i 1 ii What’s new - 2007: Open questions, expository articles, and lecture series from a mathematical blog Terence Tao May 8, 2008 1The author is supported by NSF grant CCF-0649473 and a grant from the MacArthur foun- dation. To my advisor, Eli Stein, for showing me the importance of good exposition; To my friends, for supporting this experiment; And to the readers of my blog, for their feedback and contributions. Contents 1 Open problems 1 1.1 Best bounds for capsets . 2 1.2 Noncommutative Freiman theorem . 4 1.3 Mahler’s conjecture for convex bodies . 7 1.4 Why global regularity for Navier-Stokes is hard . 10 1.5 Scarring for the Bunimovich stadium . 21 1.6 Triangle and diamond densities . 25 1.7 What is a quantum honeycomb? . 29 1.8 Boundedness of the trilinear Hilbert transform . 34 1.9 Effective Skolem-Mahler-Lech theorem . 39 1.10 The parity problem in sieve theory . 43 1.11 Deterministic RIP matrices . 55 1.12 The nonlinear Carleson conjecture . 59 2 Expository articles 63 2.1 Quantum mechanics and Tomb Raider . 64 2.2 Compressed sensing and single-pixel cameras . 71 2.3 Finite convergence principle . 77 2.4 Lebesgue differentiation theorem . 88 2.5 Ultrafilters and nonstandard analysis . 95 2.6 Dyadic models . 110 2.7 Math doesn’t suck . 121 2.8 Nonfirstorderisability . 130 2.9 Amplification and arbitrage . 133 2.10 The crossing number inequality . 142 2.11 Ratner’s theorems . 149 2.12 Lorentz group and conic sections . 155 2.13 Jordan normal form . 162 2.14 John’s blowup theorem . 166 2.15 Hilbert’s nullstellensatz . 172 2.16 Hahn-Banach, Menger, Helly . 180 2.17 Einstein’s derivation of E = mc2 . 186 v vi CONTENTS 3 Lectures 193 3.1 Simons Lecture Series: Structure and randomness . 194 3.2 Ostrowski lecture . 215 3.3 Milliman lectures . 222 Preface vii viii CONTENTS Almost nine years ago, in 1999, I began a “What’s new?” page on my UCLA home page in order to keep track of various new additions to that page (e.g. papers, slides, lecture notes, expository “short stories”, etc.). At first, these additions were simply listed without any commentary, but after a while I realised that this page was a good place to put a brief description and commentary on each of the mathematical articles that I was uploading to the page. (In short, I had begun blogging on my research, though I did not know this term at the time.) Every now and then, I received an email from someone who had just read the most recent entry on my “What’s new?” page and wanted to make some mathematical or bibliographic comment; this type of valuable feedback was one of the main reasons why I kept maintaining the page. But I did not think to try to encourage more of this feedback until late in 2006, when I posed a question on my “What’s new?” page and got a complete solution to that problem within a matter of days. It was then that I began thinking about modernising my web page to a blog format (which a few other math- ematicians had already begun doing). On 22 February 2007, I started a blog with the unimaginative name of “What’s new” at erryao.wordpress.com; I chose wordpress for a number of reasons, but perhaps the most decisive one was its recent decision to support LATEX in its blog posts. It soon became clear that the potential of this blog went beyond my original aim of merely continuing to announce my own papers and research. For instance, by far the most widely read and commented article in my blog in the first month was a non- technical article, “Quantum Mechanics and Tomb Raider” (Section 2.1), which had absolutely nothing to do with my own mathematical work. Encouraged by this, I began to experiment with other types of mathematical content on the blog; discussions of my favourite open problems, informal discussions of mathematical phenomena, principles, or tricks, guest posts by some of my colleagues, and presentations of various lectures and talks, both by myself and by others; and various bits and pieces of advice on pursuing a mathematical career and on mathematical writing. This year, I also have begun placing lecture notes for my graduate classes on my blog. After a year of mathematical blogging, I can say that the experience has been pos- itive, both for the readers of the blog and for myself. Firstly, the very act of writing a blog article helped me organise and clarify my thoughts on a given mathematical topic, to practice my mathematical writing and exposition skills, and also to inspect the references and other details more carefully. From insightful comments by experts in other fields of mathematics, I have learned of unexpected connections between different fields; in one or two cases, these even led to new research