Double node neighborhoods and families of simply connected -manifolds with

We introduce a new technique that is used to show that the complex projective plane blown up at 6, 7, or 8 points has infinitely many distinct smooth structures. None of these smooth structures admits smoothly embedded spheres with self-intersection , i.e., they are minimal. In addition, none of these smooth structures admits an underlying symplectic structure. Shortly after the appearance of a preliminary version of this article, Park, Stipsicz, and Szabo used the techniques described herein to show that the complex projective plane blown up at 5 points has infinitely many distinct smooth structures. In the final section of this paper we give a construction of such a family of examples.

     

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

     

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 .

     

Global -regularity of Schubert varieties with applications to -modules

We prove that Schubert varieties are globally -regular in the sense of Karen Smith. We apply this result to the category of equivariant and holonomic -modules on flag varieties in positive characteristic. Here recent results of Blickle are shown to imply that the simple -modules coincide with local cohomology sheaves with support in Schubert varieties. Using a local Grothendieck-Cousin complex, we prove that the decomposition of local cohomology sheaves with support in Schubert cells is multiplicity free.

     

Weakly null sequences in

We construct a weakly null normalized sequence in so that for each , the Haar basis is -equivalent to a block basis of every subsequence of . In particular, the sequence has no unconditionally basic subsequence. This answers a question raised by Bernard Maurey and H. P. Rosenthal in 1977. A similar example is given in an appropriate class of rearrangement invariant function spaces.

     

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.

     

Cyclicity of CM elliptic curves modulo

Let be an elliptic curve defined over and with complex multiplication. For a prime of good reduction, let be the reduction of modulo We find the density of the primes for which is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula and also to provide explicit error terms in the formula.

     

Enumerative tropical algebraic geometry in

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634.

The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space and holomorphic curves with certain piecewise-linear graphs there.

     

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.

     

Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric -functions

Any product of real powers of Jacobian elliptic functions can be written in the form . If all three 's are even integers, the indefinite integral of this product with respect to is a constant times a multivariate hypergeometric function with half-odd-integral 's and , showing it to be an incomplete elliptic integral of the second kind unless all three 's are 0. Permutations of c, d, and n in the integrand produce the same permutations of the variables }, allowing as many as six integrals to take a unified form. Thirty -functions of the type specified, incorporating 136 integrals, are reduced to a new choice of standard elliptic integrals obtained by permuting , , and in , which is symmetric in its first two variables and has an efficient algorithm for numerical computation.

     

On the Farrell-Jones conjecture for higher algebraic -theory

We prove the Farrell-Jones Conjecture for the algebraic -theory of a group ring in the case where the group is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature. The coefficient ring is an arbitrary associative ring with unit and the result applies to all dimensions.

     

The de Rham-Witt complex and -adic vanishing cycles

We determine the structure of the reduction modulo of the absolute de Rham-Witt complex of a smooth scheme over a discrete valuation ring of mixed characteristic with log-poles along the special fiber and show that the sub-sheaf fixed by the Frobenius map is isomorphic to the sheaf of -adic vanishing cycles. We use this result together with the main results of op. cit. to evaluate the algebraic -theory with finite coefficients of the quotient field of the henselian local ring at a generic point of the special fiber. The result affirms the Lichtenbaum-Quillen conjecture for this field.

     

Higher Weierstrass points on

We study the arithmetic properties of higher Weierstrass points on modular curves for primes . In particular, for , we obtain a relationship between the reductions modulo of the collection of -Weierstrass points on and the supersingular locus in characteristic .

     

-kernels of Lie groups

We study a filtration on the group of homotopy classes of self maps of a compact Lie group associated with homotopy groups. We determine these filtrations of and completely. We introduce two natural invariants and defined by the filtration, where is a prime number, and compute the invariants for simple Lie groups in the cases where Lie groups are -regular or quasi -regular. We apply our results to the groups of self homotopy equivalences.

     

On the -torsion of elliptic curves and elliptic surfaces in characteristic

We study the extension generated by the -coordinates of the -torsion points of an elliptic curve over a function field of characteristic . If is a non-isotrivial elliptic surface in characteristic with a -torsion section, then for 11$"> our results imply restrictions on the genus, the gonality, and the -rank of the base curve , whereas for such a surface can be constructed over any base curve . We also describe explicitly all occurring in the cases where the surface is rational or or the base curve is rational, elliptic or hyperelliptic.

     

Closed sets which are not -critical

In this paper we first observe that the complement of a countable closed subset of an -dimensional manifold has large -homology group. In the last section we use this information to prove that, under some topological conditions on the given manifold, certain families of fibers, in the total space of a fibration over , are not critical sets for some special real or -valued functions.

     

The border rank of the multiplication of matrices is seven

We prove that the bilinear map corresponding to matrix multiplication of matrices is not the limit of a sequence of bilinear maps that can be executed by performing six multiplications. This solves a longstanding problem dating back to Strassen's discovery in 1968 that the map could be executed by performing seven multiplications.

     

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