Title: A Characterization of Boundedness For Multipliers of Spherical Harmonic Expansions

Abstract: Any function f: Sᵈ → ℂ on the d-dimensional sphere has a unique expansion of the form f = Σ fₖ, where fₖ is a spherical harmonic of degree k. For a function m: (0,∞) → ℂ, define a family of operators Tᴿ on Sᵈ by setting Tᴿ f = Σ f(k/R) fₖ. Such operators are called multipliers for spherical harmonic expansions, or zonal convolution operators.

In this talk, we explore the boundedness of the operators {Tᴿ} on Lᵖ(Sᵈ). Our main result is that, for d ≥ 4 and a limited range of p, the operators {Tᴿ} are uniformly bounded on Lᵖ(Sᵈ) if and only if the Fourier multiplier operator on ℝᵈ with symbol m(|·|): ℝᵈ → ℂ is bounded on Lᵖ(ℝᵈ). This is the first transference principle of its kind for spectral multipliers on any compact manifold for p ≠ 2.

Our methods are closely connected to endpoint local smoothing inequalities for the wave equation on Sᵈ, and in this talk, we will focus on this relation and compare and contrast the difference between the analysis of multipliers for spherical harmonics using methods of Fourier integral operators as compared to the analysis of Fourier multiplier operators with a radial symbol.

Title: Equivariant Schrödinger maps in two-spatial dimensions.

Abstract: The Schrödinger map equation, sometimes referred to as the Landau-Lifshitz equation, is a continuum model describing the dynamics of the spin in ferromagnetic materials.

The main objective of this talk is to present our recent advances in understanding the dynamical behaviour of solutions to this model. We will see how a geometric approach, that has been fruitful in the study of self-similar solutions to 1D-Schrödinger maps and other related equations, can shed light in the study of equivariant Schrödinger maps in two spatial dimensions.

Title: Phase mixing for the Vlasov equation in cosmology

Abstract: The Friedmann--Lemaitre--Robertson--Walker family of spacetimes are the standard homogenous isotropic cosmological models in general relativity.  Each member of this family describes a torus, evolving from a big bang singularity and expanding indefinitely to the future, with expansion rate encoded by a suitable scale factor.  I will discuss a mixing effect which occurs for the Vlasov equation on these spacetimes when the expansion rate is suitably slow.  This is joint work with Renato Velozo Ruiz (Imperial College London).