projects and collaborations. From the feedback from readers I obtained a substantial amount of free proofreading, while also discovering what parts of my exposition were unclear or otherwise poorly worded, helping me improve my writing style in the future. It is a truism that one of the best ways to learn a subject is to teach it; and it seems that blogging about a subject comes in as a close second. In the last year (2007) alone, at least a dozen new active blogs in research math- ematics have sprung up. I believe this is an exciting new development in mathemat- ical exposition; research blogs seem to fill an important niche that neither traditional print media (textbooks, research articles, surveys, etc.) nor informal communications (lectures, seminars, conversations at a blackboard, etc.) adequately cover at present. CONTENTS ix Indeed, the research blog medium is in some ways the “best of both worlds”; informal, dynamic, and interactive, as with talks and lectures, but also coming with a permanent record, a well defined author, and links to further references, as with the print media. There are bits and pieces of folklore in mathematics, such as the difference between hard and soft analysis (Section 2.3) or the use of dyadic models for non-dyadic situa- tions (Section 2.6) which are passed down from advisor to student, or from collaborator to collaborator, but are too fuzzy and non-rigorous to be discussed in the formal liter- ature; but they can be communicated effectively and efficiently via the semi-formal medium of research blogging. On the other hand, blog articles still lack the permanence that print articles have, which becomes an issue when one wants to use them in citations. For this and other reasons, I have decided to convert some of my blog articles from 2007 into the book that you are currently reading. Not all of the 93 articles that I wrote in 2007 appear here; some were mostly administrative or otherwise non-mathematical in nature, some were primarily announcements of research papers or other articles which will appear elsewhere, some were contributed guest articles, and some were writeups of lectures by other mathematicians, which it seemed inappropriate to reproduce in a book such as this. Nevertheless, this still left me with 32 articles, which I have converted into print form (replacing hyperlinks with more traditional citations and footnotes, etc.). As a result, this book is not a perfect replica of the blog, but the mathematical content is largely the same. I have paraphrased some of the feedback from comments to the blog in the endnotes to each article, though for various reasons, ranging from lack of space to copyright concerns, not all comments are reproduced here. The articles here are rather diverse in subject matter, to put it mildly, but I have nevertheless organised them into three categories. The first category concerns vari- ous open problems in mathematics that I am fond of; some are of course more difficult than others (see e.g. the article on Navier-Stokes regularity, Section 1.4), and others are rather vague and open-ended, but I find each of them interesting, not only in their own right, but because progress on them is likely to yield insights and techniques that will be useful elsewhere. The second category are the expository articles, which vary from discussions of various well-known results in maths and science (e.g. the nullstellensatz in Section 2.15, or Einstein’s equation E = mc2 in Section 2.17), to more philosophical explorations of mathematical ideas, tricks, tools, or principles (e.g. ultrafilters in Sec- tion 2.5, or amplification in Section 2.9), to non-technical expositions of various topics in maths and science, from quantum mechanics (Section 2.1) to single-pixel cameras (Section 2.2). Finally, I am including writeups of three lecture series I gave in 2007; my Simons lecture series at MIT on structure on randomness, my Ostrowski lecture at the University of Leiden on compressed sensing, and my Milliman lectures at the University of Washington on additive combinatorics. In closing, I believe that this experiment with mathematical blogging has been gen- erally successful, and I plan to continue it in the future, and perhaps generating several more books such as this one as a result. I am grateful to all the readers of my blog for supporting this experiment, for supplying invaluable feedback and corrections, and for encouraging projects such as this book conversion. x CONTENTS A remark on notation One advantage of the blog format is that one can often define a technical term simply by linking to an external web page that contains the definition (e.g.