Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.

A variety of questions in combinatorics lead one to the task of analyzing a simplicial complex, or a more general cell complex. For example, a standard approach to investigating the structure of a partially ordered set is to instead study the topology of the associated

Traditional Morse theory deals with real valued functions f : M R and ordinary homology H (M ). The critical points of a Morse function f generate the Morse-Smale (f) over Z, using the gradient flow to define the differentials. The isomorphism H (M) imposes homological restrictions on real valued Morse functions. There is also a universal coefficient version of the Morse-Smale complex, involving the universal cover M and the fundamental group ring Z[# 1 (M )].

Given a compact PEL-type Shimura variety, a sufficiently regular weight (defined by mild and effective conditions), and a prime number p unramified in the linear data and larger than an effective bound given by the weight, we show that the étale cohomology with Zp-coefficients of the given weight vanishes away from the middle degree, and hence has no p-torsion. We do not need any other assumption (such as ones on the images of the associated Galois representations).

We present a new, intrinsic approach to Morse Theory which has interesting applications in geometry. We show that a Morse function f on a manifold determines a submanifold T of the product X \Theta X, and that (in the sense that Stokes theorem is valid) T has boundary consisting of the diagonal \Delta ae X \Theta X and a sum P = X p2Cr(f) Up \Theta Sp where Sp and Up are the stable and unstable manifolds at the critical point p. In the language of currents, @T = \Delta \Gamma P:(Stokes Theorem) This current (or kernel) equation on X \Theta X is equivalent to an operator equation d ffi T+T ffi d = I \Gamma P; ((Chain Homotopy)) where P is a chain map onto the finite complex of currents S f spanned by (integration over) the stable manifolds of f . The operator P can be defd on an exterior form ff by P(ff) = lim t!1 '

Abstract. The two-category with three-manifolds as objects, h-cobordisms as morphisms, and diffeomorphisms of these as two-morphisms, is extremely rich; from the point of view of classical physics it defines a nontrivial topological model for general relativity. A rather striking amount of work on pseudoisotopy theory [Hatcher, Waldhausen, Cohen-Carlsson-Goodwillie-Hsiang-Madsen...] can be formulated as a TQFT in this framework. The resulting theory is far from trivial even in the case of Minkowski space, when the relevant three-manifold is the standard sphere. Topological gravity extends Graeme Segal’s ideas about conformal field theory to higher dimensions. It seems to be very interesting, even in extremely restricted geometric contexts: §1 basic definitions A cobordism W: V0 → V1 between d-manifolds is a (d + 1)-dimensional manifold W together with a distinguished diffeomorphism ∂W ∼ = V op 0 V1; a diffeomorphism Φ: W → W ′ of cobordisms will be assumed consistent with this boundary data. Cob(V0, V1) is the category whose objects are such cobordisms, and whose morphisms are such diffeomorphisms. Gluing along the boundary defines a composition functor # : Cob(V ′ , V) × Cob(V, V ′ ′ ) → Cob(V, V ′ ′ ). The two-category with manifolds as objects and the categories Cob as morphisms is symmetric monoidal under disjoint union. The categories Cob are topological groupoids (all morphisms are invertible), with classifying spaces

Traditional Morse theory deals with real valued functions f: M → R and ordinary homology H∗(M). The critical points of a Morse function f generate the Morse-Smale complex CMS (f) over Z, using the gradient flow to define the differentials. The isomorphism H∗(CMS (f)) ∼ = H∗(M) imposes homological restrictions on real valued Morse functions. There is also a universal coefficient version of the Morse-Smale complex, involving the universal cover ˜ M and the fundamental group ring Z[π1(M)]. The more recent Morse theory of circle valued functions f: M → S1 is more complicated, but shares many features of the real valued theory. The critical points of a Morse function f generate the Novikov complex CNov (f) over the Novikov ring Z((z)) of formal power series with integer coefficients, using the gradient flow of the real valued Morse function f: M = f ∗R → R on the infinite cyclic cover to define the (M) is the Z((z))-coefficient homology of (M) imposes homological restrictions on circle valued Morse functions. Chapter 1 reviews real valued Morse theory. Chapters 2,3,4 introduce circle valued

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