Fall 2022

Abstract:

Let be a locally compact group, a large compact subgroup of and an arbitrary class of irreducible unitary representations of .
In this talk, before giving an extension of Bochner theorem, we’ll study the spherical Fourier transform according to the unitary dual of , construct the -spherical grassmannian using a generalized Abel transform, and give an application to the reductive Lie groups where the discrete series is not empty.

Abstract: Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a bounded linear operator. Recent works in this area consider questions such as when can a given frame for a separable Hilbert Space, , be represented by iterations of an operator on a single vector and what are necessary and sufficient conditions for a system, , to be a frame? In this talk, we will discuss the connection between frames given by iterations of a bounded operator and the theory of model spaces in the Hardy-Hilbert Space as well as necessary and sufficient conditions for a system generated by the orbit of a pair of commuting bounded operators to be a frame.

We present a detailed spectral analysis for a new class of fractal-type diamond graphs, referred to as bubble-diamond graphs, and provide a gap-labeling theorem in the sense of Bellissard for the corresponding probabilistic graph Laplacians using the technique of spectral decimation.

Labeling the gaps in the Cantor set by the integrated density of states provides a set of topological quantum numbers that reflect the branching parameter of the graph construction and the decimation structure.

The spectrum of the natural Laplacian on limit graphs is shown generically to be pure point supported on a Cantor set. However, one particular graph has a mixture of pure point and singularly continuous components.

In the last years, the study of the Dirac operator on metric graphs has generated a
growing interest. It has been widely used for modeling electronic properties of
graphene, propagation of electromagnetic waves in graphene-like photopic crystals,
ultra-cold matter in optical lattices, and many others. In this talk, we consider the
dynamic inverse problem for the one-dimensional Dirac system on finite metric tree
graphs. The main goal is to recover the topology (connectivity) of a tree, lengths of
edges, and a matrix potential function on each edge. We use the dynamic response
operator as inverse data. In addition, we present a new dynamic algorithm to solve the
forward problem for the 1-D Dirac system on general finite metric tree graphs.