DescriptionBack to top
Laplacian growth models describe the evolution of a surface separating two domains, filled with different materials or different phases of the same material. The evolution of the interface is governed by the energy imbalance along the interface, whereas the evolution of the energy is governed by a heat-like equation. There exist many versions of such models, including the Stefan and Hele-Shaw problems, and they serve as building blocks in various models of fundamental physical and biological processes: e.g., melting and solidification, condensation, crystal growth, aging of alloys, interaction of fluids with different viscosities, dynamics of membrane potentials in a network of neurons, tumor growth, etc. In addition to the applications in natural sciences, certain Laplacian growth models are also used in mathematical finance for systemic risk modeling. Due to the (typical) singularities in the solutions, to date, there exists no general well-posedness theory for Laplacian growth models. This question has received a lot of attention in the recent years, with novel probabilistic methods being combined with the more classical analytical ones to yield new existence and uniqueness results.
The meeting will bring together both reseachers who are primarily interested in the theoretical advancement of Laplacian growth models and those who approach this topic from an applied point of view. We believe that this combination will generate synergy effects for both sides: on the one hand, many challenging questions arising in Physics, Biology and Finance can be addressed by Laplacian growth models, on the other hand, in order to use these models one needs appropriate theoretical foundation, in particular, the well-posedness of solutions and convergence of numerical schemes. Another goal of the organizers is to create a bridge between the researchers who use mainly PDE and geometric analysis methods and those who primarily employ probabilistic techniques.
OrganizersBack to top
SpeakersBack to top
ScheduleBack to top
Speaker: Allan Sly (Princeton University)
Joint work with Duncan Dauvergne, Danny Nam and Dor Elboim.
Speaker: Amanda Turner (University of Leeds)
Speaker: Lionel Levine (Cornell University)
Speaker: Harald Garcke (University Regensburg)
Speaker: Inwon Kim (University of California, Los Angeles (UCLA))
Speaker: Rami Atar (Technion – Israel Institute of Technology)
Hydrodynamic limits of particle systems with selection are related to a class of free boundary problems (FBP) associated with second order parabolic PDE. This relation has been established rigorously for some selection models but remains open for others, where the difficulty lies in questions of regularity of the free boundary. We introduce a weak formulation that does not involve the notion of a free boundary, but reduces to a FBP when classical solutions exist. For the injection-branching-selection model of Brownian particles on the line with arbitrarily varying removal rates, the approach yields the characterization of limits in terms of a PDE beyond cases where classical solutions are known (or even expected) to exist. Results on a higher dimensional model of motionless branching particles will also be described, as well as a formal relation to the Stefan FBP and open problems.
Speaker: Sara Svaluto-Ferro (University of Verona)
Supercooled Stefan problems describe the evolution of the boundary between the solid and liquid phases of a substance, where the liquid is assumed to be cooled below its freezing point. Following the methodology of Delarue, Nadtochiy and Shkolnikov, we construct solutions to the one-phase one-dimensional supercooled Stefan problem through a certain McKean-Vlasov equation, which allows to define global solutions even in the presence of blow-ups. Solutions to the McKean-Vlasov equation arise as mean-field limits of particle systems interacting through hitting times, which is important for systemic risk modeling. During the presentation we will illustrate two of our main results. The first one consists in a general tightness theorem for the Skorokhod M1-topology which applies to processes that can be decomposed into a continuous and a monotone part. The second one concerns the propagation of chaos for a perturbed version of the particle system for general initial conditions.
Speaker: Scander Mustapha (Princeton University)
Speaker: Andreas Søjmark (The London School of Economics)
In this talk, we will explore a Brownian perturbation of the supercooled Stefan problem which arises naturally as a model of contagion in financial markets, conditional on the market risk that remains after full diversification. In line with the work of Delarue, Nadtochiy, and Skholnikov, we first present a probabilistic formulation of the problem. Starting from there, we then give a simple outline of a succinct argument for whether or not blow-ups will occur with probability one, depending on the initial specification. Finally, we turn to the financial application, looking at how the problem may serve as the basis for a forward-looking systemic risk indicator, aiming to capture the current market sentiment, and briefly discussing ongoing empirical work in this direction.
Speaker: Christoph Reisinger (University of Oxford)
Speaker: Kevin Yang (Harvard University)
Speaker: Irene Fonseca (Carnegie-Mellon University)
Speaker: Riccarda Rossi (University of Brescia)
Speaker: Mahir Hadzic (University College London)
Speaker: Mihaela Ignatova (Temple University)
Speaker: Etienne Tanré (Institut National de Recherche en Informatique et Automatique (INRIA))
Speaker: Eviatar Procaccia (Technion- Israel Institute of Technology)
Speaker: Yucheng Guo (Princeton University)
We study physical solutions to the three-dimensional Stefan-Gibbs-Thomson Problems under radial symmetry, of which the existence has been established in [Stefan problem with surface tension: global existence of physical solutions under radial symmetry, Nadtochiy and Shkolnikov 2023]. In this talk, I will focus on the proof that the physical solution is unique, the first such result when the free boundary is not flat, or when two phases are present. The main argument relies on a detailed analysis of the hitting probabilities for a three-dimensional Brownian motion, as well as on a novel convexity property of the free boundary obtained by comparison techniques. We also establish a variety of regularity estimates for the free boundary and for the temperature function in the course of the proof. This talk is based on joint work with Sergey Nadtochiy and Mykhaylo Shkolnikov.
Speaker: Mete Soner (Princeton University)
A numerical method for the computation of free boundaries when a stochastic representation is available will be discussed. It is based on an algorithm which we call deep empirical risk minimization developed by E, Han and Jentzen. Their approach applies generally to many stochastic optimal control problems. In the presence of free boundaries, it has to be modified to account for training based on hitting times. In this talk, I outline how this is achieved for the classical problems of optimal stopping or the obstacle problem, and for the Stefan problem for the water-ice interfaces. For the Stefan problem, we use the recent stochastic representations, the notion of physical probabilistic solutions, and level-sets parameterized by deep neural networks on the numerical side.