total 438took 0.11s

Infinitesimal Lipschitz conditions on family of analytic varietiesFeb 08 2019In this work, we extend the concept of the double of an ideal defined in \cite{G2}, to the context of modules. We also obtain the genericity of the infinitesimal Lipschitz condition A for an enlarged class of analytic spaces.

The genericity of the infinitesimal Lipschitz condition for hypersurfacesMar 21 2013Oct 11 2014We continue the development of the theory of infinitesimal Lipschitz equivalence, showing the genericity of the condition for families of hypersurfaces with isolated singularities.

Invariants of D(q,p) singularitiesAug 01 2005Feb 20 2007The basic examples of functions defining non-isolated hypersurface singularities are the A(d) singularities and the D(q,p) singularities. The A(d) singularities, up to analytic equivalence, are the product of a Morse function and the zero map, while the ... More

Infinitesimal bi-Lipschitz Equivalence of FunctionsJan 20 2016We introduce two different notions of infinitesimal bi-Lipschitz equivalence for functions, one related to bi-Lipschitz triviality of families of functions, one related to homeomorphisms which are bi-Lipschitz on the fibers of the functions in the family. ... More

Nilpotents, Integral Closure and Equisingularity conditionsJul 28 2005In earlier work, the author described various stratification conditions for a complex analytic set X in terms of the theory of integral closure of modules. However, even if an analytic set has a reduced structure, often geometric operations like intersection ... More

The Multiplicity-Polar TheoremMar 21 2007Given a family of pairs of modules parametrised by a smooth space Y, the Multiplicity-Polar Theorem relates the multiplicity of the pair of modules at a special point of the parameter to the multiplicity of the pair at a generic point. This theorem is ... More

The multiplicity of pairs of modules and hypersurface singularitiesSep 02 2005This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber ... More

The Multiplicity Polar Theorem and Isolated SingularitiesSep 13 2005Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem ... More

Segre Numbers and Hypersurface SingularitiesNov 01 1996We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, ... More

Pairs of modules and determinantal isolated singularitiesDec 31 2014Jan 01 2016We continue the development of the study of the equisingularity of isolated singularities, in the determinantal case. This version of the paper includes a substantial amount of new material (76% larger). The new material introduces the idea of the landscape ... More

Weak subintegral closure of idealsAug 22 2007Sep 12 2008We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ... More

Equisingularity and EIDSFeb 01 2016The study of Essentially Isolated Determinantal Singularites or EIDS was initiated by Ebeling and Gusein-Zade. They are non-smoothable as determinantal singularities, and in general have non-isolated singularities. Their singularities are generic in a ... More

The Lê numbers of the square of a function and their applicationsAug 08 2005L\^e numbers were introduced by Massey with the purpose of numerically controlling the topological properties of families of non-isolated hypersurface singularities and describing the topology associated with a function with non-isolated singularities. ... More

The Multiplicity Polar Theorem, collections of 1-forms and Chern numbersDec 30 2011In this work we show how the Multiplicity Polar Theorem can be used to calculate Chern numbers for a collection of 1-forms.

Equisingularity of sections, $(t^r)$ condition, and the integral closure of modulesAug 30 2005This paper uses the theory of integral closure of modules to study the sections of both real and complex analytic spaces. The stratification conditions used are the (t^) conditions introduced by Thom and Trotman. Our results include a new simple proof ... More

The local Euler obstruction and topology of the stabilization of associated determinantal varietiesNov 02 2016Nov 10 2016This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern--Schwartz--MacPherson class of such varieties. In the second part we compute ... More

The local Euler obstruction and topology of the stabilization of associated determinantal varietiesNov 02 2016This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern--Schwartz--MacPherson class of such varieties. In the second part we compute ... More

The local Euler obstruction and topology of the stabilization of associated determinantal varietiesNov 02 2016Nov 22 2016This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern--Schwartz--MacPherson class of such varieties. In the second part we compute ... More

Blow-Analytic Equivalence versus contact-bi-Lipschitz EquivalenceJan 22 2016Jan 25 2016Two blow-analytically equivalent real analytic plane function germs are sub-analytically bi-Lipschitz contact equivalent

Specialization of integral dependence for modulesOct 03 1996May 14 1998We establish the principle of specialization of integral dependence for submodules of finite colength of free modules, as part of the general algebraic-geometric theory of the Buchsbaum--Rim multiplicity. Then we apply the principle to the study of equisingularity ... More

