A tensor norm preserving unconditionality for -spaces

We show that, for each , there is an -tensor norm (in the sense of Grothendieck) with the surprising property that the -tensor product has local unconditional structure for each choice of arbitrary -spaces . In fact, is the tensor norm associated to the ideal of multiple -summing -linear forms on Banach spaces.

     

The center of the category of -modules

Let be a semi-simple connected Lie group. Let be a maximal compact subgroup of and the complexified Lie algebra of . In this paper we describe the center of the category of -modules.

     

On embedding all -manifolds into a single -manifold

For each composite number , there does not exist a single connected closed -manifold such that any smooth, simply-connected, closed -manifold can be topologically flatly embedded into it. There is a single connected closed -manifold such that any simply-connected, -manifold can be topologically flatly embedded into if is either closed and indefinite, or compact and with non-empty boundary.

     

Torsion in coinvariants of certain Cantor minimal -systems

Let be a finite abelian group. We will consider a skew product extension of a product of two Cantor minimal -systems associated with a -valued cocycle. When is non-cyclic and the cocycle is non-degenerate, it will be shown that the skew product system has torsion in its coinvariants.

     

Bimodules and -rationality of vertex operator algebras

This paper studies the twisted representations of vertex operator algebras. Let be a vertex operator algebra and an automorphism of of finite order For any , an - -bimodule is constructed. The collection of these bimodules determines any admissible -twisted -module completely. A Verma type admissible -twisted -module is constructed naturally from any -module. Furthermore, it is shown with the help of bimodule theory that a simple vertex operator algebra is -rational if and only if its twisted associative algebra is semisimple and each irreducible admissible -twisted -module is ordinary.

     

-equivalence in adjoint classical groups over fields of virtual cohomological dimension

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.

     

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.

     

Eigenvalues of Schrödinger operators with potential asymptotically homogeneous of degree

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function , the number of bound states of the operator in below . Here is a bounded potential behaving asymptotically like where is a function on the sphere. It is well known that the eigenvalues of such an operator are all nonpositive, and accumulate only at 0. If the operator on the sphere has negative eigenvalues less than , we prove that may be estimated as

Thus, in particular, if there are no such negative eigenvalues, then has a finite discrete spectrum. Moreover, under some additional assumptions including the fact that and that there is exactly one eigenvalue less than , with all others , we show that the negative spectrum is asymptotic to a geometric progression with ratio .

     

On initial segment complexity and degrees of randomness

One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define to mean that . The equivalence classes under this relation are the -degrees. We prove that if is -random, then and have no upper bound in the -degrees (hence, no join). We also prove that -randomness is closed upward in the -degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the -degrees. Unlike the -degrees, many basic properties of the -degrees are easy to prove. We show that implies , so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for , the analogue of for plain Kolmogorov complexity.

Two other interesting results are included. First, we prove that for any , a -random real computable from a --random real is automatically --random. Second, we give a plain Kolmogorov complexity characterization of -randomness. This characterization is related to our proof that implies .

     

Prime specialization in genus 0

For a prime polynomial , a classical conjecture predicts how often has prime values. For a finite field and a prime polynomial , the natural analogue of this conjecture (a prediction for how often takes prime values on ) is not generally true when is a polynomial in ( the characteristic of ). The explanation rests on a new global obstruction which can be measured by an appropriate average of the nonzero Möbius values as varies. We prove the surprising fact that this ``Möbius average,' which can be defined without reference to any conjectures, has a periodic behavior governed by the geometry of the plane curve .

The periodic Möbius average behavior implies in specific examples that a polynomial in does not take prime values as often as analogies with suggest, and it leads to a modified conjecture for how often prime values occur.

     

Profinite and pro- completions of Poincaré duality groups of dimension 3

We establish some sufficient conditions for the profinite and pro- completions of an abstract group of type (resp. of finite cohomological dimension, of finite Euler characteristic) to be of type over the field for a fixed natural prime (resp. of finite cohomological -dimension, of finite Euler -characteristic).

We apply our methods for orientable Poincaré duality groups of dimension 3 and show that the pro- completion of is a pro- Poincaré duality group of dimension 3 if and only if every subgroup of finite index in has deficiency 0 and is infinite. Furthermore if is infinite but not a Poincaré duality pro- group, then either there is a subgroup of finite index in of arbitrary large deficiency or is virtually . Finally we show that if every normal subgroup of finite index in has finite abelianization and the profinite completion of has an infinite Sylow -subgroup, then is a profinite Poincaré duality group of dimension 3 at the prime .

