We present a notion of forcing that can be used, in conjunction with other results, to show that there is a Martin-Löf random set X such that X 6≥T ∅′ and X computes every K-trivial set.

We study the L-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M = Γ\X with finite volume and arithmetic fundamental group Γ whose universal covering X is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one.

Let V be a finite set of order v. A (v,k,A.) covering design of index A. and block size k is a collection of k-element subsets, called blocks, such that every 2-subset of V occurs in at least '}.. blocks. The covering problem is to determine the minimum number of blocks, a (v, k, A.), in a covering design. It is well known that a(v,k''}..)Lr~~=~'}..ll=(v,k,A.), where rxl is the smallest integer

We construct examples of 4-dimensional manifolds M supporting a locally CAT(0)metric, whose universal covers Q M satisfy Hruska’s isolated flats condition, and contain 2-dimensional flats F with the property that @1F Š S ,! S Š @1 Q M are nontrivial knots. As a consequence, we obtain that the group 1.M/ cannot be isomorphic to the fundamental group of any compact Riemannian manifold of nonposit...

We give a new proof of the vanishing of H1(X, V ) for negative vector bundles V on normal projective varieties X satisfying rank V < dim X. Our proof is geometric, it uses a topological characterization of the affine bundles associated with non-trivial cocycles α ∈ H1(X, V ) of negative vector bundles. Following the same circle of ideas, we use the analytic characteristics of affine bundles to ...

This paper will deal with the consequences of the Uniformization Theorem, which is a major result in complex analysis and differential geometry. We will proceed by stating the theorem, which is that for any simply connected Riemann surface, there exists a biholomorphic map to one (and only one) of the following three: the Riemann sphere, the open unit disk, and the complex plane. After the theo...

Is it possible to find a formula expressing the Nielsen number N( f ) by the numbers N( ̃ f ) where ̃ f runs the set of all lifts? Such a formula seems very desirable since the difficulty of computing the Nielsen number often depends on the size of the fundamental group. Since π1 ̃ X ⊂ π1X , the computation of N( ̃ f ) may be simpler. We will translate this problem to algebra. The main result of th...

In this announcement we present results about the homotopy groups of //-spaces having the homotopy type of finite CW-complexes. We call such spaces finite H-spaces. We always assume our spaces are connected. In the sequel we always use X to denote a finite //-space. In some statements we refer to a direct sum of cyclic groups. We do not rule out the case that the sum is zero. Let X be the fibre...

A conjecture of [13] states that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics.

Table of Contents Introduction—1 §1 The Dirac operator on the Hyperbolic disk A covering of the frame bundle—4 The Dirac operator—7 Covariance of the Dirac operator—10 §2 Local Expansions Eigenfunctions for infinitesimal rotations—14 Differentiating local expansions with respect to the branch points—21 Estimates at infinity—24 §3 L 2 existence results A model for the simply connected covering D...