Termine

Given the sharp logarithmic decay of linear waves on the Kerr-AdS black hole (Holzegel, Smulevici, 2013), it is expected that the Kerr-AdS spacetime is unstable as a solution of the Einstein vacuum equations. However, the scattering construction presented here for exponentially decaying nonlinear waves on a fixed Kerr-AdS background serves as a first step to confronting the scattering problem for the full Einstein system. In this context, one may hope to derive a class of perturbations of Kerr-AdS which remain ?close? and dissipate sufficiently fast.

The fast dynamo problem concerns the amplification of magnetic fields by the motion of an electrically charged fluid. In the linear approximation, this manifests as exponential growth of the magnetic energy, at a resistivity-independent rate. In this talk, I will provide a construction of a Lipschitz, divergence-free and time-independent velocity field that is a fast dynamo on the whole space. The talk is based on joint work with Michele Coti Zelati and Massimo Sorella.

Passive scalars advected by irregular velocity fields provide a model problem for turbulent behaviour, namely the cascade of scalar variance to small scales. In this talk I will focus on the vanishing diffusivity regime for transport?diffusion: when the advecting velocity is rough, the inviscid transport equation may lose classical well-posedness and the expected conservation of the energy. Although any fixed positive diffusivity restores a unique, well-posed evolution, the limit as diffusivity tends to zero can exhibit turbulent phenomena such as anomalous dissipation and failure of the diffusive regularization to select a unique inviscid solution.

In this talk, which is meant to be an introduction to oo-categories, I will use Grothendieck's "Homotopy Hypothesis" as a narrative thread to motivate and give intuition for oo-categories. I will further try to briefly indicate how the practice of this theory actually works, and, in the end, explain how oo-categories provide a framework to revisit and refine the homotopy hypothesis. No prior knowledge of oo-categories will be assumed, though a vague idea of what they might be will be helpful.

In this talk, we will consider the massless Boltzmann equation on FLRW spacetimes with spatial topology $\mathbb{T}^3$ in the linear and decelerated expanding regimes, where the scale factor is $t^{q}$ with $q\in (0,1]$. In these backgrounds, the massless Boltzmann equation admits a non-stationary Maxwell-Jüttner equilibrium that maximises the entropy of the system. In this work, we show this equilibrium is non-linearly stable for the massless Boltzmann equation on these FLRW spacetimes in the case of hard ball interaction. We will further show a dissipation effect that occurs when the expansion rate is slow enough. This is joint work with Robert M. Strain (University of Pennsylvania) and Martin Taylor (Imperial College London).

The analysis of Dirac-type equations introduces structural challenges, which are absent in wave or Klein?Gordon equations, due to their first-order, and spinorial nature. In particular, the natural notion of energy for Dirac fields is derived from conserved currents rather than stress?energy tensors. The main focus of this talk is a spinor-adapted geometric, current-based energy framework which treats Dirac equations directly, without reduction to wave equations. I explain why this shift becomes necessary in the presence of gauge coupling and discuss how Dirac currents, gauge-covariant vector field methods, and divergence identities naturally interact with the Maxwell field, which supplies decay through its own energy hierarchy. I conclude with an overview of the current status of the Maxwell?Dirac system near Minkowski spacetime

Entropy maximisers and their stability In this talk we consider the four-waves spatially homogeneous kinetic equation arising in weak wave turbulence theory from one-dimensional microscopic oscillator chains. This is the Phonon Boltzmann Equation. I will discuss the entropy maximisation problem, the collisional invariants, and properties of solutions of the kinetic equation near the Rayleigh-Jeans (RJ) thermodynamic equilibria. This is based on joint works with Pierre Germain (Imperial College London), Joonhyun La (KIAS) and with Miguel Escobedo (Bilbao).