Specializations of Brauer classes over algebraic function fields

Let be either a number field or a field finitely generated of transcendence degree over a Hilbertian field of characteristic 0, let be the rational function field in one variable over , and let . It is known that there exist infinitely many such that the specialization induces a specialization , where has exponent equal to that of . Now let be a finite extension of and let . We give sufficient conditions on and for there to exist infinitely many such that the specialization has an extension to inducing a specialization , the residue field of , where has exponent equal to that of . We also give examples to show that, in general, such need not exist.

     

Homology manifold bordism

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact homology manifolds of dimension is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.

First, we establish homology manifold transversality for submanifolds of dimension : if is a map from an -dimensional homology manifold to a space , and is a subspace with a topological -block bundle neighborhood, and , then is homology manifold -cobordant to a map which is transverse to , with an -dimensional homology submanifold.

Second, we obtain a codimension splitting obstruction in the Wall -group for a simple homotopy equivalence from an -dimensional homology manifold to an -dimensional Poincaré space with a codimension Poincaré subspace with a topological normal bundle, such that if (and for only if) splits at up to homology manifold -cobordism.

Third, we obtain the multiplicative structure of the homology manifold bordism groups .

     

Geometric properties of the sections of solutions to the Monge-Ampère equation

In this paper we establish several geometric properties of the cross sections of generalized solutions to the Monge-Ampère equation , when the measure satisfies a doubling property. A main result is a characterization of the doubling measures in terms of a geometric property of the cross sections of . This is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations.

     

Trees and valuation rings

A subring of a division algebra is called a valuation ring of if or holds for all nonzero in . The set of all valuation rings of is a partially ordered set with respect to inclusion, having as its maximal element. As a graph is a rooted tree (called the valuation tree of ), and in contrast to the commutative case, may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra , and one main result here is a positive answer to this question where can be chosen as a quaternion division algebra over a commutative field.

     

Local boundary rigidity of a compact Riemannian manifold with curvature bounded above

This paper considers the boundary rigidity problem for a compact convex Riemannian manifold with boundary whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics on there is a -neighborhood of such that is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance -- as measured in ). More precisely, given any metric in this neighborhood with the same boundary distance function there is diffeomorphism which is the identity on such that . There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distance-function.

     

The Farrell-Jones Isomorphism Conjecture for finite covolume hyperbolic actions and the algebraic -theory of Bianchi groups

We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic -space with finite volume orbit space. We then apply this result to show that, for any Bianchi group , , , and vanish for .

     

Unipotent groups associated to reduced curves

Let be a curve defined over an algebraically closed field with 0$">. Assume that is reduced. In this paper we study the unipotent part of the Jacobian . In particular, we prove that if is large in terms of the dimension of , then is isomorphic to a product of additive groups .

     

An equivariant Brauer semigroup and the symmetric imprimitivity theorem

Suppose that is a second countable locally compact transformation group. We let denote the set of Morita equivalence classes of separable dynamical systems where is a -algebra and is compatible with the given -action on . We prove that is a commutative semigroup with identity with respect to the binary operation for an appropriately defined balanced tensor product on -algebras. If and act freely and properly on the left and right of a space , then we prove that and are isomorphic as semigroups. If the isomorphism maps the class of to the class of , then is Morita equivalent to .

     

A Palais-Smale approach to problems in Esteban-Lions domains with holes

Let be the upper half strip with a hole. In this paper, we show there exists a positive higher energy solution of semilinear elliptic equations in and describe the dynamic systems of solutions of equation in various . We also show there exist at least two positive solutions of perturbed semilinear elliptic equations in .

     

A condition for the stability of -covered on foliations of 3-manifolds

We give a sufficient condition for a codimension one, transversely orientable foliation of a closed 3-manifold to have the property that any foliation sufficiently close to it be -covered. This condition can be readily verified for many examples. Further, if an -covered foliation has a compact leaf , then any transverse loop meeting lifts to a copy of the leaf space, and the ambient manifold fibers over with as fiber.

     

On the dimension of the product of two compacta and the dimension of their intersection in general position in Euclidean space

For every two compact metric spaces and , both with dimension at most , there are dense -subsets of mappings and with .

     

Representing nonnegative homology classes of by minimal genus smooth embeddings

For any nonnegative class in , the minimal genus of smoothly embedded surfaces which represent is given for , and in some cases with , the minimal genus is also given. For the finiteness of orbits under diffeomorphisms with minimal genus , we prove that it is true for with and for with .

     

Principal curvatures of isoparametric hypersurfaces in

Let be an isoparametric hypersurface in , and the inverse image of under the Hopf map. By using the relationship between the eigenvalues of the shape operators of and , we prove that is homogeneous if and only if either or is constant, where is the number of distinct principal curvatures of and is the number of non-horizontal eigenspaces of the shape operator on .

     

Definably simple groups in o-minimal structures

Let be a group definable in an o-minimal structure . A subset of is -definable if is definable in the structure (while definable means definable in the structure ). Assume has no -definable proper subgroup of finite index. In this paper we prove that if has no nontrivial abelian normal subgroup, then is the direct product of -definable subgroups such that each is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.

     

Schauder estimates for equationswith fractional derivatives

The equation where and are fractional derivatives of order and is studied. It is shown that if , , and are Hölder-continuous and , then there is a solution such that and are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to (). Finally the solution of () with is studied.     

On the endomorphism monoids of (uniquely) complemented lattices

Let be a lattice with and . An endomorphism of is a -endomorphism, if it satisfies and . The -endomorphisms of form a monoid. In 1970, the authors proved that every monoid can be represented as the -endomorphism monoid of a suitable lattice with and . In this paper, we prove the stronger result that the lattice with a given -endomorphism monoid can be constructed as a uniquely complemented lattice; moreover, if is finite, then can be chosen as a finite complemented lattice.

     

Analytic types of plane curve singularities defined by weighted homogeneous polynomials

We classify analytically isolated plane curve singularities defined by weighted homogeneous polynomials , which are not topologically equivalent to homogeneous polynomials, in an elementary way. Moreover, in preparation for the proof of the above analytic classification theorem, assuming that either satisfies the same property as the above does or is homogeneous, then we prove easily that the weights of the above determine the topological type of and conversely. So, this gives another easy proof for the topological classification theorem of quasihomogenous singularities in , which was already known. Also, as an application, it can be shown that for a given , where is a quasihomogeneous holomorphic function with an isolated singularity at the origin or with a positive integer , analytic types of isolated hypersurface singularities defined by are easily classified where is defined just as above.

     

Infinite convolution products and refinable distributions on Lie groups

Sufficient conditions for the convergence in distribution of an infinite convolution product of measures on a connected Lie group with respect to left invariant Haar measure are derived. These conditions are used to construct distributions that satisfy where is a refinement operator constructed from a measure and a dilation automorphism . The existence of implies is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset such that for any open set containing and for any distribution on with compact support, there exists an integer such that implies If is supported on an -invariant uniform subgroup then is related, by an intertwining operator, to a transition operator on Necessary and sufficient conditions for to converge to , and for the -translates of to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of to functions supported on

     

Invariant foliations near normally hyperbolic invariant manifolds for semiflows

Let be a compact manifold which is invariant and normally hyperbolic with respect to a semiflow in a Banach space. Then in an -neighborhood of there exist local center-stable and center-unstable manifolds and , respectively. Here we show that and may each be decomposed into the disjoint union of submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.

     

Linear systems of plane curves with base points of equal multiplicity

In this article we address the problem of computing the dimension of the space of plane curves of degree with general points of multiplicity . A conjecture of Harbourne and Hirschowitz implies that when , the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple -curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all .