A new construction of -manifolds

This paper provides a topological method to construct all simply-connected, spin, smooth -manifolds with torsion-free homology using simply-connected, smooth -manifolds as building blocks. We explicitly determine the invariants that classify these -manifolds from the intersection form and specific homology classes of the -manifold building blocks.

     

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.

     

Symplectic tori in homotopy 's

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic -manifold , obtained by knot surgery on the rational elliptic surface with the left-handed trefoil knot . has the simplest homotopy type among simply-connected symplectic -manifolds known to exhibit such a property.

     

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 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.

     

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 geography of symplectic -manifolds with an arbitrary fundamental group

In this article, for each finitely presented group , we construct a family of minimal symplectic -manifolds with which cover most lattice points with large in the region . Furthermore, we show that all these -manifolds admit infinitely many distinct smooth structures.

     

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.

     

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.

     

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 .

     

Estimates for the extinction time for the Ricci flow on certain -manifolds and a question of Perelman

We show that the Ricci flow becomes extinct in finite time on any Riemannian -manifold without aspherical summands.

     

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.

     

Pseudofree -actions on surfaces

In this paper, we give a weak classification of locally linear pseudofree actions of the cyclic group of order on a surface, and prove the existence of such an action which cannot be realized as a smooth action on the standard smooth surface.

     

Correction to ``Formal degrees and adjoint -factors'

This is a correction to the paper Formal degrees and adjoint -factors, J. Amer. Math. Soc. 21 (2008), 283-304.

     

An ultrafilter with property

Let be a -supercompact cardinal. We show that carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo's theorem that carries a normal ultrafilter with the partition property.

     

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.

     

Distance between toroidal surgeries on hyperbolic knots in the -sphere

For a hyperbolic knot in the -sphere, at most finitely many Dehn surgeries yield non-hyperbolic -manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed -manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal slope, is known to be integral or half-integral. We show that the distance between two integral toroidal slopes for a hyperbolic knot, except the figure-eight knot, is at most four.

     

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.