Activities This Week

I will provide an overview of the model theoretic notions of internality and internal covers, and their relation to definable groupoids. I will then indicate a generalization of these notions to higher homotopical dimension

This is the continuation of the talk from the previous week

The classical Fourier transform is an ubiquitous operator acting on L^2(V) for a finite-dimensional quadratic space V. We study it from the point of view of representation theory. Together with other operators it forms a remarkable representation of a metaplectic group on L^2(V), that has minimal functional dimension. Minimal representation of other groups, often have models on L^2(X) for a cone X. We shall see how to define generalized Fourier transforms on L^2(X) and discuss their properties.

The Frenkel-Kontorova model is a standard model in solid state physics describing particles having nearest neighbor interactions. Mathematical analysis of this model leads to studying standard-like twist maps. In the 80’s Suris found a remarkable family of potentials for this model with integrable dynamics. In some sense this is similar to the role that ellipses play in planar billiards. In the talk we will highlight this connection, via the action-angle coordinates of the two systems. Then we will also show that an integrable pertrubation of a Suris potential has to be a Suris potential itself. This is in spirit of local results proven for the Birkhoff conjecture in billiards. The proof relies heavily on Fourier anlaysis, as well as construction of a suitable basis for L2 wihch captures the dynamics of the system. Joint work with Corentin Fierobe

In this talk, we present several dynamical and fractal-dimensional ways of characterizing the spectral measures of Schrödinger operators, such as Rajchman behavior and Hausdorff/packing dimensions, and discuss the extent to which these properties are stable under rank-one perturbations.

We begin with the concrete setting of half-line Schrödinger operators, where a theorem of Gordon shows that generic rank-one perturbations eliminate pure point spectrum, ruling out the most extreme dynamical and dimensional behavior. I will then describe constructions demonstrating that properties only slightly weaker than pure point spectrum can, in fact, be entirely stable: for certain sparse half-line models, both packing-dimension-zero and non-Rajchman behavior persist for every rank-one perturbation.

In the second part, we examine how spectral dimensions behave when passing from the whole line to the half-line. I will present an operator whose spectral measure on the line has Hausdorff dimension one, whereas every half-line restriction - under any boundary condition - has dimension zero, even though the two settings differ only by a finite-rank perturbation.