     

Square -lattice paths and

The combinatorial -Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The -Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the 'th -Catalan number is the Hilbert series for the module of diagonal harmonic alternants in variables; it is also the coefficient of in the Schur expansion of . Using -analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of and the Hilbert series of the diagonal harmonics modules.

This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several -analogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the -Catalan polynomials. We also conjecture an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of , the ``Hilbert series' , and the sign character .

     

Bounded -calculus for pseudodifferential operators and applications to the Dirichlet-Neumann operator

Operators of the form with a pseudodifferential symbol belonging to the Hörmander class , , , and certain perturbations are shown to possess a bounded -calculus in Besov-Triebel-Lizorkin and certain subspaces of Hölder spaces, provided is suitably elliptic. Applications concern pseudodifferential operators with mildly regular symbols and operators on manifolds of low regularity. An example is the Dirichlet-Neumann operator for a compact domain with -boundary.

     

Integral homology -spheres and the Johnson filtration

The mapping class group of an oriented surface of genus with one boundary component has a natural decreasing filtration , where is the kernel of the action of on the nilpotent quotient of . Using a tree Lie algebra approximating the graded Lie algebra we prove that any integral homology sphere of dimension has for some a Heegaard decomposition of the form , where and is such that . This proves a conjecture due to S. Morita and shows that the ``core' of the Casson invariant is indeed the Casson invariant.

     

Small unitary representations of the double cover of

The irreducible unitary representations of the double cover of the real group , with infinitesimal character , which are small in the sense that their annihilator in the universal enveloping algebra is maximal, are expressed as Langlands quotients of generalized principal series. In the case where is even we show that there are four such representations and in the case where is odd there is just one. The representations' smallness allows them to be written as a sum of virtual representations, leading to a character formula for their -types. We investigate the place of these small representations in the orbit method and, in the case of , show that the representation is attached to a nilpotent coadjoint orbit.The -type spectrum for the Langlands quotients is explicitly determined and shown to be multiplicity free.

     

A generalization of half-plane mappings to the ball in

Let be a normalized (, ) biholomorphic mapping of the unit ball onto a convex domain that is the union of lines parallel to some unit vector . We consider the situation in which there is one infinite singularity of on . In one case with a simple change-of-variables, we classify all convex mappings of that are half-plane mappings in the first coordinate. In the more complicated case, when is not in the span of the infinite singularity, we derive a form of the mappings in dimension .

     

Rotation topological factors of minimal -actions on the Cantor set

In this paper we study conditions under which a free minimal -action on the Cantor set is a topological extension of the action of rotations, either on the product of -tori or on a single -torus . We extend the notion of linearly recurrent systems defined for -actions on the Cantor set to -actions, and we derive in this more general setting a necessary and sufficient condition, which involves a natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one of these two types.

     

properties for Gaussian random series

Let be an arbitrary sequence of and let be a random series of the type

where is a sequence of independent Gaussian random variables and an orthonormal basis of (the finite measure space being arbitrary). By using the equivalence of Gaussian moments and an integrability theorem due to Fernique, we show that a necessary and sufficient condition for to belong to , for any almost surely is that . One of the main motivations behind this result is the construction of a nontrivial Gibbs measure invariant under the flow of the cubic defocusing nonlinear Schrödinger equation posed on the open unit disc of .

     

A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

Equipartitions of measures in

A well-known problem of B. Grünbaum (1960) asks whether for every continuous mass distribution (measure) on there exist hyperplanes dividing into parts of equal measure. It is known that the answer is positive in dimension (see H. Hadwiger (1966)) and negative for (see D. Avis (1984) and E. Ramos (1996)). We give a partial solution to Grünbaum's problem in the critical dimension by proving that each measure in admits an equipartition by hyperplanes, provided that it is symmetric with respect to a -dimensional affine subspace of . Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on Koschorke's exact singularity sequence (1981) and the remarkable properties of the essentially unique, balanced binary Gray code in dimension ; see G. C. Tootill (1956) and D. E. Knuth (2001).