The realization problem asks: When does an algebraic complex arise, up to homotopy, from a geometric complex In the case of 2- dimensional algebraic complexes, this is equiv alent to the D2 problem, which asks when homological methods can distinguish between 2 and 3 dimensional complexes. We approach the realization problem (and hence the D2 problem) by classifying all pos sible algebraic 2- complexes and showing that they are realized. We show that if a dihedral group has order 2n, then the algebraic complexes over it are parametrized by their second homology groups, which we refer to as algebraic second homotopy groups. A cancellation theorem of Swan ( 11 ), then allows us to solve the realization problem for the group D$. Let X be a finite geometric 2- complex. Standard isomorphisms give 7r2(Ar) = H2(X Z), as modules over ni(X). Schanuel's lemma may then be used to show that the stable class of n2(X) is determined by k {X). We show how 7r3(X) maybe calculated similarly. Specif ically, we show that as a module over the fundamental group, (X) = S2{ir2{X)), where S2(ir2(X)) denotes the symmetric part of the module 7r2(X) z tt2(X). As a consequence, we are able to show that when the order of n (X) is odd, the stable class of 7r3(X) is also determined by ir {X). Given a closed, connected, orientable 5- dimensional manifold, with finite fundamen tal group, we may represent it, up to homotopy equivalence, by an algebraic complex. Poincare duality induces a homotopy equivalence between this algebraic complex and its dual. We consider how similar this homotopy equivalence may be made to the identity, (through appropriate choice of algebraic complex). We show that it can be taken to be the identity on 4 of the 6 terms of the chain complex. However, by finding a homological ob struction, we show that in general the homotopy equivalence may not be written as the identity.

The thesis is concerned with the extension of the notion of primitive recursion to structures other than the natural numbers. Successor systems are generalisations of the arithmetics of Vu?kovi? [2], and as a class are closed under operations corresponding to direct products and quotient formation. Given a system ? we can also define a system a* which has successor functions Sax for each numeral a of ?. The formalisation used is derived from the free variable calculus of Goodstein [1]. Various forms of recursion are considered, none of which employ more than a small number of known functions. For example, given a function g from ? x ? to ? we can define f from ?* to ? as follows. f(0) = 0; f(Sax) = g(a,f(x)) Algebraic applications include the construction of groups and rings: actual examples range from the integers and polynomials to permutations, finite sets and ordinal numbers. Several relations which may hold between systems are investigated, as are the notions of anchored and decidable systems.*(supported by a Science Research Council grant) One chapter deals with the case of commuting successor functions, and another considers systems with only one successor. In an appendix we briefly investigate the further generalisation obtained by using non-unary successor functions. The author expresses his thanks to all concerned, especially his supervisor. Professor R. L. Goodstein. Contents of thesis: (1) Introduction, (2) The Integers, (3) Products, (4) Recursion, (5) The Star Operation, (6) Commutative systems, (7) Homomorphisms, (8) Groups, (9) Further recursion, (10) Decidable systems, (11) Single successor systems, (12) Polynomials; (A1) Small systems, (A2) Joint successor arithmetics, (A3) Polish Circles, (A4) A Formalisation of the Integers. References to abstract: [1] Goodstein, R.L., Recursive Number Theory, Amsterdam (1957) [2] Vu?kovi?, V., Partially ordered recursive arithmetics, Math.Scand. 7 (1959), 305-320.

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In this thesis we investigate the problem of recovering arithmetic structure on a field F from small quotients of its absolute Galois group. In particular, we are interested in recovering the p-adic valuation on a p-adic field from such quotients. After establishing several such results, we apply this to obtain strong versions of the Birational Section Conjecture for curves over p-adic fields. We also discuss the model-theoretic interpretation of these results, as well as begin investigating the foundations of a model-theory of schemes.

This thesis presents work relating to the rich connections between Rational Homotopy Theory and Commutative Algebra, and builds on the classical work of Quillen and Sullivan, and more recent work of Greenlees, Hess and Shamir.

Let ρ be an odd prime, and let K be a finite extension of Qp such that K contains a primitive ρ-th root of unity. Let K <ρ be the maximal ρ-extension of K with Galois group Γ <ρ of period ρ and nilpotence class < ρ. Recent results of Abrashkin describe the ramification filtration ?, and can be used to recover the structure of Γ< ρ. The group Γ< ρ is described in terms of an Fρ- Lie algebra L due to the classical equivalence of categories of Fρ-Lie algebras of nilpotent class < ρ, and ρ-groups of period ρ of the same nilpotent class. In this thesis we generalise explicit calculations of Abrashkin related to the structure of Γ< ρ.

An algorithm for detecting unstable periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp. 6172-6175] which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp. 4733-4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighbouring periods, has been shown to be highly efficient when applied to low-dimensional system. The difficulty in applying the algorithm to higher dimensional systems is mainly due to the fact that the number of stabilising transformations grows extremely fast with increasing system dimension. in this thesis, we propose to construct stabilising transformations based on the knowledge of the stability matrices of already detected periodic orbits (used as seeds). The advantage of our approach is in a substantial reduction of the number of transformations, which increases the efficiency of the detection algorithm, especially in the case of high-dimensional systems. The dependence of the number of transformations on the dimensionality of the unstable manifold rather than on system size enables us to apply, for the first time, the method of stabilising transformations to high-dimensional systems. Another important aspect of our treatment of high-dimensional flows is that we do not restrict to a Poincare surface of section. This is a particularly nice feature, since the correct placement of such a section in a high-dimensional phase space is a challenging problem in itself. The performance of the new approach is illustrated by its application to the four-dimensional kicked double rotor map, a six-dimensional system of three coupled Henon maps and to the Kuramoto-Sivashinsky system in the weakly turbulent regime.