Harmonic Analysis and Applications
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The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments. Contents Preface Introduction Lecture Notes on Quantitative Unique Continuation for Solutions of Second Order Elliptic Equations • Alexander Logunov and Eugenia Malinnikova Arithmetic Spectral Transitions: A Competition between Hyperbolicity and the Arithmetics of Small Denominators • Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang Quantitative Homogenization of Elliptic Operators with Periodic Coefficients • Zhongwei Shen Stochastic Homogenization of Elliptic Equations • Charles K. Smart T1 and Tb Theorems and Applications • Simon Bortz, Steve Hofmann, and José Luis Luna Sliding Almost Minimal Sets and the Plateau Problem • G. David Almgren’s Center Manifold in a Simple Setting • Camillo De Lellis Lecture Notes on Rectifiable Reifenberg for Measures • Aaron Naber
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