Fourth Chicago Summer School In Analysis

Minicourse by Diego Cordoba.
Active scalars driven by a 2D incompressible flow. description

This course will be devoted to study several examples of active scalars; the generalized Surface Quasi-geostrophic equation and the Incompressible Porous Media equation. We will discuss the main ideas and arguments for well-posedness, global-existence and finite time singularities.

Minicourse by Francesco Maggi.
Some key ideas from Geometric Measure Theory in action. description

The scope of these lectures is introducing students to various key ideas from Geometric Measure Theory (GMT). We will examine the most relevant features of rectifiability, and of the theories of rectifiable currents and varifolds which have been built upon it. We will explain what is a "small excess regularity criterion", an idea that originated in GMT and that is now fundamental in the analysis of many nonlinear PDE. Finally, we will illustrate these ideas in action by showing the existence of minimizers in various formulations of Plateau's problem. (This last part of the course will be based on a joint paper with Camillo De Lellis and Francesco Ghiraldin.)

Minicourse by Govind Menon.
A quick introduction to kinetic theory. description

Kinetic theory provides an effective description for the dynamics of a system of many interacting particles. The purpose of these lectures is to explain what the previous sentence means -- that is, to provide graduate students with a realistic feel for the typical questions and theorems that arise in the field. My plan is to cover the following.

1. models: the Boltzmann equation for hard spheres and sticky particle systems;
2. phenomena: approach to equlibria and self-similar solutions;
3. foundations: particle systems, the BBGKY hierarchy and closure.

Several solvable examples will be considered to build intuition for integral equations. I will also provide some background on equilibrium statistical mechanics.

The lectures are designed to be largely independent. An earlier version of lecture notes for this class may be found here.

Minicourse by Giuseppe Mingione.
Nonlinear Calderón-Zygmund theory. description

The standard Calderón-Zygmund theory deals with optimal integrability properties of solutions to linear elliptic and parabolic equations, and goes through the analysis of fundamental solutions and related singular integrals. These tools are obviously ruled out when dealing with nonlinear, possibly degenerate problems. Nevertheless, recent progresses from the last years show that it is still possible to draw a rather precise analog of the linear theory in the nonlinear case. This relatively fresh field largely intersects with so called nonlinear potential theory, i.e., a nonlinear analog of the classical potential theory. In this series of lectures I will try to outline the main recent results of these theories.

Minicourse by Alessio Porretta.
PDE methods in mean field games theory. description

Mean field games theory has been developed since 2006 by J.-M. Lasry and P.-L. Lions as amodel to describe Nash equilibria in the dynamic optimization of a large population of similar agents, where the individual strategy depends on the collective behavior through the distribution law of the states. This model leads to systems of PDEs where a backward Hamilton-Jacobi-Bellman equation is coupled with a forward Kolmogorov equation.

In this course, after a brief description of the model we will present the main features of the forward-backward systems, discussing the existence, uniqueness and stability of solutions, the long-time behavior and further questions related to optimal transport and control theory.

Minicourse by Alexis Vasseur.
Recent results on the 3D quasi-geostrophic equation. description

In oceanography, the motion of the atmosphere follows the so-called primitive equation. This corresponds the 3D Navier-Stokes equation with the effect of the rotation of the earth (Rosby effect). At large scale, this Rosby effect is very important. Asymptotically, this leads to the so-called geostrophic balance which enforces the wind velocity to be orthogonal to the gradient of the pressure in the atmosphere. Asymptotic analysis can be performed to derive the quasi-geostrophic equation model (QG), which is not as complex as the primitive equation, and not as trivial as the geostrophic balance, and still captures the large scale motion of the atmosphere. This model is extensively used in computations of oceanic and atmospheric circulation, for instance, to simulate global warming. During this talk, we will present the 3D Quasi-Geostrophic equation and its mathematical treatment.

We will first derive the equation, and describe a special easier case, known as the 2D Surface Quasi-Geostrophic equation. Finally we will construct global weak solutions for the 3D QG system in the inviscid case (without Eckman pumping), and show the existence and uniqueness of global classical solutions with large initial data in the case with Eckman pumping.