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Let X be a protective non-singular algebraic variety over an algebraically closed field k. Let f: X ⟶ X be an endomorphism of X and let F be a locally free sheaf on X. Then the vector spaces Hq [X;F] are known to be of finite rank and to be zero except for finitely many integers q. Let Φ: f*F → F be a morphism of sheaves. Then endomorphisms, eq, of each Hq [X;F] may be constructed as the composite: Hq[X;F] f*_{⟶} Hq[X;f*F] Φ*_{⟶} Hq[X;F]. Set L(f,F,Φ) = ∑ (-)qtrace(eq) . If f has a non-singular fixed point set and subject to a slight restriction on the nature of the endomorphism induced by f on the normal bundle of each fixed component, Y, formulae are obtained of the form:

L(f,F,Φ) = ∑_Y v'(Y,Φ), with the sum being taken over the set of fixed components, in the two special cases: (α) f is periodic with period prime to the characteristic of k. (β) each fixed component is a point or a curve. The definition of v'(Y;Φ) involves Chern characters and the like of the restriction of F and Φ to Y, as well as the induced action on the normal bundle. If k has characteristic p ≠ 0, in case (α) a stronger p-adic lifting of the above formula is obtained. The methods used are setting up an appropriate formalism and using the theory of derived categories in homological algebra. The algebraic geometry used is essentially that of the Borel-Serre paper (1958) vintage.

In this thesis a class of finite real spectral triples for the geometry on a fuzzy torus is introduced. The geometries are shown to be related via an action of a general integral matrix. Each geometry is shown to have four real spectral triples corresponding to the four unique spin structures found on the 2-torus. The spectrum of the Dirac operator on each geometry, and spin structure, is calculated and shown to be the quantum integer analogues of the spectrum of the Dirac operator on the corresponding commutative 2-torus. The spectrum of the noncommutative Dirac operator is then shown to converge to the spectrum of the commutative Dirac operator as the algebra becomes commutative. Finally, an outline for the proof of a fuzzy torus converging to a commutative torus, via the defined Dirac operator, is presented.

Adelic (and idelic) structures can be associated to algebraic and arithmetic varieties, and an adelic geometry can be developed as a bridge between algebraic geometry and arithmetic geometry. We study in detail adelic geometry in dimension one and two. In particular, such a theory can be seen as a generalisation of the theory of algebraic and arithmetic line bundles, so the result is a novel approach to intersection theory. The construction process of adelic objects is "from local to global" and it endows such objects with natural topologies. One of the main richnesses of adelic geometry is given by the topological interactions between adelic structures, and a deep study of them in the case of arithmetic surfaces might be crucial to the solution to higher number theory open problems.