Summer Program in Partial Differential Equations 2020
Due to the COVID-19 emergency, the 2020 Summer Program in Analysis & PDE, originally planned at UT Austin from May 26 to June 5, 2020, is postponed to new dates to be determined. The tentative idea is rescheduling it for May/June 2021. More details will be communicated around September 2020.
Postponing has been preferred to switching to an online version since among the major benefits of summer schools are the possibility of networking and meeting with peers.
This summer program provides two weeks of concentrated study of topics in Analysis for participating graduate and undergraduate students. The program is supported by the Research Training Grant (RTG) “Analysis of Partial Differential Equations”.
The first week will consist of two graduate courses of introductory character and of activities aimed at undergraduate students. The courses will be designed to be, at the same time, accessible to motivated and talented undergraduate students who are planning to apply to graduate school, and also interesting for graduate students looking to expand their mathematical preparation. Each day there will be one lecture per course, followed by a tutorial with problems on the lecture of the day, with tutoring offered by instructors and resident graduate students and postdoctoral associates. The daily activities will be concluded by a seminar/panel aimed at undergraduate students.
The second week will consist of advanced short courses aiming at introducing specific, cutting-edge research themes in various area of Analysis.
The multidisciplinary character of the program will expose participants to topics that are commonly perceived as belonging to “different” areas of Analysis. One goal of the program is to educate the participants to see unity across these different areas and to approach Analysis from a more comprehensive viewpoint.
The entire two-week program will take place at the University of Texas at Austin from Tuesday, May 26th until Friday, June 5th, 2020.
Registration is mandatory as it will be possible to accommodate only a limited number of participants. Financial support for undergraduate and graduate students is available.
Please register here. You will be asked which part of the program you plan to attend. You will be given the possibility of uploading a statement of interest, a research statement and a CV/resume, and to indicate the contact information of a letter writer.
The deadline for the application is March 31st, 2020.
ORGANIZERS
Phil Isett, Francesco Maggi and Salvatore Stuvard
COURSES — WEEK ONE — May 26 – 30
Phil Isett: Distribution theory with Applications
Abstract: We will develop the basics of distribution theory and give applications to linear PDE and Fourier analysis. Topics to be addressed include definition of distributions, basic operations involving distributions, pullback of distributions, approximation by smooth functions, applications to classical analysis, fundamental solutions and representation formula for various linear PDE, and a proof of the Fourier inversion formula.
Francesco Maggi & Salvatore Stuvard: An introduction to Plateau’s problem with currents
Abstract: Plateau’s problem, the minimization of area among surfaces with boundary given by a fixed curve, is one of the most fundamental problems in Analysis. By regarding surfaces as “higher dimensional distributions”, the theory of currents paves the way towards a rigorous mathematical treatment of Plateau’s problem, leading to the proof of existence of solutions, and allowing the qualitative and quantitative analysis of their geometric properties. In this course, we will introduce the theory of currents from an elementary viewpoint, illustrating the main theorems by a series of worked problems and exercises, and in light of their applications to Plateau’s problem.
Week one schedule: TBA
COURSES — WEEK TWO — June 1 – 5
Tarek Elgindi, (UC San Diego) On the Well-posedness of the Incompressible Euler Equations
Abstract: We will start with a crash course on the classical well-posedness theory for the incompressible Euler equation including arguments for local and global existence. We will then discuss various constructions of infinite time blow-up for the 2d Euler equations, results on 1d models, and then some ideas toward finite-time blow up for the 3d Euler equation.
Robert Haslhofer, (University of Toronto): Mean curvature flow of surfaces
Abstract: Mean curvature flow is the most natural evolution equation in extrinsic geometry, and shares many features with Hamilton’s Ricci flow from intrinsic geometry. In this lecture series, I will provide an introduction to the mean curvature flow of surfaces, with a focus on the analysis of singularities. We will see that the surfaces evolve uniquely through neck singularities and non-uniquely through conical singularities. Studying these questions, we will also learn many general concepts and methods, such as monotonicity formulas, epsilon-regularity, blowup analysis, and weak solutions, that are of great importance in the analysis of a wide range of PDEs.
Connor Mooney (UC Irvine): Hilbert’s 19th problem revisited
Abstract: Hilbert’s 19th problem asks whether the solutions of regular problems in the calculus of variations are always analytic. This problem was solved in the 1950s by De Giorgi and Nash. However, various generalizations of the problem, motivated by applications to fluids, elasticity, and geometry, are actively studied today. We will discuss several of these problems, some of which were solved recently and others which remain open.
Sung-Jin Oh (UC Berkeley): On the Cauchy problem for quasilinear dispersive PDEs
Abstract: Quasilinear dispersive PDEs often arise in fluid dynamics and plasma physics as effective models. The goal of this lecture series is to provide an introduction to the theory of local well-/ill-posedness of the Cauchy problem for such equations. In the first part, I will cover classical concepts that are relevant to wellposedness, such as hyperbolicity, energy estimate, Mizohata condition, local smoothing estimate etc. In the second part, I will discuss some complementary ill(!)-posedness results. A particular emphasis will be given to the phenomenon of degenerate dispersion, which is a very strong instability mechanism for conservative quasilinear dispersive PDE.
Week two schedule: TBA