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

     

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.

     

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.

     

Exponential sums on , II

We prove a vanishing theorem for the -adic cohomology of exponential sums on . In particular, we obtain new classes of exponential sums on that have a single nonvanishing -adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.

     

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.

     

On -adic Hermitian Eisenstein series

In this paper we generalize the notion of -adic modular form to the Hermitian modular case and prove a formula that shows a coincidence between certain -adic Hermitian Eisenstein series and the genus theta series associated with Hermitian matrix with determinant .

     

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.

     

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.

     

Analytic -adic cell decomposition and integrals

We prove a conjecture of Denef on parameterized -adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.

     

A van der Corput lemma for the -adic numbers

We prove a version of van der Corput's lemma for polynomials over the -adic numbers.

     

Generalized spherical functions on reductive -adic groups

Let be the group of rational points of a connected reductive -adic group and let be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of . In this paper we show that spaces of generalized spherical functions are finite dimensional. We compute dimensions of spaces of generalized spherical functions on a Zariski open dense set of infinitesimal characters. As a consequence, we get that on that Zariski open dense set of infinitesimal characters, the dimension of the space of generalized spherical functions is constant on each connected component of infinitesimal characters. We also obtain the formula for the generalized spherical functions by integrals of Eisenstein type. On the Zariski open dense set of infinitesimal characters that we have mentioned above, these integrals then give the formula for all the generalized spherical functions. At the end, let as mention that among others we prove that there exists a Zariski open dense subset of infinitesimal characters such that the category of smooth representations of with fixed infinitesimal character belonging to this subset is semi-simple.

     

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.

     

Spectral pictures of and

The spectral pictures of products and of Banach space operators are compared; in particular when one of them is `of index zero'.

     

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

     

-theory of -pseudo-differential algebras

We are concerned with the so-called -pseudo-differential calculus. We describe the spectrum of the unital and commutative -algebra given by the norm closure of the space of -order pseudo-differential operators modulo compact operators; other related algebras are also considered. Finally, their -theory is computed.

     

-adic formal series and primitive polynomials over finite fields

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over -adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , the Hansen-Mullen conjecture that there exists a primitive polynomial over of degree with the -th ( coefficient fixed in advance except when if is odd and when if is even.

     

Sharply -transitive groups with point stabilizer of exponent or

Using the fact that all groups of exponent are nilpotent, we show that every sharply -transitive permutation group whose point stabilizer has exponent or is finite.