Block representation type of reduced enveloping algebras

Let be an algebraically closed field of characteristic , a connected, reductive -group, , and the reduced enveloping algebra of associated with . Assume that is simply-connected, is good for and has a non-degenerate -invariant bilinear form. All blocks of having finite and tame representation type are determined.

     

Milnor classes of local complete intersections

Let be a compact local complete intersection defined as the zero set of a section of a holomorphic vector bundle over the ambient space. For each connected component of the singular set of , we define the Milnor class in the homology of . The difference between the Schwartz-MacPherson class and the Fulton-Johnson class of is shown to be equal to the sum of over the connected components of . This is done by proving Poincaré-Hopf type theorems for these classes with respect to suitable tangent frames. The -degree component coincides with the Milnor numbers already defined by various authors in particular situations. We also give an explicit formula for when is a non-singular component and satisfies the Whitney condition along .

     

Finely -harmonic functions of a Markov process

Let be a Borel right process and a fixed excessive measure. Given a finely open nearly Borel set we define an operator which we regard as an extension of the restriction to of the generator of . It maps functions on to (locally) signed measures on not charging -semipolars. Given a locally smooth signed measure we define to be (finely) -harmonic on provided on and denote the class of such by . Under mild conditions on we show that is equivalent to a local ``Poisson' representation of . We characterize by an analog of the mean value property under secondary assumptions. We obtain global Poisson type representations and study the Dirichlet problem for elements of under suitable finiteness hypotheses. The results take their nicest form when specialized to Hunt processes.

     

Coloring

If and , then define the graph to be the graph whose vertex set is with two vertices being adjacent iff there are distinct such that . For various and and various , typically or , the graph can be properly colored with colors. It is shown that in some cases such a coloring can also have the additional property that if is an isometric embedding, then the restriction of to is a bijection onto .

     

Weak amenability of triangular Banach algebras

Let and be unital Banach algebras, and let be a Banach -module. Then becomes a triangular Banach algebra when equipped with the Banach space norm . A Banach algebra is said to be -weakly amenable if all derivations from into its dual space are inner. In this paper we investigate Arens regularity and -weak amenability of a triangular Banach algebra in relation to that of the algebras , and their action on the module .

     

Rational -equivariant homotopy theory

We give an algebraicization of rational -equivariant homotopy theory. There is an algebraic category of `` -systems' which is equivalent to the homotopy category of rational -simply connected -spaces. There is also a theory of ``minimal models' for -systems, analogous to Sullivan's minimal algebras. Each -space has an associated minimal -system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.

     

Product systems over right-angled Artin semigroups

We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid . We show that the existing notions of product systems fit into our categorical framework, as do the -graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid . In particular, we obtain necessary and sufficient conditions under which a collection of -graphs form the coordinate graphs of a -graph.     

Monge's transport problem on a Riemannian manifold

Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures , find the measure-preserving map between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold -- and assuming absolute continuity of -- an optimal map will be shown to exist. Aspects of its uniqueness are also established.

     

Sufficient conditions for zero-one laws

We generalize a result of Bateman and Erdos concerning partitions, thereby answering a question of Compton. From this result it follows that if is a class of finite relational structures that is closed under the formation of disjoint unions and the extraction of components, and if it has the property that the number of indecomposables of size is bounded above by a polynomial in , then has a monadic second order - law. Moreover, we show that if a class of finite structures with the unique factorization property is closed under the formation of direct products and the extraction of indecomposable factors, and if it has the property that the number of indecomposables of size at most is bounded above by a polynomial in , then this class has a first order - law. These results cover all known natural examples of classes of structures that have been proved to have a logical - law by Compton's method of analyzing generating functions.

     

Summing inclusion maps between symmetric sequence spaces

In 1973/74 Bennett and (independently) Carl proved that for the identity map id: is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a -concave symmetric Banach sequence space the identity map is absolutely -summing, i.e., for every unconditionally summable sequence in the scalar sequence is contained in . Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator on with values in a -concave symmetric Banach sequence space is a multiplier from into . Furthermore, we prove an asymptotic formula for the -th approximation number of the identity map , where denotes the linear span of the first standard unit vectors in , and apply it to Lorentz and Orlicz sequence spaces.

     

Induced operators on symmetry classes of tensors

Let be an -dimensional Hilbert space. Suppose is a subgroup of the symmetric group of degree , and is a character of degree 1 on . Consider the symmetrizer on the tensor space

defined by and . The vector space

is a subspace of , called the symmetry class of tensors over associated with and . The elements in of the form are called decomposable tensors and are denoted by . For any linear operator acting on , there is a (unique) induced operator acting on satisfying

In this paper, several basic problems on induced operators are studied.

     

Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group

Applied to a continuous surjection of completely regular Hausdorff spaces and , the Stone-Cech compactification functor yields a surjection . For an -fold covering map , we show that the fibres of , while never containing more than points, may degenerate to sets of cardinality properly dividing . In the special case of the universal bundle of a -group , we show more precisely that every possible type of -orbit occurs among the fibres of . To prove this, we use a weak form of the so-called generalized Sullivan conjecture.

     

An analogue of minimal surface theory in

We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semi-simple Lie groups (e.g. ), which contains minimal surfaces in and constant mean curvature surfaces in . A Weierstrass type representation formula and a Chern-Osserman type inequality for such surfaces are given.

     

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 .

     

Dual decompositions of 4-manifolds

This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index . In dimensions results of Smale (trivial ) and Wall (general ) describe analogous decompositions up to diffeomorphism in terms of homotopy type of skeleta or chain complexes. In dimension 4 we show the same data determines decompositions up to 2-deformation of their spines. In higher dimensions spine 2-deformation implies diffeomorphism, but in dimension 4 the fundamental group of the boundary is not determined. Sample results: (1.5) Two 2-complexes are (up to 2-deformation) spines of a dual decomposition of the 4-sphere if and only if they satisfy the conclusions of Alexander-Lefshetz duality ( and ). (3.3) If is 1-connected then there is a ``pseudo' handle decomposition without 1-handles, in the sense that there is a pseudo collar (a relative 2-handlebody with spine that 2-deforms to ) and is obtained from this by attaching handles of index .

     

On a refinement of the generalized Catalan numbers for Weyl groups

Let be an irreducible crystallographic root system with Weyl group , coroot lattice and Coxeter number , spanning a Euclidean space , and let be a positive integer. It is known that the set of regions into which the fundamental chamber of is dissected by the hyperplanes in of the form for and is equinumerous to the set of orbits of the action of on the quotient . A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of , are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case and .

     

Existence of curves with prescribed topological singularities

Throughout this paper we study the existence of irreducible curves on smooth projective surfaces with singular points of prescribed topological types . There are necessary conditions for the existence of the type for some fixed divisor on and suitable coefficients , and , and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case . An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up of the form , deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in , and K3-surfaces.

     

Automorphisms of finite order on Gorenstein del Pezzo surfaces

In this paper we shall determine all actions of groups of prime order with on Gorenstein del Pezzo (singular) surfaces of Picard number 1. We show that every order- element in ( , being the minimal resolution of ) is lifted from a projective transformation of . We also determine when is finite in terms of , and the number of singular members in . In particular, we show that either for some , or for every prime , there is at least one element of order in (hence is infinite).

     

Topological dynamics on moduli spaces II

Let be an orientable genus 0$"> surface with boundary . Let be the mapping class group of fixing . The group acts on the space of -gauge equivalence classes of flat -connections on with fixed holonomy on . We study the topological dynamics of the -action and give conditions for the individual -orbits to be dense in .