Serre duality for non-commutative -bundles

Let be a smooth scheme of finite type over a field , let be a locally free -bimodule of rank , and let be the non-commutative symmetric algebra generated by . We construct an internal functor, , on the category of graded right -modules. When has rank 2, we prove that is Gorenstein by computing the right derived functors of . When is a smooth projective variety, we prove a version of Serre Duality for using the right derived functors of .

     

Maximal holonomy of infra-nilmanifolds with -dimensional quaternionic Heisenberg geometry

Let be the quaternionic Heisenberg group of real dimension and let denote the maximal order of the holonomy groups of all infra-nilmanifolds with -geometry. We prove that . As an application, by applying Kim and Parker's result, we obtain that the minimum volume of a -dimensional quaternionic hyperbolic manifold with cusps is at least

     

Boundary Hölder and estimates for local solutions of the tangential Cauchy-Riemann equation

We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood of a point when is a generic -concave manifold of real codimension in , where . Our method is to first derive a homotopy formula for in when is the intersection of with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any -closed form on without shrinking. We obtain Hölder and estimates up to the boundary for the solution operator. RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage d'un point d'une sous-variété générique -concave de codimension quelconque de . Nous construisons une formule d'homotopie pour le sur , lorsque est l'intersection de et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme -fermée sur . Nous en déduisons des estimations et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur .

     

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.

     

Linking numbers in rational homology -spheres, cyclic branched covers and infinite cyclic covers

We study the linking numbers in a rational homology -sphere and in the infinite cyclic cover of the complement of a knot. They take values in and in , respectively, where denotes the quotient field of . It is known that the modulo- linking number in the rational homology -sphere is determined by the linking matrix of the framed link and that the modulo- linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate `modulo  ' and `modulo  '. When the finite cyclic cover of the -sphere branched over a knot is a rational homology -sphere, the linking number of a pair in the preimage of a link in the -sphere is determined by the Goeritz/Seifert matrix of the knot.

     

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.

     

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.

     

Operators on spaces preserving copies of Schreier spaces

It is proved that an operator , compact metrizable, a separable Banach space, for which the -Szlenk index of is greater than or equal to , , is an isomorphism on a subspace of isomorphic to , the Schreier space of order . As a corollary, one obtains that a complemented subspace of with Szlenk index equal to contains a subspace isomorphic to .

     

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 .

     

The -module structure of -modules

Let be a regular ring, essentially of finite type over a perfect field . An -module is called a unit -module if it comes equipped with an isomorphism , where denotes the Frobenius map on , and is the associated pullback functor. It is well known that then carries a natural -module structure. In this paper we investigate the relation between the unit -structure and the induced -structure on . In particular, it is shown that if is algebraically closed and is a simple finitely generated unit -module, then it is also simple as a -module. An example showing the necessity of being algebraically closed is also given.

     

Convergence of double Fourier series and -classes

The double Fourier series of functions of the generalized bounded variation class are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class defined here, have quadrant limits at every point and, for there exist at most countable sets and such that, for and is continuous at . It is shown that the previously studied class contains essentially discontinuous functions unless the sequence satisfies a strong condition.

     

Hyperpolygon spaces and their cores

Given an -tuple of positive real numbers , Konno (2000) defines the hyperpolygon space , a hyperkähler analogue of the Kähler variety parametrizing polygons in with edge lengths . The polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural -action, and the union of the precompact orbits is called the core. We study the components of the core of , interpreting each one as a moduli space of pairs of polygons in with certain properties. Konno gives a presentation of the cohomology ring of ; we extend this result by computing the -equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

     

On one-dimensional self-similar tilings and -tiles

Let be an integer base, a digit set and the set of radix expansions. It is well known that if has nonvoid interior, then can tile with some translation set ( is called a tile and a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of ; (ii) for a given , characterize so that is a tile.

We show that for a given pair , there is a unique self-replicating translation set , and it has period for some . This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for when are distinct primes. The only other known characterization is for , due to Lagarias and Wang. The proof for the case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the product-form digit set of Odlyzko.

     

Simple birational extensions of the polynomial algebra

The Abhyankar-Sathaye Problem asks whether any biregular embedding can be rectified, that is, whether there exists an automorphism such that is a linear embedding. Here we study this problem for the embeddings whose image is given in by an equation , where and . Under certain additional assumptions we show that, indeed, the polynomial is a variable of the polynomial ring (i.e., a coordinate of a polynomial automorphism of ). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings . Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial as above, a criterion for when .

     

Degeneration of linear systems through fat points on surfaces

In this paper we introduce a technique to degenerate surfaces and linear systems through fat points in general position on surfaces. Using this degeneration we show that on generic surfaces it is enough to prove that linear systems with one fat point are non-special in order to obtain the non-speciality of homogeneous linear systems through fat points in general position. Moreover, we use this degeneration to obtain a result for homogeneous linear systems through fat points in general position on a general quartic surface in .

     

Harmonic maps and 2-cycles, realizing the Thurston norm

Let be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, , of a harmonic map with Morse-type singularities delivers the Thurston norm of its homology class .

In particular, for a map with connected fibers and any well-positioned oriented surface in the homology class of a fiber, we show that the Thurston number satisfies an inequality

Here the variation is can be expressed in terms of the -invariants of the fiber components, and the twist measures the complexity of the intersection of with a particular set of ``bad" fiber components. This complexity is tightly linked with the optimal ``-height" of , being lifted to the -induced cyclic cover .

Based on these invariants, for any Morse map , we introduce the notion of its twist . We prove that, for a harmonic , if and only if .

     

Average size of -Selmer groups of elliptic curves, I

In this paper, we study a class of elliptic curves over with -torsion group , and prove that the average order of the -Selmer groups is bounded.

     

Generalized interpolation in with a complexity constraint

In a seminal paper, Sarason generalized some classical interpolation problems for functions on the unit disc to problems concerning lifting onto of an operator that is defined on ( is an inner function) and commutes with the (compressed) shift . In particular, he showed that interpolants (i.e., such that ) having norm equal to exist, and that in certain cases such an is unique and can be expressed as a fraction with . In this paper, we study interpolants that are such fractions of functions and are bounded in norm by (assuming that , in which case they always exist). We parameterize the collection of all such pairs and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

     

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 .

     

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 .