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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 tothe (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.