Bath Summer 2024

Speakers

Practical information

Day: Thursday 9th May 2024 (afternoon).

Venue: Wolfson Lecture Theatre, 1st floor of 4 West, University of Bath (Travel advice).

Organiser: Veronique Fischer.

Registration: Please fill in this form. 

If you require financial support, please contact Veronique Fischer by 26th April. 

Social dinner: 18.45 at Browns Bath, Old Police Station, Orange Grove, BA1 1LP.

Programme

13.15 - 14.05: Simon Myerson,  Bounds for spectral projectors on the three dimensional torus

Abstract: This is joint work work Pierre Germain. We study L^2 to L^p operator norms of spectral projectors for the Euclidean Laplacian on the three-dimensional torus, in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window limit we are left with L^p norms of eigenfunctions of the Laplacian, as considered for instance by Bourgain. We prove new cases of our previous conjecture concerning the size of these norms. We use methods  from number theory: the geometry of numbers, the circle method and exponential sum bounds due to Müller.

14.15 - 15:05: Hajer Bahouri, Dispersion phenomena and analysis of nonlinear evolution equations

Abstract: The  first  aim of this talk is to review dispersion phenomena, which express that waves with different frequencies move at different velocities, with applications to nonlinear evolution equations. When a loss of dispersion occurs, as for compact Riemannian manifolds or for the Heisenberg group, some alternative approches allow to obtain weak  informations about the behavior of the associated solutions. The second aim of the talk consists then to introduce these alternatives, known as  Fourier restriction results, Kato-smoothing effect and cutting methods.

Coffee break

15:30 - 16:20: Monica Musso, Leapfrogging vortex rings for Euler equations

Abstract: We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called "binormal curvature flow". Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings, and of a helical filament, associated to a translating-rotating helix. In this talk I will consider the case of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila, M. del Pino  and J. Wei.

16:30 - 17:30: Søren Mikkelsen, Semiclassical functional calculus on nilpotent Lie groups

Abstract: One of the fundamental tools in spectral theory is the functional calculus. This allow us to define functions of self-adjoint operators. However, this theory is abstract and do not immediately give explicit forms for kernels. In the case of pseudo-differential operators and smooth functions the situation is improved. For this case it is well established that a smooth function of a self-adjoint pseudo-differential operator is again a pseudo-differential operator. In this talk we will present a semiclassical functional calculus for suitable subelliptic operators on nilpotent Lie groups. As an application of this calculus, we obtain Weyl asymptotics for large classes of subelliptic operators. 

Acknowledgement

The visits of Hajer Bahouri and Simon Myerson to the University of Bath are partially supported by Research Project Grant 2020-037 Quantum limits for subelliptic operators funded by the Leverhulme Trust.