Spring 2024

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Abstract: Over 20 years ago, the seminal paper by Bourgain and Goldstein introduced several fundamental ideas, including large deviation estimates (LDTs), into the analysis of Schrödinger operators on the integer lattice with potentials defined by irrational rotations. Goldstein and Schlag combined these LDTs with an Avalanche Principle expansion of SL(2,R) cocycles, linking the Hölder regularity of the integrated density to the distribution of zeros of determinants in finite volumes. More than a decade ago Avila presented his global theory of SL(2,C) cocylces, which hinged on the quantization property of the acceleration. In this talk, we introduce a joint work with Schlag, in which we establish a connection between Avila’s acceleration and Goldstein and Schlag’s zero count, and how it leads to improvements of some open problems.

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Abstract: Two classic questions – the Erdos distinct distance problem, which asks about the least number of distinct distances determined by points in the plane, and its continuous analog, the Falconer distance problem – both focus on distance. Here, distance can be thought of as a simple two point configuration. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as three point configurations. In this talk I will go through some of the history of such point configuration questions, show how geometric averaging operators arise naturally and give some recent results.

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Abstract: Quantized phase retrieval addresses the problem of recovering a signal, up to unimodular constants, from a set of coarsely quantized phaseless linear measurements. We introduce a signal recovery algorithm called ICQPhase, which addresses this problem using iterative consistent reconstruction. We discuss general background on consistent reconstruction methods in signal processing. Then we show that ICQPhase achieves mean squared error of order in a variety of experimental settings, where m is the number of quantized phaseless measurements, and we also outline initial theoretical progress and open problems. This is join work with Dylan Domel-White.

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Abstract: Allan Cormack’s pioneering work published in 1963 and 1964 provided the mathematical foundations of computerized tomography (CT) and thereby the first practical method to “see into” an object without physically breaking it open. Now, 50 years later, the research field of imaging is thriving and still pushes researchers to overcome new theoretical and practical challenges. One underlying assumption in the work of Cormack is that the searched-for quantity is stationary during the acquisition of the X-ray data. However, this assumption is violated in many modern applications, e.g. due to physiological motion of a patient or while imaging engines at working stage. Such a dynamic behavior leads to inconsistent data sets and the application of standard image reconstruction techniques lead to motion artefacts which can significantly impede a reliable diagnostics. Consequently, suitable models and algorithms have to be developed in order to provide artefact free images. In particular, microlocal analysis offers a powerful tool to analyze dynamic imaging problems, thus providing deep insights about the nature of the problem. Based on such analysis, we derive numerical schemes to reduce artefacts in the reconstructed image and to improve the overall image quality. We further present an alternative strategy which treats the dynamic behavior as uncertainty in the forward model. Both reconstruction approaches are evaluated on simulated and real CT-data with different dynamic behavior.

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Abstract: The development of technologies leads to new applications, new challenges, and new issues in the field of Imaging. Two of the main challenges, namely model inexactness and motion, are tackled in this talk. Dynamic inverse problems have been vastly studied in the last decade with the initial aim to reduce the artefacts observed in a CT scan due to the movements of the patient and have been developed into broader setups and applications. Motion leads in general to model inexactness. For instance, in computerized tomography (CT), the movement of the patient alters the nature of the integration curves and therefore, intrinsically, the model itself. Since the motion is in general unknown, it implies that the model is not exactly known. However, the model inexactness can be more specific with other applications. A good example is Compton scattering imaging. Modelling the Compton scattering effect leads to many challenges such as non-linearity of the forward model, multiple scattering and high level of noise for moving targets. While the non-linearity is addressed by a necessary linear approximation of the first-order scattering with respect to the sought-for electron density, the multiple-order scattering stands for a substantial and unavoidable part of the spectral data which is difficult to handle due to highly complex forward models. Last but not least, the stochastic nature of the Compton effect may involve a large measurement noise, in particular when the object under study is subject to motion, and therefore time and motion must be taken into account. To tackle these different issues, we study in this talk two data-driven techniques, namely the regularized sequential subspace optimization and a Bayesian method based on the generalized Golub-Kahan bidiagonalization. We then explore the possibilities to mimic and improve the stochastic approach with deep neural networks. The results are illustrated by simulations.

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Abstract: The distribution of prime numbers is intimately connected to the Riemann zeta function. While there are important open questions related to the zeta function, many of its most basic analytic properties are inherited from the lattice structure of the integers via Fourier analysis. Similarly, if one wishes to study the distribution of primes satisfying congruence conditions, a key tool is the analysis of certain periodic functions on the integers. In this talk, we’ll explain this story and its generalizations to some higher dimensional lattices. We’ll connect this to a beautiful theorem in number theory called the Chebotarev density theorem, and, if there’s time, we’ll discuss some recent work of the speaker on so-called effective Chebotarev density theorems.

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