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 .

     

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.

     

Stability theory, permutations of indiscernibles, and embedded finite models

We show that the expressive power of first-order logic over finite models embedded in a model is determined by stability-theoretic properties of . In particular, we show that if is stable, then every class of finite structures that can be defined by embedding the structures in , can be defined in pure first-order logic. We also show that if does not have the independence property, then any class of finite structures that can be defined by embedding the structures in , can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let be a set of indiscernibles in a model and suppose is elementarily equivalent to where is -saturated. If is stable and is saturated, then every permutation of extends to an automorphism of and the theory of is stable. Let be a sequence of -indiscernibles in a model , which does not have the independence property, and suppose is elementarily equivalent to where is a complete dense linear order and is -saturated. Then -types over are order-definable and if is -saturated, every order preserving permutation of can be extended to a back-and-forth system.

     

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.

     

Local differentiability of distance functions

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets for which the distance function is continuously differentiable everywhere on an open ``tube' of uniform thickness around . Here a corresponding local theory is developed for the property of being continuously differentiable outside of on some neighborhood of a point . This is shown to be equivalent to the prox-regularity of at , which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation. Additional characterizations are provided in terms of being locally of class or such that is convex around for some 0$">. Prox-regularity of at corresponds further to the normal cone mapping having a hypomonotone truncation around , and leads to a formula for by way of . The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting.

     

Tracially AF -algebras

Inspired by a paper of S. Popa and the classification theory of nuclear -algebras, we introduce a class of -algebras which we call tracially approximately finite dimensional (TAF). A TAF -algebra is not an AF-algebra in general, but a ``large' part of it can be approximated by finite dimensional subalgebras. We show that if a unital simple -algebra is TAF then it is quasidiagonal, and has real rank zero, stable rank one and weakly unperforated -group. All nuclear simple -algebras of real rank zero, stable rank one, with weakly unperforated -group classified so far by their -theoretical data are TAF. We provide examples of nonnuclear simple TAF -algebras. A sufficient condition for unital nuclear separable quasidiagonal -algebras to be TAF is also given. The main results include a characterization of simple rational AF-algebras. We show that a separable nuclear simple TAF -algebra satisfying the Universal Coefficient Theorem and having and is isomorphic to a simple AF-algebra with the same -theory.

     

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.

     

On syzygies of abelian varieties

In this paper we prove the following result: Let be a complex torus and a normally generated line bundle on ; then, for every , the line bundle satisfies Property of Green-Lazarsfeld.

     

Intersection theory on non-commutative surfaces

Consider a non-commutative algebraic surface, , and an effective divisor on , as defined by Van den Bergh. We show that the Riemann-Roch theorem, the genus formula, and the self intersection formula from classical algebraic geometry generalize to this setting.

We also apply our theory to some special cases, including the blow up of in a point, and show that the self intersection of the exceptional divisor is . This is used to give an example of a non-commutative surface with a commutative which cannot be blown down, because its self intersection is rather than . We also get some results on Hilbert polynomials of modules on .

     

Geometry of Banach spaces having shrinking approximations of the identity

Let and let be a compact set of scalars. We introduce property of Banach spaces by the requirement that whenever is a bounded net converging weak to in and . Using with 1$">, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to -, -, and -ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.

     

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.     

The metric projection onto the soul

We study geometric properties of the metric projection of an open manifold with nonnegative sectional curvature onto a soul . is shown to be up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of , and study the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of .

     

Partial differential equations with matricial coefficients and generalized translation operators

Let be the Bessel operator with matricial coefficients defined on by

where is a diagonal matrix and let be an matrix-valued function. In this work, we prove that there exists an isomorphism on the space of even , -valued functions which transmutes and . This allows us to define generalized translation operators and to develop harmonic analysis associated with . By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.

     

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.

     

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 .

     

Homology decompositions for classifying spaces of compact Lie groups

Let be a prime number and be a compact Lie group. A homology decomposition for the classifying space is a way of building up to mod homology as a homotopy colimit of classifying spaces of subgroups of . In this paper we develop techniques for constructing such homology decompositions. Jackowski, McClure and Oliver (Homotopy classification of self-maps of BG via -actions, Ann. of Math. 135 (1992), 183-270) construct a homology decomposition of by classifying spaces of -stubborn subgroups of . Their decomposition is based on the existence of a finite-dimensional mod acyclic --complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the -stubborn decomposition of which does not use this geometric construction.

     

Mean convergence of orthogonal Fourier series of modified functions

We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the -metric for any 2. $"> We prove also that if is a uniformly bounded ONS which is complete in all the spaces , then there exists a rearrangement of the natural numbers such that the system has the strong -property for all 2$">; that is, for every and for every and 0 $">there exists a function which coincides with except on a set of measure less than and whose Fourier series with respect to the system converges in      

-theory of projective Stiefel manifolds

Using the Hodgkin spectral sequence we calculate , the complex -theory of the projective Stiefel manifold , for even. For odd, we are only able to calculate , but this is sufficient to determine the order of the complexified Hopf bundle over .

     

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 .

     

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.