Small time asymptotics for stochastic evolution equationsDec 03 2010We obtain a large deviation principle describing the small time asymptotics of the solution of a stochastic evolution equation with multiplicative noise. Our assumptions are a condition on the linear drift operator that is satisfied by generators of analytic ... More

The Erdos discrepancy problemSep 17 2015Nov 10 2015We 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 ... More

Mixing for progressions in non-abelian groupsDec 11 2012May 31 2013We study the mixing properties of progressions $(x,xg,xg^2)$, $(x,xg,xg^2,xg^3)$ of length three and four in a model class of finite non-abelian groups, namely the special linear groups $SL_d(F)$ over a finite field $F$, with $d$ bounded. For length three ... More

The asymptotic distribution of a single eigenvalue gap of a Wigner matrixMar 07 2012Aug 31 2012We 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 ... More

Freiman's theorem for solvable groupsJun 18 2009Feb 19 2010Freiman's theorem asserts, roughly speaking, if that a finite set in a torsion-free abelian group has small doubling, then it can be efficiently contained in (or controlled by) a generalised arithmetic progression. This was generalised by Green and Ruzsa ... More

Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schrödinger equationJul 16 2008Feb 23 2009We consider the focusing mass-critical NLS $iu_t + \Delta u = - |u|^{4/d} u$ in high dimensions $d \geq 4$, with initial data $u(0) = u_0$ having finite mass $M(u_0) = \int_{\R^d} |u_0(x)|^2 dx < \infty$. It is well known that this problem admits unique ... More

Obstructions to uniformity, and arithmetic patterns in the primesMay 19 2005In this expository article, we describe the recent approach, motivated by ergodic theory, towards detecting arithmetic patterns in the primes, and in particular establishing that the primes contain arbitrarily long arithmetic progressions. One of the ... More

A counterexample to an endpoint bilinear Strichartz inequalitySep 29 2006The endpoint Strichartz estimate $\| e^{it\Delta} f \|_{L^2_t L^\infty_x(\R \times \R^2)} \lesssim \|f\|_{L^2_x(\R^2)}$ is known to be false by the work of Montgomery-Smith, despite being only ``logarithmically far'' from being true in some sense. In ... More

A variant of the hypergraph removal lemmaMar 24 2005Nov 16 2005Recent work of Gowers and Nagle, R\"odl, Schacht, and Skokan has established a hypergraph removal lemma, which in turn implies some results of Szemer\'edi and Furstenberg-Katznelson concerning one-dimensional and multi-dimensional arithmetic progressions ... More

Geometric renormalization of large energy wave mapsNov 16 2004There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization ... More

An uncertainty principle for cyclic groups of prime orderAug 29 2003Jul 22 2004Let $G$ be a finite abelian group, and let $f: G \to \C$ be a complex function on $G$. The uncertainty principle asserts that the support $\supp(f) := \{x \in G: f(x) \neq 0\}$ is related to the support of the Fourier transform $\hat f: G \to \C$ by the ... More

Global well-posedness of the Benjamin-Ono equation in H^1(R)Jul 22 2003Jun 26 2004We 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 ... More

Some recent progress on the Restriction conjectureMar 12 2003We survey recent developments on the Restriction conjecture.

Global regularity of wave maps I. Small critical Sobolev norm in high dimensionOct 07 2000Dec 02 2000We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty, not present ... More

Ill-posedness for one-dimensional wave maps at the critical regularityNov 29 1998Mar 29 1999We show that the wave map equation in $\R^{1+1}$ is in general ill-posed in the critical space $\dot H^{1/2}$, and the Besov space $\dot B^{1/2,1}_2$. The problem is attributed to the bad behaviour of the one-dimensional bilinear expression $D^{-1}(f ... More

A converse extrapolation theorem for translation invariant operatorsDec 01 1999We prove the converse of Yano's extrapolation theorem for translation invariant operators.

