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In this talk we will review some recent results on stochastic dispersive equations, focusing on critical stochastic Schrödinger and Zakharov equations. We will first show the construction and conditional uniqueness of multi-bubble Bourgain-Wang type blow-up solutions and non-pure multi-solitons to focusing mass-critical (stochastic) nonlinear Schrödinger equations. In particular, the refined uniqueness is derived in the low asymptotic regime. Furthermore, we will present the noise regularization effect on blow-up and scattering dynamics for stochastic Zakharov system particularly in the 4D energy-critical case.

The last stone to the solution of Hilbert’s nineteenth problem was put by E. De Giorgi and J. Nash by proving that solutions of elliptic and parabolic equations with rough coefficients are Hölder continuous. Their result was reformulated in terms of a Harnack inequality by J. Moser. Recently, this theory was developed in the context of kinetic equations. The goal of this series of lectures is to present it.

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This talk focuses on a fundamental concept in the field of partial differential equations — unique continuation principles. Such a principle describes the propagation of the zeros of solutions to PDEs. Specifically, it answers the question: what condition is required to guarantee that if a solution to a PDE vanishes on a certain subset of the spatial domain, then it must also vanish on a larger subset of the domain. Motivated by Hardy’s uncertainty principle, Escauriaza, Kenig, Ponce, and Vega were able to show in a series of papers that if a linear Schrödinger solution decays sufficiently fast at two different times, the solution must be trivial. In this talk, we will discuss unique continuation properties of solutions to higher-order Schrödinger equations and variable-coefficient Schrödinger equations, and extend the classical Escauriaza-Kenig-Ponce-Vega type of result to these models. This is based on joint works with S. Federico-Z. Li, and Z. Lee.

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The numerical reconstruction of solutions to kinetic models remains a formidable task. Challenges include numerical stability, conservation, accuracy, and complexity reduction. In this talk we will discuss the numerical treatment of two fundamental kinetic models: the Vlasov-Poisson system, and the Landau equation. Specifically, we will introduce several techniques that are currently in development. These include a discontinuous Galerkin, a representation theory, and a semi-Lagrangian approach. Other relevant aspects, such as positivity preservation, error estimates, Schur’s lemma, low-rank methods, implementation challenges, and the extension to the relativistic Landau will be discussed. The goal is for this presentation is to be a conversational and friendly introduction to the subject. We will focus on the “philosophy” behind these techniques as opposed to their technical aspects.

The last stone to the solution of Hilbert’s nineteenth problem was put by E. De Giorgi and J. Nash by proving that solutions of elliptic and parabolic equations with rough coefficients are Hölder continuous. Their result was reformulated in terms of a Harnack inequality by J. Moser. Recently, this theory was developed in the context of kinetic equations. The goal of this series of lectures is to present it.

You will only need to wear very comfortable clothes, bring a yoga mat, and possibly a small blanket to cover yourself during the final relaxation. A cushion or yoga block could also be useful if sitting on the mat is uncomfortable for you.

The yoga lessons will take place in the auditorium, though we could also explore the possibility of occasionally practicing outdoors.

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In this talk I want to explain as simply as I can the reason why some stochastic partial

differential equations, like e.g. KPZ, need a procedure called renormalisation.

This consists in a modification of the equation by adding new non-linearities that typically 

contain diverging constants. This is strongly related to a similar phenomenon in

quantum field theory and its explanation in the context of SPDEs is probably

more intuitive. Unfortunately a complete description of the procedure requires

an algebraic machinery that most probabilists prefer to avoid. However the

beauty of this technique resides also in the deep interplay between analysis,

algebra and probability.

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You will only need to wear very comfortable clothes, bring a yoga mat, and possibly a small blanket to cover yourself during the final relaxation. A cushion or yoga block could also be useful if sitting on the mat is uncomfortable for you.

The yoga lessons will take place in the auditorium, though we could also explore the possibility of occasionally practicing outdoors.

Zoom Link