Talks

Recent advances in General Relativity point toward unanticipated, and dynamic, relations between ultracompact objects and the universe they inhabit. The possibility for strongly gravitating systems, like astrophysical black holes (BHs) and their embedding cosmology, to directly interact has been dubbed "cosmological coupling." We focus on recent results from the DOE Stage IV Dark Energy (DE) Spectroscopic Instrument (DESI), which strongly suggest that DE is dynamical. Using typical empirical models for the cosmic star-formation rate density as a proxy to BH production, we show that the DESI-inferred time-evolution of DE is consistent with cosmologically coupled stellar-collapse BHs as the source of DE. The predicted cosmological expansion rate today, H_0 = 69.94 +/- 0.81 km/Mpc/s, is in excellent agreement with H_0 = 69.58 +/- 1.58 km/Mpc/s recently reported by the Chicago-Carnegie Hubble Program using Cepheid, Tip of the Red Giant Branch, and J-Region Asymptotic Giant Branch stellar distance-ladder calibrations. With DESI Redshift Space Distortions and Year 3 datasets on the horizon, we highlight exciting prospects for further observational confrontation in the near term.

In this lecture I will explain a second type of iteration, introduced by La'szlo' and myself, which produces continuous solutions of incompressible Euler in a fashion which shares a lot of similarities with the theorem of Nash explained in Lecture 3.

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.

Many physical phenomena may be modeled by first order symmetric hyperbolic equations with degenerate dissipative or diffusive terms. This is the case in gas dynamics, where the mass is conserved during the evolution, but the momentum balance includes a diffusion (viscosity) or damping (relaxation) term.
Such so-called partially dissipative or diffusive systems have been extensively studied by S .Kawashima in his PhD thesis from 1984. For a rather general class of systems he pointed out a simple necessary condition for the global existence of solutions in the vicinity of constant solutions.
This condition that is nowadays named (SK) condition has been revisited in a number of research works. In particular, K. Beauchard and E. Zuazua proposed in 2010 an explicit method for constructing a Lyapunov functional allowing to refine Kawashima’s results and to establish global existence results in some situations that were not covered before. Very recently, advances have been made by T. Crin-B...

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 ...