Spring 2023

Abstract: The first part of the talk will be concerned with the spectral analysis of a new class of fractal-type diamond graphs and provide a gap-labeling theorem in the sense of Bellissard. We will show that labeling the spectral gaps 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 second part of the talk is a gentle introduction to several research projects I worked on. We will review topics ranging from analysis on fractals, through quantum walks on graphs, to perfect quantum state transfer.

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Abstract: We expand on a well-known result of Daubechies, Cahill, and Casazza which says that the infinite dimensional phase-retrieval problem is never stable with respect to certain natural choices of metric. Using their proof, we bootstrap all the way up to the operator recovery problem for compact operators on a Hilbert space.

The next part of the talk is a novel and far simpler proof which generalizes the result all the way to bounded operators on a Hilbert space, and dispenses with some of the criteria required in the proof by Daubechies, Cahill, and Casazza. We also give an interpretation of the result in terms of the non-existence of certain types of Banach Frames.

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Abstract:    Some theorems are discovered, proved and, one hopes, remembered, invoked and cited. Others are proved and reproved, again and again. The latter type is getting increasing attention. We’ll explore this phenomenon through 2+1 theorems in geometry and linear algebra, and ponder reasons to reprove.

The talk will include audience participation. Bring a sheet of paper and prepare to draw a large triangle and an angle or two (protractor optional). Save the backside for row reducing a matrix, or aim your device to SageMath.

Abstract:   The Heil-Ramanathan-Topiwala (HRT) Conjecture posits that any finite sequence of vectors arising from time-frequency shifts is linearly independent. In this talk, we will provide an overview of established results and delve into recent attempts to tackle unresolved cases. By examining the intricacies of time-frequency shifts, we seek to uncover deeper insights into the HRT Conjecture and its implications for the broader field of harmonic analysis.

To find the most efficient shape of a noise-absorbing wall to dissipate the acousti- cal energy of a sound wave, we consider a frequency model described by the Helmholtz equation with damping on the boundary modeled by a complex-valued Robin boundary condition. We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall ensures the infimum of the acoustic energy. Then we also introduce a larger compact class of (ε, ∞) – or uniform domains with possibly non-Lipschitz (for ex- ample, fractal) boundaries on which an optimal shape exists, giving the minimum of the energy. The boundaries are described as the supports of Radon measures ensuring their Hausdorff dimension in the segment [n−1, n) . A by-product of our proof is that the class of bounded (ε, ∞) -domains with fixed ε is stable under Hausdorff convergence. Another related result is the Mosco convergence of Robin-type energy functionals on converging domains.

In this talk, we consider seismic operators with two scanning geometries: zero-offset (the source and receiver are at the same point and translated over the surface of the earth) and common-offset (the source and receiver are offset a fixed distance from each other and translated together). We explain the basics of the problem and show how the data can be modeled using a Radon transform that integrates over surfaces determined by the background velocity. We then analyze the model with a linearly increasing background velocity in two spatial dimensions. We show for zero offset data that a standard reconstruction operator (the normal operator) will reliably image features (wavefront set) of the object and not add artifacts. We prove that this is stable under sufficiently small perturbations of the background velocity or of the offset. Therefore, if the nonconstant sound speed is close enough to being linearly increasing then its normal operator will also recover features and not add artifacts. We provide a reconstruction from simulated data to demonstrate our results.