Inverse theorems for sets and measures of polynomial growthJul 05 2015Oct 01 2015We give a structural description of the finite subsets $A$ of an arbitrary group $G$ which obey the polynomial growth condition $|A^n| \leq n^d |A|$ for some bounded $d$ and sufficiently large $n$, showing that such sets are controlled by (a bounded number ... More

Finite time blowup for Lagrangian modifications of the three-dimensional Euler equationJun 27 2016Nov 14 2016In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as \begin{align*} \partial_t \omega + {\mathcal L}_u \omega &= 0\\ u &= \delta \tilde \eta^{-1} \Delta^{-1} \omega ... More

Structure and randomness in combinatoricsJul 29 2007Aug 03 2007Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable ... More

The dichotomy between structure and randomness, arithmetic progressions, and the primesDec 06 2005Dec 31 2005A famous theorem of Szemer\'edi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a fundamental dichotomy ... More

Upper semicontinuity of the lamination hullSep 30 2016Let $K \subseteq \mathbb{R}^{2 \times 2}$ be a compact set, let $K^{rc}$ be its rank-one convex hull, and let $L(K)$ be its lamination convex hull. It is shown that the mapping $K \to \overline{L(K)}$ is not upper semicontinuous on the diagonal matrices ... More

The Nonequilibrium Dynamics of Driven Line LiquidsJun 24 1992We study the nonequilibrium dynamics of line liquids as realized in the nonlinear motion of flux lines of a superconductor driven by an applied electric current. Our analysis suggests a transition in the dynamics of the lines from a smooth, laminar phase ... More

Measuring Dynamical K/$π$ and p/$π$ Fluctuations in Au+Au Collisions from the STAR ExperimentJan 17 2011Results from new measurements of dynamical $K/\pi$ and $p/\pi$ ratio fluctuations are presented. Dynamical fluctuations in global conserved quantities such as baryon number, strangeness, or charge may be observed near a QCD critical point. The STAR experiment ... More

Recent progress on the restriction conjectureNov 12 2003The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and other hypersurfaces. ... More

From rotating needles to stability of waves; emerging connections between combinatorics, analysis and PDEAug 14 2000We survey the interconnections between geometric combinatorics (such as the Kakeya problem), arithmetic combinatorics (such as the classical problem of determining which sets contain arithmetic progressions), oscillatory integrals (such as the Bochner-Riesz, ... More

Local well-posedness of the Yang-Mills equation in the Temporal Gauge below the energy normMay 07 2000Nov 28 2009We show that the Yang-Mills equation in three dimensions is locally well-posed in the Temporal gauge for initial data in H^s x H^{s-1} for s > 3/4, if the norm of the initial data is sufficiently small. The main new ingredients are a splitting of the ... More

Endpoint bilinear restriction theorems for the cone, and some sharp null form estimatesSep 14 1999Apr 13 2000Recently Wolff obtained a nearly sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and generalize to the case when one of the subsets is large. As a consequence, we ... More

The weak-type (1,1) of L \log L homogeneous convolution operatorsMar 29 1999Sep 06 1999We show that a homogeneous convolution kernel on an arbitrary homogeneous group which is L \log L on the unit annulus is bounded on L^p for 1 < p < \infty and is of weak-type (1,1), generalizing the result of Seeger. The proof is in a similar spirit to ... More

Cancellation for the multilinear Hilbert transformMay 24 2015May 29 2015For any natural number $k$, consider the $k$-linear Hilbert transform $$ H_k( f_1,\dots,f_k )(x) := \operatorname{p.v.} \int_{\bf R} f_1(x+t) \dots f_k(x+kt)\ \frac{dt}{t}$$ for test functions $f_1,\dots,f_k: {\bf R} \to {\bf C}$. It is conjectured that ... More

Outliers in the spectrum of iid matrices with bounded rank perturbationsDec 21 2010Sep 05 2014It is known that if one perturbs a large iid random matrix by a bounded rank error, then the majority of the eigenvalues will remain distributed according to the circular law. However, the bounded rank perturbation may also create one or more outlier ... More

A remark on primality testing and decimal expansionsFeb 22 2008Apr 18 2010We show that for any fixed base $a$, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base $a$ expansion; the case $a=2$ was already established by Cohen-Selfridge and Sun, using ... More

Norm convergence of multiple ergodic averages for commuting transformationsJul 08 2007Oct 24 2007Let $T_1, ..., T_l: X \to X$ be commuting measure-preserving transformations on a probability space $(X, \X, \mu)$. We show that the multiple ergodic averages $\frac{1}{N} \sum_{n=0}^{N-1} f_1(T_1^n x) ... f_l(T_l^n x)$ are convergent in $L^2(X,\X,\mu)$ ... More

A quantitative version of the Besicovitch projection theorem via multiscale analysisJun 18 2007May 15 2008By 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 ... More

