Preprints

I have organized my preprints into eight categories:

• Kakeya, Restriction, and Bochner-Riesz problems.  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.  Generally, these are areas of harmonic analysis with a strong geometric and combinatorial flavor.
• Honeycombs and Puzzles.  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.
• Multilinear operators.  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.
• Partial Differential Equations.  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.  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.
• Compressed sensing.  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.  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 joint work with Emmanuel Candes, Justin Romberg, Mark Rudelson, and Roman Vershynin.
• Miscellaneous Harmonic Analysis.  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 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.
• Arithmetic Combinatorics and Number theory.  This is my work on the combinatorics of addition, multiplication, and arithmetic progressions, and connections with number theory (for instance, in the distribution of the primes) and ergodic theory (via Szemeredi’s theorem), or with random matrices.  This includes my work with Tim Austin, Jean Bourgain, Kevin Costello, Tanja Eisner, Ben Green, Nets Katz, Van Vu, and Tamar Ziegler.
• Totally Miscellaneous.  This is anything not in the above seven categories; this includes my work with Andrew Millard and Peter Hall.

You can also look directly at my collection of expository notes and of the slides (these also appear, organized by category, in the above eight pages), or my collection of books.  Here is my Google Scholar page.

• All my recent papers have been archived in the Math ArXiV (which is why the downloads have labels such as math.CA/9910039).  You can get e-mail notification of all preprints sent to this server in your field of interest by following these instructions.
• My very early papers are only available in either dvi or compressed Postscript (ps.Z) format.  You may have to "uncompress" a ps.Z file before being able to view it.  DVI files are smaller, and thus quicker to download, than Postscript files, but there may be problems reading them (e.g. certain fonts may not be found). If the DVI file is difficult to read, try the postscript file.
• If you are getting a “404 file not found error”, the problem may be with the file address (the address should use forward slashes / instead of backward slashes \).  If the problem concerns an expository note (“short story”), you can try accessing the directory directly.
• Here's a DVI viewer for Microsoft Windows.
• Note that these files are for personal use only, as most of them are copyrighted by the journals indicated. A large number of these papers were done with the support of NSF grant DMS-9706764, the Clay Mathematical Institute, and/or grants from the Sloan, Packard, and Macarthur foundations.
• Please e-mail me if there are any problems with accessing or reading these files!