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1.
Let be a closed, oriented -manifold. A folklore conjecture states that admits a symplectic structure if and only if admits a fibration over the circle. We will prove this conjecture in the case when is irreducible and its fundamental group satisfies appropriate subgroup separability conditions. This statement includes -manifolds with vanishing Thurston norm, graph manifolds and -manifolds with surface subgroup separability (a condition satisfied conjecturally by all hyperbolic -manifolds). Our result covers, in particular, the case of 0-framed surgeries along knots of genus one. The statement follows from the proof that twisted Alexander polynomials decide fiberability for all the -manifolds listed above. As a corollary, it follows that twisted Alexander polynomials decide if a knot of genus one is fibered.
2.
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 .
3.
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.
4.
Niels Lauritzen Ulf Raben-Pedersen Jesper Funch Thomsen 《Journal of the American Mathematical Society》2006,19(2):345-355
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.
5.
CheeWhye Chin 《Journal of the American Mathematical Society》2004,17(3):723-747
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 .
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7.
Kaoru Hiraga Atsushi Ichino Tamotsu Ikeda 《Journal of the American Mathematical Society》2008,21(1):283-304
We give a conjectural formula for the formal degree of a discrete series representation in terms of the adjoint -factor. Our conjecture is supported by various examples and is compatible with the Weyl dimension formula. Using twisted endoscopy, we also verify the conjecture for a stable discrete series representation of over a non-archimedean local field of characteristic zero.
8.
Nicolas Burq Gilles Lebeau Fabrice Planchon 《Journal of the American Mathematical Society》2008,21(3):831-845
We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on for any smooth (compact) domain . The main ingredient in the proof is an spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.
9.
Amit Kulshrestha R. Parimala 《Transactions of the American Mathematical Society》2008,360(3):1193-1221
Let be a field of characteristic not whose virtual cohomological dimension is at most . Let be a semisimple group of adjoint type defined over . Let denote the normal subgroup of consisting of elements -equivalent to identity. We show that if is of classical type not containing a factor of type , . If is a simple classical adjoint group of type , we show that if and its multi-quadratic extensions satisfy strong approximation property, then . This leads to a new proof of the -triviality of -rational points of adjoint classical groups defined over number fields.
10.
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.
11.
Alexandru D. Ionescu Stephen Wainger 《Journal of the American Mathematical Society》2006,19(2):357-383
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.
12.
J. Marshall Ash Stefan Catoiu 《Transactions of the American Mathematical Society》2008,360(2):959-987
For , a one-parameter family of symmetric quantum derivatives is defined for each order of differentiation as are two families of Riemann symmetric quantum derivatives. For , symmetrization holds, that is, whenever the th Peano derivative exists at a point, all of these derivatives of order also exist at that point. The main result, desymmetrization, is that conversely, for , each symmetric quantum derivative is a.e. equivalent to the Peano derivative of the same order. For and , each th symmetric quantum derivative coincides with both corresponding th Riemann symmetric quantum derivatives, so, in particular, for and , both th Riemann symmetric quantum derivatives are a.e. equivalent to the Peano derivative.
13.
Thierry Giordano Hiroki Matui Ian F. Putnam Christian F. Skau 《Journal of the American Mathematical Society》2008,21(3):863-892
We show that every minimal, free action of the group on the Cantor set is orbit equivalent to an AF-relation. As a consequence, this extends the classification of minimal systems on the Cantor set up to orbit equivalence to include AF-relations, -actions and -actions.
14.
A. Jaikin-Zapirain 《Journal of the American Mathematical Society》2006,19(1):91-118
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
15.
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
16.
Lisa Orloff Clark Astrid an Huef 《Proceedings of the American Mathematical Society》2008,136(2):623-634
Suppose is a second countable, locally compact, Hausdorff, principal groupoid with a fixed left Haar system. We define a notion of integrability for groupoids and show is integrable if and only if the groupoid -algebra has bounded trace.
17.
William B. Johnson Bernard Maurey Gideon Schechtman 《Journal of the American Mathematical Society》2007,20(1):25-36
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.
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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.
20.
Christopher B. Croke 《Proceedings of the American Mathematical Society》2008,136(2):715-717
We consider Riemannian metrics on the -sphere for such that the distance between any pair of antipodal points is bounded below by 1. We show that the volume can be arbitrarily small. This is in contrast to the -dimensional case where Berger has shown that .