Two remarks on the generalised Korteweg de-Vries equationJun 09 2006Jan 20 2009We make two observations concerning the generalised Korteweg de Vries equation $u_t + u_{xxx} = \mu (|u|^{p-1} u)_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for $L^2$-critical equation ... More

Scattering for the quartic generalised Korteweg-de Vries equationMay 14 2006We show that the quartic generalised KdV equation $$ u_t + u_{xxx} + (u^4)_x = 0$$ is globally wellposed for data in the critical (scale-invariant) space $\dot H^{-1/6}_x(\R)$ with small norm (and locally wellposed for large norm), improving a result ... More

Global behaviour of nonlinear dispersive and wave equationsAug 11 2006We survey recent advances in the analysis of the large data global (and asymptotic) behaviour of nonlinear dispersive equations such as the nonlinear wave (NLW), nonlinear Schr\"odinger (NLS), wave maps (WM), Schr\"odinger maps (SM), generalised Korteweg-de ... More

Product set estimates for non-commutative groupsJan 18 2006Oct 26 2011We develop the Pl\"unnecke-Ruzsa and Balog-Szemer\'edi-Gowers theory of sum set estimates in the non-commutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freiman-type inverse theorem for ... More

An integration approach to the Toeplitz square peg problemNov 22 2016The "square peg problem" or "inscribed square problem" of Toeplitz asks if every simple closed curve in the plane inscribes a (non-degenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ... More

A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equationsNov 13 2006Jan 27 2014We study the asymptotic behavior of large data solutions to Schr\"odinger equations $i u_t + \Delta u = F(u)$ in $\R^d$, assuming globally bounded $H^1_x(\R^d)$ norm (i.e. no blowup in the energy space), in high dimensions $d \geq 5$ and with nonlinearity ... More

A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potentialMay 11 2008May 28 2008We 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 Schr\"odinger equations $i u_t + \Delta u = |u|^{p-1} u + Vu$ in $\R^d$, where $V \in C^\infty_0(\R^d)$ is a ... More

Finite time blowup for a supercritical defocusing nonlinear wave systemFeb 25 2016May 13 2016We consider the global regularity problem for defocusing nonlinear wave systems $$ \Box u = (\nabla_{{\bf R}^m} F)(u) $$ on Minkowski spacetime ${\bf R}^{1+d}$ with d'Alambertian $\Box := -\partial_t^2 + \sum_{i=1}^d \partial_{x_i}^2$, the field $u: {\bf ... More

The sum-product phenomenon in arbitrary ringsJun 16 2008Feb 23 2009The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense "close" to a finite subring of $R$. This phenomenon has been analysed intensively ... More

Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable setsNov 13 2012Jan 03 2013Let $P: \F \times \F \to \F$ be a polynomial of bounded degree over a finite field $\F$ of large characteristic. In this paper we establish the following dichotomy: either $P$ is a moderate asymmetric expander in the sense that $|P(A,B)| \gg |\F|$ whenever ... More

Energy Dependence of Short and Long-Range Multiplicity Correlations in Au+Au Collisions from STARFeb 26 2007A general overview of the measurement of long-range multiplicity correlations measured by the STAR experiment in Au+Au collisions at RHIC is presented. The presence of long-range correlations can provide insight into the early stages, and the type of ... More

Noncommutative sets of small doublingJun 11 2011Apr 03 2012A corollary of Kneser's theorem, one sees that any finite non-empty subset $A$ of an abelian group $G = (G,+)$ with $|A + A| \leq (2-\eps) |A|$ can be covered by at most $\frac{2}{\eps}-1$ translates of a finite group $H$ of cardinality at most $(2-\eps)|A|$. ... More

A remark on partial sums involving the Mobius functionAug 29 2009Oct 05 2009Let $<\P > \subset \N$ be a multiplicative subsemigroup of the natural numbers $\N = \{1,2,3,...\}$ generated by an arbitrary set $\P$ of primes (finite or infinite). We given an elementary proof that the partial sums $\sum_{n \in < \P >: n \leq x} \frac{\mu(n)}{n}$ ... More

Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutionsJun 16 2009Aug 06 2009In 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 ... More

Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy classAug 04 2008Aug 06 2009Using the harmonic map heat flow and the function spaces of Tataru and the author, we establish a large data local well-posedness result in the energy class for wave maps from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$. This ... More

A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equationOct 08 2007May 21 2009The global regularity problem for the periodic Navier-Stokes system asks whether to every smooth divergence-free initial datum $u_0: (\R/\Z)^3 \to \R^3$ there exists a global smooth solution u. In this note we observe (using a simple compactness argument) ... More

What is good mathematics?Feb 13 2007Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemer\'edi's theorem is presented.

