Lusternik-Schnirelmann category of

First we give an upper bound of , the L-S category of a principal -bundle for a connected compact group with a characteristic map . Assume that there is a cone-decomposition of in the sense of Ganea that is compatible with multiplication. Then we have for , if is compressible into with trivial higher Hopf invariant . Second, we introduce a new computable lower bound, for . The two new estimates imply , where is a category weight due to Rudyak and Strom.

     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.

     

Singular solutions of parabolic -Laplacian with absorption

We consider, for and , the -Laplacian evolution equation with absorption

We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in , and satisfy for all . We prove the following:

(i)

When , there does not exist any such singular solution.

(ii)

When , there exists, for every , a unique singular solution that satisfies as .

Also, as , where is a singular solution that satisfies as .

Furthermore, any singular solution is either or for some finite positive .

     

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.

     

On the Diophantine equation : Higher-order recurrences

Let be a field of characteristic and let be a linear recurring sequence of degree in defined by the initial terms and by the difference equation

with . Finally, let be an element of . In this paper we are giving fairly general conditions depending only on on , and on under which the Diophantine equation

has only finitely many solutions . Moreover, we are giving an upper bound for the number of solutions, which depends only on . This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

     

Surjectivity for Hamiltonian -spaces in -theory

Let be a compact connected Lie group, and a Hamiltonian -space with proper moment map . We give a surjectivity result which expresses the -theory of the symplectic quotient in terms of the equivariant -theory of the original manifold , under certain technical conditions on . This result is a natural -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian -spaces. We discuss this lemma in detail and highlight the differences between the -theory and rational cohomology versions of this lemma.

We also introduce a -theoretic version of equivariant formality and prove that when the fundamental group of is torsion-free, every compact Hamiltonian -space is equivariantly formal. Under these conditions, the forgetful map is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in admits an equivariant extension in .

     

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.

     

-continuity properties of the symmetric -stable process

Let be a domain of finite Lebesgue measure in and let be the symmetric -stable process killed upon exiting . Each element of the set of eigenvalues associated to , regarded as a function of , is right continuous. In addition, if is Lipschitz and bounded, then each is continuous in and the set of associated eigenfunctions is precompact.

     

On -adic intermediate Jacobians

For an algebraic variety of dimension with totally degenerate reduction over a -adic field (definition recalled below) and an integer with , we define a rigid analytic torus together with an Abel-Jacobi mapping to it from the Chow group of codimension algebraic cycles that are homologically equivalent to zero modulo rational equivalence. These tori are analogous to those defined by Griffiths using Hodge theory over . We compare and contrast the complex and -adic theories. Finally, we examine a special case of a -adic analogue of the Generalized Hodge Conjecture.

     

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 .

     

All -local finite groups of rank two for odd prime

In this paper we give a classification of the rank two -local finite groups for odd . This study requires the analysis of the possible saturated fusion systems in terms of the outer automorphism group of the possible -radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic -local finite group, getting some new examples of these for .

     

-estimates for the linearized Monge-Ampère equation

Let be a strictly convex domain and let be a convex function such that    det in . The linearized Monge-Ampère equation is

where det is the matrix of cofactors of . We prove that there exist and depending only on , and such that

for all solutions to the equation .

     

Weighted estimates in for Laplace's equation on Lipschitz domains

Let , , be a bounded Lipschitz domain. For Laplace's equation in , we study the Dirichlet and Neumann problems with boundary data in the weighted space , where , is a fixed point on , and denotes the surface measure on . We prove that there exists such that the Dirichlet problem is uniquely solvable if , and the Neumann problem is uniquely solvable if . If is a domain, one may take . The regularity for the Dirichlet problem with data in the weighted Sobolev space is also considered. Finally we establish the weighted estimates with general weights for the Dirichlet and regularity problems.

     

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.

     

Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator may be defined as

where is an appropriate distance function and is the inverse Fourier transform. The behavior of on is described for , a rough distance function. We conjecture that this operator is bounded on when and , and unbounded when . This conjecture is verified for large ranges of .

     

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.

     

A sharp weak type inequality for martingale transforms and other subordinate martingales

If is a martingale difference sequence, a sequence of numbers in , and a positive integer, then

Here denotes the best constant. If , then as was shown by Burkholder. We show here that for the case 2$">, and that is also the best constant in the analogous inequality for two martingales and indexed by , right continuous with limits from the left, adapted to the same filtration, and such that is nonnegative and nondecreasing in . In Section 7, we prove a similar inequality for harmonic functions.

     

The -theory of is undecidable

The three quantifier theory of , the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of that lies between the two and three quantifier theories with but includes function symbols.

Theorem. The two quantifier theory of , the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on ) is undecidable.

The same result holds for various lattices of ideals of which are natural extensions of preserving join and infimum when it exits.

     

Legendrian contact homology in

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form , where is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of and, more generally, invariants of self transverse immersions into up to restricted regular homotopies. When , this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.

     

-bounds for the Beurling-Ahlfors transform

Let denote the Beurling-Ahlfors transform defined on , . The celebrated conjecture of T. Iwaniec states that its norm where . In this paper the new upper estimate

is found.