An uncountable family of nonorbit equivalent actions of

For each , we construct an uncountable family of free ergodic measure preserving actions of the free group on the standard probability space such that any two are nonorbit equivalent (in fact, not even stably orbit equivalent). These actions are all ``rigid' (in the sense of Popa), with the IIfactors mutually nonisomorphic (even nonstably isomorphic) and in the class

     

On the size of -fold sum and product sets of integers

In this paper, we show that for all 1$"> there is a positive integer such that if is an arbitrary finite set of integers, 2$">, then either N^{b}$"> or N^{b}$">. Here (resp. ) denotes the -fold sum (resp. product) of . This fact is deduced from the following harmonic analysis result obtained in the paper. For all 2$"> and 0$">, there is a 0$"> such that if satisfies , then the -constant of (in the sense of W. Rudin) is at most .

     

Independence of of monodromy groups

Let be a smooth curve over a finite field of characteristic , let be a number field, and let be an -compatible system of lisse sheaves on the curve . For each place of not lying over , the -component of the system is a lisse -sheaf on , whose associated arithmetic monodromy group is an algebraic group over the local field . We use Serre's theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show that when the -compatible system is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is ``independent of '. More precisely, after replacing by a finite extension, there exists a connected split reductive algebraic group over the number field such that for every place of not lying over , the identity component of the arithmetic monodromy group of is isomorphic to the group with coefficients extended to the local field .

     

The moduli space of quadratic rational maps

Let be the space of quadratic rational maps , modulo the action by conjugation of the group of Möbius transformations. In this paper a compactification of is defined, as a modification of Milnor's , by choosing representatives of a conjugacy class such that the measure of maximal entropy of has conformal barycenter at the origin in and taking the closure in the space of probability measures. It is shown that is the smallest compactification of such that all iterate maps extend continuously to , where is the natural compactification of coming from geometric invariant theory.

     

Zeta function of representations of compact -adic analytic groups

Let be an FAb compact -adic analytic group and suppose that 2$"> or and is uniform. We prove that there are natural numbers and functions rational in such that

     

boundedness of discrete singular Radon transforms

We prove that if is a Calderón-Zygmund kernel and is a polynomial of degree with real coefficients, then the discrete singular Radon transform operator

extends to a bounded operator on , . This gives a positive answer to an earlier conjecture of E. M. Stein and S. Wainger.

     

The threshold for random -SAT is

Let be a random -CNF formula formed by selecting uniformly and independently out of all possible -clauses on variables. It is well known that if , then is unsatisfiable with probability that tends to 1 as . We prove that if , where , then is satisfiable with probability that tends to 1 as .

Our technique, in fact, yields an explicit lower bound for the random -SAT threshold for every . For our bounds improve all previously known such bounds.

     

Duality of Hardy and BMO spaces associated with operators with heat kernel bounds

Let be the infinitesimal generator of an analytic semigroup on with suitable upper bounds on its heat kernels. Auscher, Duong, and McIntosh defined a Hardy space by means of an area integral function associated with the operator . By using a variant of the maximal function associated with the semigroup , a space of functions of BMO type was defined by Duong and Yan and it generalizes the classical BMO space. In this paper, we show that if has a bounded holomorphic functional calculus on , then the dual space of is where is the adjoint operator of . We then obtain a characterization of the space in terms of the Carleson measure. We also discuss the dimensions of the kernel spaces of BMO when is a second-order elliptic operator of divergence form and when is a Schrödinger operator, and study the inclusion between the classical BMO space and spaces associated with operators.

     

Curvature and injectivity radius estimates for Einstein 4-manifolds

Let denote an Einstein -manifold with Einstein constant, , normalized to satisfy . For , a metric ball, we prove a uniform estimate for the pointwise norm of the curvature tensor on , under the assumption that the -norm of the curvature on is less than a small positive constant, which is independent of , and which in particular, does not depend on a lower bound on the volume of . In case , we prove a lower injectivity radius bound analogous to that which occurs in the theorem of Margulis, for compact manifolds with negative sectional curvature, . These estimates provide key tools in the study of singularity formation for -dimensional Einstein metrics. As one application among others, we give a natural compactification of the moduli space of Einstein metrics with negative Einstein constant on a given .

     

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point the local real-analytic CR automorphisms of fixing can be parametrized real-analytically by their jets at . As a direct application, we derive a Lie group structure for the topological group . Furthermore, we also show that the order of the jet space in which the group embeds can be chosen to depend upper-semicontinuously on . As a first consequence, it follows that given any compact real-analytic hypersurface in , there exists an integer depending only on such that for every point germs at of CR diffeomorphisms mapping into another real-analytic hypersurface in are uniquely determined by their -jet at that point. Another consequence is the following boundary version of H. Cartan's uniqueness theorem: given any bounded domain with smooth real-analytic boundary, there exists an integer depending only on such that if is a proper holomorphic mapping extending smoothly up to near some point with the same -jet at with that of the identity mapping, then necessarily .

Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.

     

Universal characteristic factors and Furstenberg averages

Let be an ergodic probability measure-preserving system. For a natural number we consider the averages

where , and are integers. A factor of is characteristic for averaging schemes of length (or -characteristic) if for any nonzero distinct integers , the limiting behavior of the averages in (*) is unaltered if we first project the functions onto the factor. A factor of is a -universal characteristic factor (-u.c.f.) if it is a -characteristic factor, and a factor of any -characteristic factor. We show that there exists a unique -u.c.f., and it has the structure of a -step nilsystem, more specifically an inverse limit of -step nilflows. Using this we show that the averages in (*) converge in . This provides an alternative proof to the one given by Host and Kra.

     

Sieving by large integers and covering systems of congruences

An old question of Erdos asks if there exists, for each number , a finite set of integers greater than and residue classes for whose union is . We prove that if is bounded for such a covering of the integers, then the least member of is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number , the complement in of any union of residue classes , for distinct , has density at least for sufficiently large. Here is a positive number depending only on . Either of these new results implies another conjecture of Erdos and Graham, that if is a finite set of moduli greater than , with a choice for residue classes for which covers , then the largest member of cannot be . We further obtain stronger forms of these results and establish other information, including an improvement of a related theorem of Haight.

     

Conformal restriction: The chordal case

We characterize and describe all random subsets of a given simply connected planar domain (the upper half-plane , say) which satisfy the ``conformal restriction' property, i.e., connects two fixed boundary points ( and , say) and the law of conditioned to remain in a simply connected open subset of is identical to that of , where is a conformal map from onto with and . The construction of this family relies on the stochastic Loewner evolution processes with parameter and on their distortion under conformal maps. We show in particular that SLE is the only random simple curve satisfying conformal restriction and we relate it to the outer boundaries of planar Brownian motion and SLE.     

Long arithmetic progressions in sumsets: Thresholds and bounds

For a set of integers, the sumset consists of those numbers which can be represented as a sum of elements of :

Closely related and equally interesting notion is that of , which is the collection of numbers which can be represented as a sum of different elements of :

The goal of this paper is to investigate the structure of and , where is a subset of . As application, we solve two conjectures by Erdös and Folkman, posed in 1960s.

     

Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi

In 1978 De Giorgi formulated the following conjecture. Let be a solution of in all of such that and 0$"> in . Is it true that all level sets of are hyperplanes, at least if ? Equivalently, does depend only on one variable? When , this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for . The question, however, remains open for . The results for and 3 apply also to the equation for a large class of nonlinearities .

     

Zero distribution of Müntz extremal polynomials in

Let be a sequence of distinct positive numbers. Let and let denote the extremal Müntz polynomial in with exponents . We investigate the zero distribution of . In particular, we show that if

then the normalized zero counting measure of converges weakly as to

while if or , the limiting measure is a Dirac delta at or , respectively.

     

Quasisymmetric groups

One of the first problems in the theory of quasisymmetric and convergence groups was to investigate whether every quasisymmetric group that acts on the sphere , , is a quasisymmetric conjugate of a Möbius group that acts on . This was shown to be true for by Sullivan and Tukia, and it was shown to be wrong for by Tukia. It also follows from the work of Martin and of Freedman and Skora. In this paper we settle the case of by showing that any -quasisymmetric group is -quasisymmetrically conjugated to a Möbius group. The constant is a function .

     

Intermediate subfactors with no extra structure

If are type II factors with and we show that restrictions on the standard invariants of the elementary inclusions , , and imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto and do not commute, then is or . In the former case is the fixed point algebra for an outer action of on and the angle is , and in the latter case the angle is and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.

     

The honeycomb model of tensor products II: Puzzles determine facets of the Littlewood-Richardson cone

The set of possible spectra of zero-sum triples of Hermitian matrices forms a polyhedral cone, whose facets have been already studied by Knutson and Tao, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. We give a complete determination of these facets; there is one for each triple of Grassmannian Schubert cycles intersecting in a unique point. In particular, the list of inequalities determined by Belkale to be sufficient is in fact minimal.

We introduce puzzles, which are new combinatorial gadgets to compute Grassmannian Schubert calculus, and seem to have much interest in their own right. As the proofs herein indicate, the Hermitian sum problem is very naturally studied using puzzles directly, and their connection to Schubert calculus is quite incidental to our approach. In particular, we get new, puzzle-theoretic, proofs of the results of Horn, Klyachko, Helmke and Rosenthal, Totaro, and Belkale.

Along the way we give a characterization of ``rigid' puzzles, which we use to prove a conjecture of W. Fulton: ``if for a triple of dominant weights of the irreducible representation appears exactly once in , then for all , appears exactly once in .'