Talks

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...

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.

There is a well-known discrepancy in mathematical fluid mechanics between phenomena that we can observe and phenomena on which we have theorems. The challenge for the mathematician is then to formulate an existence theory of solutions to the equations of hydrodynamics which is able to reflect observation. The most important such observation, forming the backbone of turbulence theory, is anomalous dissipation. In the talk, we survey some of the recent developments concerning weak solutions in this context.

We start by demonstrating that numerical methods do not necessarily converge to entropy or admissible weak solutions of the Euler and Navier-Stokes equations of fluid dynamics on mesh refinement due to appearance of eddies at smaller and smaller scales. As an alternative, we revisit the concept of statistical solutions which are time-parametrized probability measures, consistent with the fluid evolution. We empirically show that the same numerical methods converge to a statistical solution and also derive verifiable sufficient conditions under which this convergence can be made rigorous. Numerical experiments illustrating interesting properties of statistical solutions are also presented. We conclude by showing how state of the art generative AI models (conditional diffusion) can significantly lower the cost of computing statistical solutions while maintaining accuracy.

Causality is a core concept in both General Relativity (GR) and Quantum Information Theory (QIT), yet it manifests differently in each domain. In GR, causal cones appear as a defining property of spacetime. Conversely, in QIT, causality relates to the abstract flow of information in quantum processes, independent of spacetime. This raises a crucial question: under what conditions can an abstract quantum process be realised within spacetime? The question is especially intriguing for quantum processes with indefinite causal structure, like the Quantum Switch, which resist classical causal descriptions. In this talk, I will present no-go theorems that reveal fundamental limitations on the realisability of such processes in spacetime and, thus, more generally, on the interplay between GR and QIT. This is based on joint work with V. Vilasini (Physical Review Letters, 133 080201, 2024).

Recent research on quantum reference frames (QRFs) has shown that whether a system is in a superposed state of locations, momenta, and other properties can depend on the quantum reference frame relative to which it is being described. Whether an event is localized in spacetime or not can change under QRF transformations, in that case so-called quantum-controlled diffeomorphisms. This raises a critical question: can quantum reference frame transformations render indefinite causal order definite? In this talk, I propose a relativistic definition of causal order based on worldline coincidences and proper time differences, establishing it as an operationally meaningful observable in both general relativity and quantum mechanics. Using this definition, we can analyse the indefiniteness of causal order in the optical and gravitational quantum switch on equal footing. This analysis suggests an operational rather than a spacetime-based understanding of events. I will compare these findings to other recent results and conclude with broader implications for events in non-classical contexts

Multipartite quantum channels realisable in a spacetime obey the no-superluminal-signalling constraints imposed by relativistic causality. But what about the converse: Can every channel that exhibits no superluminal signalling also be realised through relativistically valid dynamics? To our knowledge, only special cases of this question have been studied. For bipartite channels, the answer has been found to be negative in general (Beckman et al., 2001), though we will argue that counterexamples must necessarily involve a form of fine-tuning. Another special case of the question has been extensively explored under the name of nonlocal quantum computation in the context of position-based cryptography. We will pose and motivate the question in generality, conjecture a positive answer for all but the fine-tuned channels, and present results towards proving it, drawing on insights from nonlocal quantum computation and the new field of causally faithful circuit decompositions of unitary transformations (see also Tuesday). Beyond their relevance to spacetime realisability, the circuit decompositions involved in addressing the question also find applications in quantum causal modelling.