The ergodic and combinatorial approaches to Szemerédi's theoremApr 20 2006A famous theorem of Szemer\'edi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the original combinatorial ... More

Global regularity of wave maps II. Small energy in two dimensionsNov 22 2000Jan 10 2001We show that wave maps from Minkowski space $\R^{1+n}$ to a sphere $S^{m-1}$ are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$, in all dimensions $n \geq 2$. This generalizes the results ... More

Localisation and compactness properties of the Navier-Stokes global regularity problemAug 04 2011May 31 2012In this paper we establish a number of implications between various qualitative and quantitative versions of the global regularity problem for the Navier-Stokes equations, in the periodic, smooth finite energy, smooth $H^1$, Schwartz, or mild $H^1$ categories, ... More

Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equationJun 17 2009Jun 27 2009Let $d \geq 3$. We consider the global Cauchy problem for the generalised Navier-Stokes system \partial_t u + (u \cdot \nabla) u &= - D^2 u - \nabla p \nabla \cdot u &= 0 u(0,x) &= u_0(x) for $u: \R^+ \times \R^d \to \R^d$ and $p: \R^+ \times \R^d \to ... More

Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearityMar 07 2016Apr 13 2016We consider the global regularity problem for nonlinear wave systems $$ \Box u = f(u) $$ on Minkowski spacetime ${\bf R}^{1+d}$ with d'Alambertian $\Box := -\partial_t^2 + \sum_{i=1}^d \partial_{x_i}^2$, where the field $u \colon {\bf R}^{1+d} \to {\bf ... More

The logarithmically averaged Chowla and Elliott conjectures for two-point correlationsSep 17 2015Jul 29 2016Let $\lambda$ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts that $$ \sum_{n \leq x} \lambda(a_1 n + b_1) \lambda(a_2 n+b_2) = o(x) $$ as $x \to \infty$, for any fixed natural numbers $a_1,a_2,b_1,b_2$ ... More

Global regularity of wave maps VII. Control of delocalised or dispersed solutionsAug 06 2009Aug 08 2009This is the final paper in the series \cite{tao:heatwave}, \cite{tao:heatwave2}, \cite{tao:heatwave3}, \cite{tao:heatwave4} that establishes global regularity for two-dimensional wave maps into hyperbolic targets. In this paper we establish the remaining ... More

An inverse theorem for the bilinear $L^2$ Strichartz estimate for the wave equationApr 19 2009A standard bilinear $L^2$ Strichartz estimate for the wave equation, which underlies the theory of $X^{s,b}$ spaces of Bourgain and Klainerman-Machedon, asserts (roughly speaking) that if two finite-energy solutions to the wave equation are supported ... More

The high exponent limit $p \to \infty$ for the one-dimensional nonlinear wave equationJan 22 2009Feb 20 2009We investigate the behaviour of solutions $\phi = \phi^{(p)}$ to the one-dimensional nonlinear wave equation $-\phi_{tt} + \phi_{xx} = -|\phi|^{p-1} \phi$ with initial data $\phi(0,x) = \phi_0(x)$, $\phi_t(0,x) = \phi_1(x)$, in the high exponent limit ... More

Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spacesMay 30 2008Aug 08 2009We show that wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$ are globally smooth in time if the initial data is smooth, conditionally on some reasonable claims concerning the local theory of such wave maps, ... More

Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric dataJun 07 2006We establish global regularity for the logarithmically energy-supercritical wave equation $\Box u = u^5 \log(2+u^2)$ in three spatial dimensions for spherically symmetric initial data, by modifying an argument of Ginibre, Soffer and Velo \cite{gsv} for ... More

Arithmetic progressions and the primes - El Escorial lecturesNov 11 2004We 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.

