A note on -estimates for stable integrals with drift

Let be of the form where is a symmetric stable process of index with . We obtain various -estimates for the process . In particular, for and any measurable, nonnegative function we derive the inequality

As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation for any initial value .

     

On the essential commutant of QC

Let (QC) (resp. ) be the -algebra generated by the Toeplitz operators QC (resp. ) on the Hardy space of the unit circle. A well-known theorem of Davidson asserts that (QC) is the essential commutant of . We show that the essential commutant of (QC) is strictly larger than . Thus the image of in the Calkin algebra does not satisfy the double commutant relation. We also give a criterion for membership in the essential commutant of (QC).

     

Morava -theory of filtered colimits

Morava -theory is a much-studied theory in algebraic topology, but it is not a homology theory in the usual sense, because it fails to preserve coproducts (resp. filtered homotopy colimits). The object of this paper is to construct a spectral sequence to compute the Morava -theory of a coproduct (resp. filtered homotopy colimit). The -term of this spectral sequence involves the derived functors of direct sum (resp. filtered colimit) in an appropriate abelian category. We show that there are at most (resp. ) of these derived functors. When , we recover the known result that homotopy commutes with an appropriate version of direct sum in the -local stable homotopy category.

     

-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.

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

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 .

     

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.

     

Quantum symmetric derivatives

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.

     

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 .

     

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).

     

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.

     

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.

     

Branch structure of -holomorphic curves near periodic orbits of a contact manifold

Let be a three-dimensional contact manifold, and a finite-energy pseudoholomorphic map from the punctured disc in that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. Examples of this behaviour, which are studied in some detail, include pseudoholomorphic maps into , where denotes a rational ellipsoid (contact structure induced by the standard complex structure on ), as well as contact structures arising from non-standard circle-fibrations of the three-sphere.

     

Vertex operator algebras, extended diagram, and McKay's observation on the Monster simple group

We study McKay's observation on the Monster simple group, which relates the -involutions of the Monster simple group to the extended diagram, using the theory of vertex operator algebras (VOAs). We first consider the sublattices of the lattice obtained by removing one node from the extended diagram at each time. We then construct a certain coset (or commutant) subalgebra associated with in the lattice VOA . There are two natural conformal vectors of central charge in such that their inner product is exactly the value predicted by Conway (1985). The Griess algebra of coincides with the algebra described in his Table 3. There is a canonical automorphism of of order . Such an automorphism can be extended to the Leech lattice VOA , and it is in fact a product of two Miyamoto involutions. In the sequel (2005) to this article, the properties of will be discussed in detail. It is expected that if is actually contained in the Moonshine VOA , the product of two Miyamoto involutions is in the desired conjugacy class of the Monster simple group.

     

Quantum cohomology and the -Schur basis

We prove that structure constants related to Hecke algebras at roots of unity are special cases of -Littlewood-Richardson coefficients associated to a product of -Schur functions. As a consequence, both the 3-point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to are shown to be -Littlewood-Richardson coefficients. From this, Mark Shimozono conjectured that the -Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas -Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual -Schur functions defined on weights of -tableaux that, given Shimozono's conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.

     

Iwasawa theory for -local spectra

The Iwasawa algebra is a power series ring in one variable over the -adic integers. It has long been studied by number theorists in the context of -extensions of number fields. It also arises, however, as a ring of operations in -adic topological -theory. In this paper we study -local stable homotopy theory using the structure theory of modules over the Iwasawa algebra. In particular, for odd we classify -local spectra up to pseudo-equivalence (the analogue of pseudo-isomorphism for -modules) and give an Iwasawa-theoretic classification of the thick subcategories of the weakly dualizable spectra.

     

Compatible valuations and generalized Milnor -theory

Given a field and a subgroup of there is a minimal group for which there exists an -compatible valuation whose units are contained in . Assuming that has finite index in and contains for prime, we describe in computable -theoretic terms.

     

The -stable pieces of the wonderful compactification

Let be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification of into finite many -stable pieces, which was introduced by Lusztig. In this paper, we will investigate the closure of any -stable piece in . We will show that the closure is a disjoint union of some -stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many -orbits.

     

Local and global -regularity

A bounded domain is called -regular if the plurisubharmonic envelope of every continuous function on extends continuously to . We show using Gauthier's Fusion Lemma that a domain is locally -regular if and only if it is -regular.