interpret certain PDEs as gradient flows with respect to the Wasserstein metric, and comment on why that could be useful, following Felix Otto’s seminal paper “The geometry of dissipative evolution equations: the porous medium equation”.

The macroscopic phenomenon of diffusion arises from the random motion of particles. Thus, it is not surprising that there is a deep connection between Brownian motion and heat and other diffusion equations. We will examine the relation between the Brownian motion and the heat equation as well as look at some examples of PDE that can be solved by running the Brownian motion.

The Grad Mercier equation is a non-local PDE with applications in plasma physics. In this talk, I will first present a survey of the current existence, uniqueness, and regularity results on the Grad Mercier PDE. Then, I will show that the free boundary associated with the equation is a set of finite n-1 Hausdorff measure. This is based on joint work with Luis Caffarelli, Daniel Restrepo, and Ignacio Tomasetti.

Yes, it’s true – stochastic optimal control problems and Hamilton Jacobi (HJ) equations pervade the applied sciences and play a key role in modern PDE theory. However: most students have never been introduced to optimal control and first-order Hamilton-Jacobi PDE associated to them, let alone their more sophisticated stochastic counterpart (with 2nd order HJ equations). We will introduce the topic by answering these questions: What is an optimal control problem, as opposed to a calculus-of-variations problem? How do such problems yield Hamilton-Jacobi equations? When does calculus of variations reach its limits, and modern PDE theory [i.e. the viscosity solution] begin? Come to find out!