Classification of Almost Quarter-Pinched Manifolds
We show that if a simply connected manifold is almost quarter pinched then it is diffeomorphic to a CROSS (a compact rank one symmetric space) or a sphere.
math.DG↗
arXiv subjects
Terence Tao
Publications and source records attributed to Terence Tao.
On random $\pm 1$ matrices: Singularity and Determinant
This papers contains two results concerning random $n \times n$ Bernoulli matrices. First, we show that with probability tending to one the determinant has absolute value $\sqrt {n!} \exp(O(\sqrt(n log n)))$. Next, we prove a new upper bound $.939^n$ on the probability that the matrix is singular. We also give some generalizations to other random matrix models.
math.CO↗
Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions
Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah, Bahouri-Gérard and Nakanishi have established global wellposedness, regularity, and scattering in the energy class for the energy-critical nonlinear wave equation $\Box u = u^5$ in $\R^{1+3}$, together with a spacetime bound $$ \| u \|_{L^4_t L^{12}_x(\R^{1+3})} \leq M(E(u))$$ for some finite quantity M(E(u)) depending only on the energy E(u) of u. We reprove this result, and show that this quantity obeys a bound of at most exponential type in the energy, and specifically $M(E) \leq C (1+E)^{C E^{105/2}}$ for some absolute constant C > 0. The argument combines the quantitative local potential energy decay estimates of these previous papers with arguments used by Bourgain and the author for the analogous nonlinear Schrödinger equation.
math.AP↗
A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential
We study the asymptotic behavior of large data solutions in the energy space $H := H^1(\R^d)$ in very high dimension $d \geq 11$ to defocusing Schrödinger equations $i u_t + Δu = |u|^{p-1} u + Vu$ in $\R^d$, where $V \in C^\infty_0(\R^d)$ is a real potential (which could contain bound states), and $1+\frac{4}{d} < p < 1+\frac{4}{d-2}$ is an exponent which is energy-subcritical and mass-supercritical. In the spherically symmetric case, we show that as $t \to +\infty$, these solutions split into a radiation term that evolves according to the linear Schrödinger equation, and a remainder which converges in $H$ to a compact attractor $K$, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in $H$. The main novelty of this result is that $K$ is a \emph{global} attractor, being independent of the initial energy of the initial data; in particular, no matter how large the initial data is, all but a bounded amount of energy is radiated away in the limit.
math.AP↗
John-type theorems for generalized arithmetic progressions and iterated sumsets
A classical theorem of Fritz John allows one to describe a convex body, up to constants, as an ellipsoid. In this article we establish similar descriptions for generalized (i.e. multidimensional) arithmetic progressions in terms of proper (i.e. collision-free) generalized arithmetic progressions, in both torsion-free and torsion settings. We also obtain a similar characterization of iterated sumsets in arbitrary abelian groups in terms of progressions, thus strengthening and extending recent results of Szemerédi and Vu.
math.CO↗
A quantitative version of the Besicovitch projection theorem via multiscale analysis
By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has measure zero) and a standard companion result, namely that any planar set with at least two projections of measure zero is purely unrectifiable. We illustrate these results by providing an explicit (but weak) upper bound on the average projection of the $n^{th}$ generation of a product Cantor set.
math.CA↗
Linear Equations in Primes
Consider a system Ψof non-constant affine-linear forms ψ_1,...,ψ_t: Z^d -> Z, no two of which are linearly dependent. Let N be a large integer, and let K be a convex subset of [-N,N]^d. A famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N -> \infty, for the number of integer points n in K for which the integers ψ_1(n),...,ψ_t(n) are simultaneously prime. This implies many other well-known conjectures, such as the Hardy-Littlewood prime tuples conjecture, the twin prime conjecture, and the (weak) Goldbach conjecture. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms ψ_1,...,ψ_t are affinely related; this excludes the important ``binary'' cases such as the twin prime or Goldbach conjectures, but does allow one to count ``non-degenerate'' configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the Inverse Gowers-norm conjecture GI(s) and the Mobius and Nilsequences Conjecture MN(s), where s \in {1,2,...} is the complexity of the system and measures the extent to which the forms ψ_i depend on each other. For s = 1 these are essentially classical, and the authors recently resolved the cases s = 2. Our results are therefore unconditional in the case s = 2, and in particular we can obtain the expected asymptotics for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally for any (non-degenerate) problem involving two linear equations in four prime unknowns.
math.NT↗
The Dantzig selector: Statistical estimation when $p$ is much larger than $n$
In many important statistical applications, the number of variables or parameters $p$ is much larger than the number of observations $n$. Suppose then that we have observations $y=Xβ+z$, where $β\in\mathbf{R}^p$ is a parameter vector of interest, $X$ is a data matrix with possibly far fewer rows than columns, $n\ll p$, and the $z_i$'s are i.i.d. $N(0,σ^2)$. Is it possible to estimate $β$ reliably based on the noisy data $y$? To estimate $β$, we introduce a new estimator--we call it the Dantzig selector--which is a solution to the $\ell_1$-regularization problem \[\min_{\tilde{\b eta}\in\mathbf{R}^p}\|\tildeβ\|_{\ell_1}\quad subject to\quad \|X^*r\|_{\ell_{\infty}}\leq(1+t^{-1})\sqrt{2\log p}\cdotσ,\] where $r$ is the residual vector $y-X\tildeβ$ and $t$ is a positive scalar. We show that if $X$ obeys a uniform uncertainty principle (with unit-normed columns) and if the true parameter vector $β$ is sufficiently sparse (which here roughly guarantees that the model is identifiable), then with very large probability, \[\|\hatβ-β\|_{\ell_2}^2\le C^2\cdot2\log p\cdot \Biggl(σ^2+\sum_i\min(β_i^2,σ^2)\Biggr).\] Our results are nonasymptotic and we give values for the constant $C$. Even though $n$ may be much smaller than $p$, our estimator achieves a loss within a logarithmic factor of the ideal mean squared error one would achieve with an oracle which would supply perfect information about which coordinates are nonzero, and which were above the noise level. In multivariate regression and from a model selection viewpoint, our result says that it is possible nearly to select the best subset of variables by solving a very simple convex program, which, in fact, can easily be recast as a convenient linear program (LP).
