Arithmetic Combinatorics and Number theory.This includes my work with Pascal Auscher, Jon Bennet, Tony Carbery, Michael Christ, Michael Cowling, Steve Hofman, Alex Iosevich, Nets Katz, Camil Muscalu, Christoph Thiele, Andreas Seeger, Brani Vidakovic, and Jim Wright. This is my catch-all page for harmonic analysis, wavelet, or functional analysis papers which are not directly related to multilinear operators, to the Kakeya/Restriction/Bochner-Riesz family of conjectures, or to sparse recovery. This is joint work with Emmanuel Candes, Justin Romberg, Mark Rudelson, and Roman Vershynin. It turns out that an l^1 minimization (or “basis pursuit”) approach works remarkably well, as long as the measurements obey suitable “uniform uncertainty principles”. This is work in applied harmonic analysis, signal processing, coding theory, and statistics, all centered around the issue of how to recover a sparse or compressible signal as best as possible if only a small number of (possibly noisy) measurements are made. Many of these papers are joint work with one or more of my co-authors Ioan Bejenaru, Michael Christ, Jim Colliander, Phillipe LeFloch, Andrew Hassell, Mark Keel, Rowan Killip, Sergiu Klainerman, Igor Rodnianski, Gigliola Staffilani, Eitan Tadmor, Hideo Takaoka, Gang Tian, Jared Wunsch, Monica Visan, and Xiaoyi Zhang. This is mostly work on non-linear dispersive and wave equations (and their associated linear and multilinear estimates), but also includes some work on elliptic theory and inverse scattering. This is joint work with Michael Christ, Ciprian Demeter, Loukas Grafakos, Xiaochun Li, Camil Muscalu, Jill Pipher, Erin Terwilleger, and Christoph Thiele dealing with multilinear operators such as the bilinear Hilbert transform and Carleson's maximal operator, and their generalizations a characteristic feature of such operators is that one is forced to decompose the phase plane in rather adaptive ways. This is joint work with Allen Knutson and Chris Woodward on the honeycomb and puzzle combinatorial models for computing sums of Hermitian matrices, tensor product multiplicities, and Schubert calculus intersection numbers. Generally, these are areas of harmonic analysis with a strong geometric and combinatorial flavor. This includes my work with Nets Katz, Izabella Laba, Gerd Mockenhaupt, Adam Sikora, Ana Vargas, and Luis Vega on the various variants of the Kakeya, Restriction, Bochner-Riesz, local smoothing, and L^p null form estimate problems. Kakeya, Restriction, and Bochner-Riesz problems.I have organized my preprints into eight categories:
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