Global well-posedness of the Benjamin-Ono equation in H^1(R)

We show that the Benjamin-Ono equation is globally well-posed in $H^s(\R)$ for $s \geq 1$. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in $H^s$ for any $s$. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.Comment: 21 pages, no figures. Minor mathematical errors correcte

Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data

In any dimension $n \geq 3$, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear Schr\"odinger equation $i u_t + \Delta u = |u|^{\frac{4}{n-2}} u$ in $\R \times \R^n$ exist globally and scatter to free solutions; this generalizes the three and four dimensional results of Bourgain and Grillakis. Furthermore we have bounds on various spacetime norms of the solution which are of exponential type in the energy, which improves on the tower-type bounds of Bourgain. In higher dimensions $n \geq 6$ some new technical difficulties arise because of the very low power of the non-linearity.Comment: 23 pages, no figures, to appear, New York J. Math. This is the final versio

Arithmetic progressions and the primes - El Escorial lectures

We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length $k$ in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.Comment: 38 pages, no figures, submitted, El Escorial conference proceedings. These lectures are oriented towards a harmonic analysis audienc

A pseudoconformal compactification of the nonlinear Schr\"odinger equation and applications

We interpret the lens transformation (a variant of the pseudoconformal transformation) as a pseudoconformal compactification of spacetime, which converts the nonlinear Schr\"odinger equation (NLS) without potential with a nonlinear Schr\"odinger equation with attractive harmonic potential. We then discuss how several existing results about NLS can be placed in this compactified setting, thus offering a new perspective to view this theory.Comment: 17 pages, no figures, to appear, New York J. Math. This is the final versio

Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions

In the previous papers in this series, the global regularity conjecture for wave maps from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic space \H^m was reduced to the problem of constructing a minimal-energy blowup solution which is almost periodic modulo symmetries in the event that the conjecture fails. In this paper, we show that this problem can be reduced further, to that of showing that solutions at the critical energy which are either frequency-delocalised, spatially-dispersed, or spatially-delocalised have bounded ``entropy''. These latter facts will be demonstrated in the final paper in this series.Comment: 36 pages, no figures. Will not be published in current form, pending future reorganisation of the heatwave projec

The asymptotic distribution of a single eigenvalue gap of a Wigner matrix

We show that the distribution of (a suitable rescaling of) a single eigenvalue gap $\lambda_{i+1}(M_n)-\lambda_i(M_n)$ of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin-Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin-Mehta law required either an averaging in the eigenvalue index parameter $i$, or fixing the energy level $u$ instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function $N_{(-\infty,x)}(\tilde M_n)$ (where $\tilde M_n$ is a suitably rescaled version of $M_n$) with the event that there is no spectrum in an interval $[x,x+s]$, in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.Comment: 21 pages, no figures, submitted, Prob. Thy. and Related Fields. This is the final version, incorporating the referee comment

The Erdos discrepancy problem

We show that for any sequence $f: {\bf N} \to \{-1,+1\}$ taking values in $\{-1,+1\}$, the discrepancy $$\sup_{n,d \in {\bf N}} \left|\sum_{j=1}^n f(jd)\right|$$ of $f$ is infinite. This answers a question of Erd\H{o}s. In fact the argument also applies to sequences $f$ taking values in the unit sphere of a real or complex Hilbert space. The argument uses three ingredients. The first is a Fourier-analytic reduction, obtained as part of the Polymath5 project on this problem, which reduces the problem to the case when $f$ is replaced by a (stochastic) completely multiplicative function ${\bf g}$. The second is a logarithmically averaged version of the Elliott conjecture, established recently by the author, which effectively reduces to the case when ${\bf g}$ usually pretends to be a modulated Dirichlet character. The final ingredient is (an extension of) a further argument obtained by the Polymath5 project which shows unbounded discrepancy in this case.Comment: 29 pages, no figures. Formatted using the Discrete Analysis style fil

Equivalence of the logarithmically averaged Chowla and Sarnak conjectures

Let $\lambda$ denote the Liouville function. The Chowla conjecture asserts that $$\sum_{n \leq X} \lambda(a_1 n + b_1) \lambda(a_2 n+b_2) \dots \lambda(a_k n + b_k) = o_{X \to \infty}(X)$$ for any fixed natural numbers $a_1,a_2,\dots,a_k$ and non-negative integer $b_1,b_2,\dots,b_k$ with $a_ib_j-a_jb_i \neq 0$ for all $1 \leq i < j \leq k$, and any $X \geq 1$. This conjecture is open for $k \geq 2$. As is well known, this conjecture implies the conjecture of Sarnak that $$\sum_{n \leq X} \lambda(n) f(n) = o_{X \to \infty}(X)$$ whenever $f : {\bf N} \to {\bf C}$ is a fixed deterministic sequence and $X \geq 1$. In this paper, we consider the weaker logarithmically averaged versions of these conjectures, namely that $$\sum_{X/\omega \leq n \leq X} \frac{\lambda(a_1 n + b_1) \lambda(a_2 n+b_2) \dots \lambda(a_k n + b_k)}{n} = o_{\omega \to \infty}(\log \omega)$$ and $$\sum_{X/\omega \leq n \leq X} \frac{\lambda(n) f(n)}{n} = o_{\omega \to \infty}(\log \omega)$$ under the same hypotheses on $a_1,\dots,a_k,b_1,\dots,b_k$ and $f$, and for any $2 \leq \omega \leq X$. Our main result is that these latter two conjectures are logically equivalent to each other, as well as to the "local Gowers uniformity" of the Liouville function. The main tools used here are the entropy decrement argument of the author used recently to establish the $k=2$ case of the logarithmically averaged Chowla conjecture, as well as the inverse conjecture for the Gowers norms, obtained by Green, Ziegler, and the author.Comment: 28 pages, 1 figure, submitted, Number Theory - Diophantine problems, uniform distribution and applications, Festschrift in honour of Robert F. Tichy's 60th birthda

Perelman's proof of the Poincar\'e conjecture: a nonlinear PDE perspective

We discuss some of the key ideas of Perelman's proof of Poincar\'e's conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.Comment: 42 pages, unpublishe

A quantitative ergodic theory proof of Szemer\'edi's theorem

A famous theorem of Szemer\'edi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general $k$ there are essentially four known proofs of this fact; Szemer\'edi's original combinatorial proof using the Szemer\'edi regularity lemma and van der Waerden's theorem, Furstenberg's proof using ergodic theory, Gowers' proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers' more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is ``elementary'' in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.Comment: 52 pages (but a 20 page abridged version is available at http://www.math.ucla.edu/~tao/preprints/Expository/short_furstenberg.dvi), submitted, Electron. J. Combin. Some references added from previous versio