Organized by:
- John Anderson (Stony Brook)
- Mihalis Dafermos (Princeton)
- Theodore Drivas (Stony Brook)
- Jonathan Luk (Stanford)
Recent years have seen many exciting advances in the study of PDEs. Experience has shown that ideas developed in the context of certain PDEs can often be adapted to say something interesting about other equations as well. Even when this is not the case, probing the adaptability of techniques to different equations can help find new techniques, or at the least find interesting new directions for study. The size and specialization of the community studying PDEs means that there are limited opportunities for different groups to have substantial interactions, despite the clear benefits.
This workshop aims to provide an opportunity for such in depth interactions among mathematicians interested in fundamental questions about PDEs in classical physics. It will feature minicourses by experts which provide answers to such questions, and which provide an introduction to the powerful techniques involved. These minicourses will be geared towards advanced graduate students as well as other established researchers who may specialize in different areas. Each week will feature about three such minicourses.
One of the main goals of the workshop is to encourage discussions outside of the minicourses among the many participants. For this reason, afternoons will often be free to help facilitate such interactions, and participants are encouraged to stay for multiple weeks of the three week program. The program will also feature sessions run by special lecturers with the sole aim of finding interesting directions for further study. Various social activities will also be organized to further facilitate interactions among the participants.
Minicourse lecturers: Jacob Bedrossian, Tarek Elgindi, Peter Hintz, Juhi Jang, Eugenia Malinnikova, Sung-Jin Oh, Igor Rodnianski, Luis Silvestre, Jared Speck, Mikhail Vishik, more TBD
Special sessions: Alexander Shnirelman, more TBD
Lecturer: Tarek Elgindi
Title: Introduction to the study of the 2d Euler equation
Abstract: We will discuss various aspects of the 2d Euler equation with an emphasis on steady states. We will discuss their construction and stability properties. We will also discuss the long-time behavior of nearby solutions.
Lecturer: Peter Hintz
Title: Spacetime engineering via black hole gluing.
Abstract: The Einstein vacuum equations of General Relativity admit explicit solutions describing a single black hole. Recent years have seen remarkable progress in the understanding of the stability properties of these solutions under small perturbations. In my lectures, I will explain methods for the construction of spacetimes containing several black holes via gluing techniques. The goal is to describe some of the ideas behind the construction of spacetimes describing extreme mass ratio mergers — the collision of a very light black hole with a unit mass black hole, followed by the relaxation of the resulting single black hole back to a stationary state.
Lecturer: Juhi Jang
Title: Singularities and stability in self-gravitating fluids
Abstract: We will discuss recent progress on dynamics of self-gravitating fluids with focus on singular solutions describing physical phenomena including expansion and implosion, and their stability.
Lecturer: Jared Speck
Title: Shocks in multi-dimensional compressible fluids
Abstract: In the recent past, there have been dramatic advances in the rigorous mathematical theory of shocks for the multi-dimensional compressible Euler equations. A lot of the progress has relied on geometric methods that were developed to study Einstein’s equations. In this mini-course, I will provide an overview of the field and highlight techniques that have proven fruitful for solving problems without symmetry, irrotationality, or isentropicity assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe open problems.
Special sessions: TBD