Results 1  10
of
13
A sharp bilinear restriction estimate for paraboloids, Geom
 Func. Anal
"... Abstract. Recently Wolff [28] 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. Except for an endpoint, this answers a conjecture of Machedon ..."
Abstract

Cited by 40 (7 self)
 Add to MetaCart
Abstract. Recently Wolff [28] 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. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.
Restriction and Kakeya phenomena for finite fields
 DUKE MATH. J
, 2004
"... The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. I ..."
Abstract

Cited by 25 (0 self)
 Add to MetaCart
The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. In many cases the Euclidean arguments carry over easily to the finite setting (and are, in fact, somewhat cleaner), but there
On the size of Kakeya sets in finite fields
 J. AMS
, 2008
"... Abstract. A Kakeya set is a subset of � n, where � is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least Cn · q n, where Cn depends only on n. This answers a question of Wolff [Wol99]. 1. ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
Abstract. A Kakeya set is a subset of � n, where � is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least Cn · q n, where Cn depends only on n. This answers a question of Wolff [Wol99]. 1.
From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and pde
 Notices Amer. Math. Soc
, 2000
"... In 1917 S. Kakeya posed the Kakeya needle problem: What is the smallest area required to rotate a unit line segment (a “needle”) by 180 degrees in the plane? Rotating around the midpoint requires ..."
Abstract

Cited by 22 (5 self)
 Add to MetaCart
In 1917 S. Kakeya posed the Kakeya needle problem: What is the smallest area required to rotate a unit line segment (a “needle”) by 180 degrees in the plane? Rotating around the midpoint requires
Recent progress on the Kakeya conjecture
 PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS (EL ESCORIAL
, 2000
"... We survey recent developments on the Kakeya problem. ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
We survey recent developments on the Kakeya problem.
SOME RECENT PROGRESS ON THE RESTRICTION CONJECTURE
, 2003
"... Abstract. We survey recent developments on the Restriction conjecture. ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Abstract. We survey recent developments on the Restriction conjecture.
Mixed norm estimates for a restricted Xray transform in R^4 and R^5
 INTERNAT. MATH. RESEARCH NOTICES
"... ..."
BOOK REVIEWS BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... The literature of mathematics comprises millions of works, published ones as well as ones deposited in electronic archives. The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thous ..."
Abstract
 Add to MetaCart
The literature of mathematics comprises millions of works, published ones as well as ones deposited in electronic archives. The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year [28]. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. In addition, many works also formulate unsolved problems, often in the form of precise conjectures. How essential is it for the development of mathematical science to draw the readers’ attention unceasingly to open problems? Maybe it would suffice to publish only new results? The firstrank mathematicians of the present time give a definitive answer to this question. In his preface to the first Russian edition [20] of the book under review, 1 V. I. Arnold reminisced: “I. G. Petrovskiĭ, who was one of my teachers in Mathematics, taught me that the most important thing that a student should learn from his supervisor is that some question is still open. Further choice of the problem from