Fall 2024

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Abstract: We develop a method to restore an unknown sine-Gordone like nonlinear partial differential equation (PDE) from its linearization around an unknown stationary solution. The main idea is to use the Goldstone theorem to establish a correspondence between eigenfunctions of the Schrodinger equation and the nonlinearity of the PDE.

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Abstract: In 1964 Kadec established the exact value of the Paley-Wiener constant which bounds the perturbation of harmonics in the orthonormal basis of exponential functions for the system to remain a Riesz basis. The result has many extensions such as the theorems by Katznelson, Avdonin, and Balan. All of these extensions, however, apply to systems of exponential functions on an interval. In this talk I will show how to extend Kadec’s proof to systems of vectors generated by sampling an orbit of a vector under an isometric group representation. The talk is based on joint work with B. Miller.

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We present a definition of a short-time Fourier transform in terms of the eigenfunctions of the Neumann Laplacian and the corresponding heat kernel on the Sierpinski Gasket. We show that that for this is well-defined and leads to properties akin to that of the classical short-time Fourier transform. Furthermore, this leads to an inversion formula that allows one to express in terms of its space-frequency shifts. We also discuss potential extensions to general post-critically finite fractals.

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Abstract: Frames, like orthonormal bases, provide a continuous, linear, and stable reconstruction formula for vectors in a Hilbert space. Unlike orthonormal bases, frames allow for redundancy, which makes them more flexible for both theoretical work and practical applications. One such application is phase retrieval, which is used in fields like X-ray crystallography and coherent diffraction imaging, where only the intensity of each linear measurement is available, and the phase information is lost. In this talk, I will introduce the concepts of frames and phase retrieval, then discuss a real-world scenario where sensors are set up to clip any measurement that exceeds a threshold due to saturation. This problem, known as declipping or saturation recovery, focuses on reconstructing a vector from such clipped measurements. This talk will focus on methods for recovering vectors using phase retrieval and saturation recovery techniques..

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