Rivière-Fabes Symposium on Analysis and PDE

On the (anisotropic) fractional Calderón problem *

Abstract: In many measurement settings in engineering, the sciences and medicine, only indirect and non-invasive information is available on the underlying system. Mathematically, these lead to inverse problems. The Calderón problem is a prototypical example: One seeks to reconstruct the unknown conductivity of a conducting body through voltage and current measurements at the surface of the object. In these lectures I will give an introduction to a nonlocal version of it, the fractional Calderón problem. I will discuss how nonlocality leads to novel structures and mechanisms. In particular, I will focus on uniqueness results in the anisotropic setting and on connections between the classical local and the fractional Calderón problems.

*Same title & abstract for both talks

Hilbert's sixth problem: derivation of the Boltzmann and fluid equations *

Abstract: We present recent works with Zaher Hani and Xiao Ma, in which we derive the Boltzmann equation from the hard sphere dynamics in the Boltzmann-Grad limit, for the full time range in which the (strong) solution to the Boltzmann equation exists. This is done in the Euclidean setting in any dimension $d\geq 2$, and in the periodic setting in dimensions $d\in\{2,3\}$. As a corollary, we also derive the corresponding fluid equations from the hard sphere dynamics. This resolves Hilbert's Sixth Problem pertaining to the derivation of hydrodynamic equations from colliding particle systems, via the Boltzmann equation as the intermediate step.

*Same title & abstract for both talks

Talk I: Machine Learning in PDE: Discovering new, unstable solutions

Abstract: In this talk I will explain several recent results combining machine learning techniques and more traditional mathematics. The overarching theme is the interplay between modern (ML) and classical methods in order to discover new solutions of certain PDE with low or very low numerical error. I will also describe how to turn the numerical approximate solutions into a rigorous proof via computer-assisted methods, leading in some cases to singularity formation, or to the existence of certain special solutions (e.g. traveling waves). In particular, unstable solutions are now amenable to be discovered.

Talk II: Implosions for compressible Euler, compressible Navier-Stokes and nonlinear Schrödinger equations

Abstract: In the last years there have been remarkable proofs of imploding singularities for the compressible Euler and Navier-Stokes equations, and via a Madelung transformation, for the defocusing nonlinear Schrödinger equation as well. In this talk I will introduce these types of singularities and prove their existence, combining classical and modern mathematics' ingredients such as computer-assisted proofs. I will also outline future exciting directions and open problems.

Talk 1: Plateau’s laws for soap films, the Allen-Cahn equation, and a hierarchy of Plateau-type problems

In the physics and engineering literature, the stability of soap films, for example under drainage phenomena, is studied by modeling them as compounds of two dimensional minimal surfaces joined together by three-dimensional domains (containing positive amounts of liquid), called {\it Plateau borders} along the classical Plateau-type singularities. In the joint works with Michael Novack (LSU) and Daniel Restrepo (Johns Hopkins University) presented in this talk we provide a rigorous description of Plateau borders consistent with the applied literature in the context of area minimization. Moreover, we study the approximation of our new capillarity model for soap films by means of the Allen-Cahn energy functional, thus “solving” the well-known incompatibility between Plateau laws and stable solutions to the Allen–Cahn equation and describing a hierarchy of Plateau problems. Central to our analysis is a measure-theoretic revision of the topological notion of homotopic spanning that has been behind much recent progress on the classical Plateau problem.

Talk 2: Asymptotic behavior of a diffused volume preserving mean curvature flow

We consider a diffused interface version of the volume-preserving mean curvature flow in the Euclidean space, and prove, in every dimension and under natural assumptions on the initial datum, exponential convergence towards single "diffused balls". The relation with the problem of determining the long time behavior of the standard volume-preserving mean curvature flow is also discussed. This is based on joint works with Matteo Bonforte (U. Autonoma Madrid) and Daniel Restrepo (Johns Hopkins U.); see https://arxiv.org/pdf/2202.11583 and https://arxiv.org/pdf/2407.18868.

• 3:00 – 3:30 pm - Coffee and Registration (120 Vincent Hall)
• 3:30 – 4:30 pm - Francesco Maggi (16 Vincent Hall)
• 4:30 – 5:00 pm - Coffee Break (120 Vincent Hall)
• 5:00 – 6:00 pm - Francesco Maggi (16 Vincent Hall)

• 8:30 – 9:00 am - Coffee and light breakfast (120 Vincent Hall)
• 9:00 – 10:00 am - Yu Deng (16 Vincent Hall)
• 10:00 – 11:30 am - Coffee and Poster Session I (120 Vincent Hall / Open Hallway)
• 11:30 – 12:30 pm - Angkana Ruland (16 Vincent Hall)
• 12:30 – 2:00 pm - Lunch Break – on your own
• 2:00 – 3:00 pm - Javier Gomez Serrano (16 Vincent Hall)
• 3:00 – 3:30 pm - Coffee Break (120 Vincent Hall)
• 3:30 – 4:30 pm - Angkana Ruland (16 Vincent Hall)
• 4:30 – 5:30 pm - Poster Session II (120 Vincent Hall / Open Hallway)
• 6:00 pm - Reception (Campus Club, Coffman Memorial Union, 4th Floor)
   

• 8:30 – 9:00 am - Coffee and light breakfast (120 Vincent Hall)
• 9:00 – 10:00 am - Javier Gomez Serrano (16 Vincent Hall)
• 10:00 – 10:30 am - Coffee Break (120 Vincent Hall)
• 10:30 - 11:30 am - Yu Deng (16 Vincent Hall)