math.ST↗
Rejoinder: The Dantzig selector: Statistical estimation when $p$ is much larger than $n$
Rejoinder to ``The Dantzig selector: Statistical estimation when $p$ is much larger than $n$'' [math/0506081]
math.ST↗
The cubic nonlinear Schrödinger equation in two dimensions with radial data
We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation $iu_t + Δu = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.
math.AP↗
Random Matrices: The circular Law
Let $\a$ be a complex random variable with mean zero and bounded variance $σ^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being i.i.d. copies of $\a$. Let $λ_{1}, ..., λ_{n}$ be the eigenvalues of $\frac{1}{σ\sqrt n}N_{n}$. Define the empirical spectral distribution $μ_{n}$ of $N_{n}$ by the formula $$ μ_n(s,t) := \frac{1}{n} # \{k \leq n| \Re(λ_k) \leq s; \Im(λ_k) \leq t \}.$$ The Circular law conjecture asserts that $μ_{n}$ converges to the uniform distribution $μ_\infty$ over the unit disk as $n$ tends to infinity. We prove this conjecture under the slightly stronger assumption that the $(2+η)þ$-moment of $\a$ is bounded, for any $η>0$. Our method builds and improves upon earlier work of Girko, Bai, Götze-Tikhomirov, and Pan-Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.
math.PR↗
Why are solitons stable?
The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, \emph{solitary wave} or \emph{soliton} solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions is relatively straightforward, but the fact that they are \emph{stable} requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the \emph{critical} case in which the nonlinearity is at exactly the right power to potentially allow for a self-similar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and Grillakis-Shatah-Strauss, to the more recent asymptotic stability and blowup analysis of Martel-Merle and Merle-Raphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations.
math.AP↗
New bounds for Szemeredi's Theorem, I: Progressions of length 4 in finite field geometries
Let F be a fixed finite field of characteristic at least 5. Let G = F^n be the n-dimensional vector space over F, and write N := |G|. We show that if A is a subset of G with size at least c_F N(log N)^{-c}, for some absolute constant c > 0 and some c_F > 0, then A contains four distinct elements in arithmetic progression. This is equivalent, in the usual notation of additive combinatorics, to the assertion that r_4(G) <<_F N(log N)^{-c}.
math.CO↗
Multi-linear multipliers associated to simplexes of arbitrary length
In this article we prove that the $n$-linear operator whose symbol is the characteristic function of the simplex $Δ_n = ξ_1 < ... < ξ_n$ is bounded from $L^2 \times ... \times L^2$ into $L^{2/n}$, generalizing in this way our previous work on the "bi-est" operator (which corresponds to the case $n=3$) as well as Lacey-Thiele theorem on the bi-linear Hilbert transform (which corresponds to the case $n=2$).
math.CA↗
The Walsh model for $M_2^{*}$ Carleson
We study the Walsh model of a certain maximal truncation of Carleson's operator, related to the Return Times Theorem from Ergodic Theory.
math.CA↗
The distribution of polynomials over finite fields, with applications to the Gowers norms
In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F^n -> F is poorly-distributed only if P is determined by the values of a few polynomials of lower degree, in which case we say that P has small rank. We give several applications of this result, paying particular attention to consequences for the theory of the so-called Gowers norms. We establish an inverse result for the Gowers U^{d+1}-norm of functions of the form f(x)= e_F(P(x)), where P : F^n -> F is a polynomial of degree less than F, showing that this norm can only be large if f correlates with e_F(Q(x)) for some polynomial Q : F^n -> F of degree at most d. The requirement deg(P) < |F| cannot be dropped entirely. Indeed, we show the above claim fails in characteristic 2 when d = 3 and deg(P)=4, showing that the quartic symmetric polynomial S_4 in F_2^n has large Gowers U^4-norm but does not correlate strongly with any cubic polynomial. This shows that the theory of Gowers norms in low characteristic is not as simple as previously supposed. This counterexample has also been discovered independently by Lovett, Meshulam, and Samorodnitsky. We conclude with sundry other applications of our main result, including a recurrence result and a certain type of nullstellensatz.
math.CO↗
A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fields
We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F_2^n, improving the previously known bounds in such theorems. For instance, if A is a subset of F_2^n such that |A+A| <= K|A| (thus A has small additive doubling), we show that there exists an affine subspace V of F_2^n of cardinality |V| >> K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. Under the assumption that A contains at least |A|^3/K quadruples with a_1 + a_2 + a_3 + a_4 = 0 we obtain a similar result, albeit with the slightly weaker condition |V| >> K^{-O(K)}|A|.
math.CO↗
Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation
We establish that the quadratic non-linear Schrödinger equation $$ iu_t + u_{xx} = u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\R)$ when $s \geq -1$ and ill-posed when $s < -1$. Previous work of Kenig, Ponce and Vega had established local well-posedness for $s > -3/4$. The local well-posedness is achieved by an iteration using a modification of the standard $X^{s,b}$ spaces. The ill-posedness uses an abstract and general argument relying on the high-to-low frequency cascade present in the non-linearity, and a computation of the first non-linear iterate.
math.AP↗