Talks

In this lecture I will explain the ideas of Nash's surprising construction, in the 1950s, of many C^1 isometric embeddings of Riemannian manifolds as hypersurfaces of the Euclidean space. We will then touch upon an open problem in the area, due to to Borisov and Gromov, about the threshold Hoelder regularity for which the Nash phenomenon is possible. This problem turns out to be intimately linked to the Onsager conjecture and we will survey the results proved so far about it.

In the recent resolution of the strong Onsager’s conjecture in L^3 framework, so-called wavelet-inspired Nash’s iteration has been developed. In this lecture, we will sketch this new method and provide the necessary background. The lecture will be based on our recent joint work with Matthew Novack.

We consider the complete Navier--Stokes--Fourier system governing the time evolution of a general compressible, viscous, and heat conducting fluid. The fluid motion is driven by non--conservative boundary conditions. We show that, unlike the conservative systems, the non--conservative ones admit in general a bounded absorbing set. Asymptotic compactness of global in time solutions on this set is then established. The result has several corollaries including the existence of a stationary statistical solutions, convergence of ergodic averages and convergence to equailibrium solutions for small perturbations.

The lectures will introduce perturbations of Euler's equation by highly irregular paths where the perturbations are such that the solution preserves a range of physically relevant quantities. Using formal computations, we shall see that, when d=2, a purely Lagrangian formulation of the equation seems to be within reach.

However, special care is needed to give rigorous meaning to the noisy terms of the equation and in these lectures, we will consider the framework of rough paths. We will see how the so-called 'Sewing Lemma' can be used to define integrals as Riemann sums w.r.t paths of low regularity and how to use this result to construct rough path integrals. Then we will derive very precise a priori estimates for differential equations driven by rough paths and these estimates will be used to prove well-posedness of equations where the drift term satisfies an Osgood regularity. Moreover, we will study flows generated by the differential equations and see that the flows are volume ...

Dark matter and neutrinos are elusive neutral matter filling up our Universe. The PandaX (Particle and Astrophysical Xenon) experiment, located in the China Jinping Underground Laboratory, is dedicated to searching for dark matter particles and studying the fundamental properties of neutrinos. The current running experiment, PandaX-4T, contains a sensitive time-projection-chamber with a 3.7-ton liquid xenon target. In this talk, after an overview, I will present recent results from PandaX-4T in dark matter direct detection, double beta decay of 136Xe, as well as solar neutrinos. I will also present a concrete plan for the next-generation xenon observatory in CJPL, PandaX-xT.

In this lecture I will explain the first work of La'szlo' and myself, which introduced for the first time ideas from differential inclusions in the study of the incompressible Euler equations. These ideas allowed to produce far-reaching generalizations of pioneering results by Scheffer and Shnirelman, showing the abundance of counterintuitive bounded solutions.

We establish existence of infinitely many stationary solutions as well as  ergodic stationary solutions to the three dimensional Navier--Stokes and Euler equations in the deterministic as well as stochastic setting, driven by an additive noise. The solutions belong to the regularity class $C(\mathbb{R};H^{\vartheta})\cap C^{\vartheta}(\mathbb{R};L^{2})$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. Moreover, we are able to make conclusions regarding the vanishing viscosity limit. The result is based on a new stochastic version of the convex integration method which provides uniform moment bounds locally in the aforementioned function spaces.

We say that an ordinary or partial differential equation is regularized by noise if the addition of a suitable noise term restores well-posedness or improves regularity of the solution to the equation. Regularization by noise is by now well understood for ODEs and linear transport-type PDEs, but it is less understood for nonlinear PDEs like Euler and Navier-Stokes equations, with many open questions.

In the first part of this series of lectures, we consider the case of ODEs and associated transport equations with irregular drift: we show that the addition of an additive Brownian noise restores well-posedness of the ODE; we introduce the corresponding transport noise and show that this noise restores well-posedness for the associated transport equation. In the second part, we focus our attention on the effect of a particular transport-type noise, which is divergence-free, Gaussian, white in time and poorly correlated in space (nonsmooth Kraichnan noise). The associated linear transpo...