Linear series over real and -adic fields

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve of genus to with prescribed ramification also yields weaker results when working over the real numbers or -adic fields. Specifically, let be such a field: we see that given , , , and satisfying , there exists smooth curves of genus together with points such that all maps from to can, up to automorphism of the image, be defined over . We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case , , and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

     

Functional calculus and -regularity of a class of Banach algebras

Suppose that is a -dynamical system such that is of polynomial growth. If is finite dimensional, we show that any element in has slow growth and that is -regular. Furthermore, if is discrete and is a ``nice representation' of , we define a new Banach -algebra which coincides with when is finite dimensional. We also show that any element in has slow growth and is -regular.

     

On mixing and completely mixing properties of positive -contractions of finite von Neumann algebras

Akcoglu and Suchaston proved the following result: Let be a positive contraction. Assume that for the sequence converges weakly in . Then either or there exists a positive function , such that . In the paper we prove an extension of this result in a finite von Neumann algebra setting, and as a consequence we obtain that if a positive contraction of a noncommutative -space has no nonzero positive invariant element, then its mixing property implies the completely mixing property.

     

A minimal pair of -degrees

We construct a minimal pair of -degrees. We do this by showing the existence of an unbounded nondecreasing function which forces -triviality in the sense that is -trivial if and only if for all , .

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

On quasi-complete intersections of codimension

In this paper, we prove that if , , is a locally complete intersection of pure codimension and defined scheme-theoretically by three hypersurfaces of degrees , then for using liaison theory and the Arapura vanishing theorem for singular varieties. As a corollary, a smooth threefold is projectively normal if is defined by three quintic hypersurfaces.

     

Eigenvalues of the Laplacian acting on -forms and metric conformal deformations

Let be a compact connected orientable Riemannian manifold of dimension and let be the -th positive eigenvalue of the Laplacian acting on differential forms of degree on . We prove that the metric can be conformally deformed to a metric , having the same volume as , with arbitrarily large for all .

Note that for the other values of , that is and , one can deduce from the literature that, 0$">, the -th eigenvalue is uniformly bounded on any conformal class of metrics of fixed volume on .

For , we show that, for any positive integer , there exists a metric conformal to such that, , , that is, the first eigenforms of are all exact forms.

     

A characterisation of in terms of forcing

We show that ``saturation' of the universe with respect to forcing over with partial orders on is equivalent to the existence of .

     

Multiplicative bijections of

Let be a compact Hausdorff space which satisfies the first axiom of countability, let and let , be the set of all continuous functions from to If , ,is a bijective multiplicative map, then there exist a homeomorphism and a continuous map such that for all and for all

     

-bounded groups and other topological groups with strong combinatorial properties

We construct several topological groups with very strong combinatorial properties. In particular, we give simple examples of subgroups of (thus strictly -bounded) which have the Menger and Hurewicz properties but are not -compact, and show that the product of two -bounded subgroups of may fail to be -bounded, even when they satisfy the stronger property . This solves a problem of Tkacenko and Hernandez, and extends independent solutions of Krawczyk and Michalewski and of Banakh, Nickolas, and Sanchis. We also construct separable metrizable groups of size continuum such that every countable Borel -cover of contains a -cover of .

     

Hyperelliptic curves over of every -rank without extra automorphisms

We prove that for any pair of integers such that or 0$">, there exists a (hyper)elliptic curve over of genus and -rank whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties over of dimension and -rank such that .

     

for commutative rings with identity

Let , , , , be the usual operators on classes of rings: and for isomorphic and homomorphic images of rings and , , respectively for subrings, direct, and subdirect products of rings. If is a class of commutative rings with identity (and in general of any kind of algebraic structures), then the class is known to be the variety generated by the class . Although the class is in general a proper subclass of the class for many familiar varieties . Our goal is to give an example of a class of commutative rings with identity such that . As a consequence we will describe the structure of two partially ordered monoids of operators.

     

A dual graph construction for higher-rank graphs, and -theory for finite 2-graphs

Given a -graph and an element of , we define the dual -graph, . We show that when is row-finite and has no sources, the -algebras and coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the -theory of when is finite and strongly connected and satisfies the aperiodicity condition.

     

Remarks on spectra and multipliers for convolution operators

We prove that the spectrum of a convolution operator on a locally compact group by a self-adjoint -function is the same on and and consequently on all spaces, if and only if a Beurling algebra contains non-analytic functions on operating on into .

     

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.

     

Compact perturbations of -accretive operators in Banach spaces

This paper continues a discussion that arose twenty years ago, concerning the perturbation of an -accretive operator by a compact mapping in Banach spaces. Indeed, if is -accretive and is compact, then the boundary condition for and implies that is in the closure of the range of . Perhaps the most interesting aspect of this result is the proof itself, which does not appeal to the classical degree theory argument used for this type of problem.

     

Fiber products, Poincaré duality and -ring spectra

For a Poincaré duality space and a map , consider the homotopy fiber product . If is orientable with respect to a multiplicative cohomology theory , then, after suitably regrading, it is shown that the -homology of has the structure of a graded associative algebra. When is the diagonal map of a manifold , one recovers a result of Chas and Sullivan about the homology of the unbased loop space .

     

Smooth actions of

Given a smooth action of on a -dimensional differentiable manifold , for each we associate with ``almost all" oriented orbits of dimension an element of .

     

Projections in operator ranges

If is a Hilbert space, is a positive bounded linear operator on and is a closed subspace of , the relative position between and establishes a notion of compatibility. We show that the compatibility of is equivalent to the existence of a convenient orthogonal projection in the operator range with its canonical Hilbertian structure.

     

Topological entropy and AF subalgebras of graph -algebras

Let be the canonical AF subalgebra of a graph -algebra associated with a locally finite directed graph . For Brown and Voiculescu's topological entropy of the canonical completely positive map on , is known to hold for a finite graph , where is the loop entropy of Gurevic and is the block entropy of Salama. For an irreducible infinite graph , the inequality has recently been known. It is shown in this paper that

where is the graph with the direction of the edges reversed. Some irreducible infinite graphs 1)$"> with are also examined.