Talks

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 ...

Scaling up the Hilbert space of a quantum computer is essential to demonstrate quantum advantage over classical computing. Other than packing more quantum information carrier in a quantum computing system, one avenue for increasing the quantum computing Hilbert space is by encoding more logical states per quantum information carrier (making it a qudit), rather than a traditional 2-state qubit encoding. Trapped ions typically exhibit multiple (meta)stable electronic energy states that are suitable for logical state encodings and are a suitable platform for qudit applications. In this talk, I present our work on realizing a 25-level trapped 137Ba+ ion qudit, which is the maximal encoding allowed by the single-shot readout protocol that we employ. We show that the harnessing of these additional logical states per ion can be done by just adding a few more lasers to the control system, without any trapping architecture modification compared to a qubit system. I will also present the experimental challenges and limitations, fundamental and technical, relevant to realizing a 25-level trapped 137Ba+ ion qudit.

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.