In some forms of superstring theory particular theta-functions come up as partition functions. The associated Abelian varieties come either from certain cohomology groups of the underlying universe or, in more recent theories (e.g. [Witten] [Mo-Wi]), are linked to their K-groups. Expressed in mathematical terms, one canonically associates to the cohomology or the K-theory of an even dimensional...

An undirected graph is called integral, if all of its eigenvalues are integers. Let Γ = Zm1 ⊗· · ·⊗Zmr be an abelian group represented as the direct product of cyclic groups Zmi of order mi such that all greatest common divisors gcd(mi,mj) ≤ 2 for i 6= j. We prove that a Cayley graph Cay(Γ, S) over Γ is integral, if and only if S ⊆ Γ belongs to the the Boolean algebra B(Γ) generated by the subg...

The set of integral ows on a nite graph ? is naturally an integral lattice 1 (?) in the Euclidean space Ker((1) of harmonic real-valued functions on the edge set of ?. Various properties of ? (bipartite character, girth, complexity, separability) are shown to correspond to properties of 1 (?) (parity, minimal norm, determinant, decomposability). The dual lattice of 1 (?) is identiied to the int...

Let N(d,G,X) be the number of degree d number fields K with Galois group G and whose discriminant DK satisfies |DK | ≤ X. Under standard conjectures in diophantine geometry, we show that N(4, A4, X) X2/3+ , and that there are N3+ monic, quartic polynomials with integral coefficients of height ≤ N whose Galois groups are smaller than S4, confirming a question of Gallagher. Unconditionally we hav...

In this paper we consider Z-actions, d ≥ 1, by automorphisms of compact connected abelian groups which contain at least one expansive automorphism (such actions are called algebraic Z-actions of expansive rank one). If α is such a Z-action on an infinite compact connected abelian group X, then every expansive element α of this action has a dense group ∆αn(X) of homoclinic points. For different ...

The set of integral flows on a finite graph Γ is naturally an integral lattice Λ1(Γ) in the Euclidean space Ker(∆1) of harmonic real-valued functions on the edge set of Γ. Various properties of Γ (bipartite character, girth, complexity, separability) are shown to correspond to properties of Λ1(Γ) (parity, minimal norm, determinant, decomposability). The dual lattice of Λ1(Γ) is identified to th...

The conventional path integral expression for the Yang-Mills transition amplitude with flat measure and gauge-fixing built in via the Faddeev-Popov method has been claimed to fall short of guaranteeing gauge invariance in the nonperturbative regime. We show, however, that it yields the gauge invariant partition function where the projection onto gauge invariant wave functions is explicitly perf...

The goal of local class field theory is to classify abelian Galois extensions of a local field K. Several definitions of local fields are in use. In this thesis, local fields, which will be defined explicitly in Section 2, are fields that are complete with respect to a discrete valuation and have a finite residue field. A prototypical first example is Qp, the completion of Q with respect to the...

We study certain function algebras and their operator algebra completions on r-discrete abelian groupoids, the corresponding conditional expectations, maximal abelian subalgebras (masa) and eigen-functionals. We give a semidirect product decomposition for an abelian groupoid. This is done through a matched pair and leads to a C*-diagonal (for a special case). We use this decomposition to study ...

We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field extension. In particular, this yields new and – we feel – simpler proofs of theorems of L. Claborn and C.R. Leedham-Green. Luther Claborn received his PhD from U. Mi...