A quantitative ergodic theory proof of Szemerédi's theoremMay 13 2004May 23 2004A 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 ... More

Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equationNov 29 1998Oct 20 2004The endpoint Strichartz estimates for the Schr\"odinger equation are known to be false in two dimensions. However, if one averages the solution in $L^2$ in the angular variable, we show that the homogeneous endpoint and the retarded half-endpoint estimates ... More

A new bound for finite field Besicovitch sets in four dimensionsApr 19 2002Sep 10 2002Let $F$ be a finite field with characteristic greater than two. Define a \emph{Besicovitch set} in $F^4$ to be a set $P \subseteq F^4$ containing a line in every direction. The \emph{Kakeya conjecture} asserts that $|P| \approx |F|^4$. A result of Wolff ... More

Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theoryOct 24 2013Mar 29 2014Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned ... More

A sharp bilinear restriction estimate for paraboloidsOct 07 2002Dec 13 2002Recently Wolff obtained a sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of ``elliptic surfaces'' such as paraboloids and spheres. Except for an ... More

On the universality of potential well dynamicsJul 08 2017Oct 30 2017Given a smooth potential function $V : \mathbf{R}^m \to \mathbf{R}$, one can consider the ODE $\partial_t^2 u = -(\nabla V)(u)$ describing the trajectory of a particle $t \mapsto u(t)$ in the potential well $V$. We consider the question of whether the ... More

Fuglede's conjecture is false in 5 and higher dimensionsJun 08 2003Jun 09 2003We give an example of a set $\Omega \subset \R^5$ which is a finite union of unit cubes, such that $L^2(\Omega)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\Omega|^{1/2}} e^{2\pi i \xi_j \cdot x}: \xi_j \in \Lambda \}$ for some discrete ... More

Finite time blowup for Lagrangian modifications of the three-dimensional Euler equationJun 27 2016In the language of differential geometry, the incompressible inviscid Euler equations can be written in vorticity-vector potential form as \begin{align*} \partial_t \omega + {\mathcal L}_u \omega &= 0\\ u &= \delta \tilde \eta^{-1} \Delta^{-1} \omega ... More

Equivalence of the logarithmically averaged Chowla and Sarnak conjecturesMay 16 2016Let $\lambda$ 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 ... More

Failure of the $L^1$ pointwise and maximal ergodic theorems for the free groupMay 18 2015Let $F_2$ denote the free group on two generators $a,b$. For any measure-preserving system $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$ on a finite measure space $X = (X,{\mathcal X},\mu)$, any $f \in L^1(X)$, and any $n \geq 1$, define the averaging operators ... More

The Elliott-Halberstam conjecture implies the Vinogradov least quadratic nonresidue conjectureOct 26 2014Jan 13 2015For each prime $p$, let $n(p)$ denote the least quadratic nonresidue modulo $p$. Vinogradov conjectured that $n(p) = O(p^\eps)$ for every fixed $\eps>0$. This conjecture follows from the generalised Riemann hypothesis, and is known to hold for almost ... More

Finite time blowup for an averaged three-dimensional Navier-Stokes equationFeb 03 2014Apr 01 2015The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ can be expressed in the form $\partial_t u = \Delta u + B(u,u)$, where $B$ is a certain bilinear operator on divergence-free vector fields $u$ obeying the cancellation property $\langle ... More

Sumset and inverse sumset theorems for Shannon entropyJun 24 2009Oct 26 2009Let $G = (G,+)$ be an additive group. The sumset theory of Pl\"unnecke and Ruzsa gives several relations between the size of sumsets $A+B$ of finite sets $A, B$, and related objects such as iterated sumsets $kA$ and difference sets $A-B$, while the inverse ... More

Why are solitons stable?Feb 18 2008Feb 20 2008The 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} ... More

Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspectiveOct 29 2006We 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.

Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy classJun 22 2008Aug 06 2009Using the harmonic map heat flow, we construct an energy class for wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$, and then show (conditionally on a large data well-posedness claim for such wave maps) that ... More

Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensionsJan 09 2006Jun 20 2008Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah, Bahouri-G\'erard and Nakanishi have established global wellposedness, regularity, and scattering in the energy class for the energy-critical nonlinear wave equation $\Box u = u^5$ ... More

The weak-type $(1,1)$ of Fourier integral operators of order $-(n-1)/2$Jan 23 2002Jun 11 2002Let $T$ be a Fourier integral operator on $\R^n$ of order $-(n-1)/2$. It was shown by Seeger, Sogge, and Stein that $T$ mapped the Hardy space $H^1$ to $L^1$. In this note we show that $T$ is also of weak-type $(1,1)$. The main ideas are a decomposition ... More