Complex Multiplication of Exactly Solvable Calabi-Yau Varieties

We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi-Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology of the Calabi-Yau manifold, and the conformal field theoretic quantities of the underlying string emerge from the number theoretic structure induced on the varieties by the complex multiplication symmetry. The geometric structure that provides a conceptual interpretation of the relation between geometry and the conformal field theory is discrete, and turns out to be given by the torsion points on the abelian varieties.Comment: 44 page

On the 2-part of class groups and Diophantine equations

This thesis contains several pieces of work related to the 2-part of class groups and Diophantine equations. We first give an overview of some techniques known in computing the 2-part of the class groups of quadratic number fields, including the use of the Rédei symbol and Rédei reciprocity in the study of the 8-rank of the class groups of quadratic fields. We review the construction of governing fields for the 8-rank by Corsman and extend a proof of Smith on the distribution of the 8-rank for imaginary quadratic fields, to real quadratic fields, conditional on the general Riemann hypothesis. In joint work with Peter Koymans, Djordjo Milovic, and Carlo Pagano, we improve a previous lower bound by Fouvry and Klüners, on the density of the solvability of the negative Pell equation over the set of squarefree positive integers with no prime factors congruent to 3 mod 4. We show how Rédei reciprocity allows us to apply techniques introduced by Smith to obtain this improvement. In joint work with Djordjo Milovic, using Kuroda's formula, we study the average behaviour of the unit group index in certain families of totally real biquadratic fields Q(√p,√d) parametrised by the prime p. In joint work with Christine McMeekin and Djordjo Milovic, we study certain cyclic totally real number fields K, in which we attach a quadratic symbol spin(a,σ) to each odd prime ideal a and each non-trivial σ in Gal(K/Q). We prove a formula for the density of primes ideals p such that spin(p,σ) = 1 for all non-trivial σ in Gal(K/Q). Finally, we study integral points on the quadratic twists E_D:y²=x³-D²x of the congruent number curve. We show that the number of non-torsion integral points on E_D is << (3.8)^{\rank E_D(Q)} and its average is bounded above by 2. We deduce that the system of simultaneous Pell equations aX²-bY²=d, bY²-cZ²=d for pairwise coprime positive integers a,b,c,d, has at most << (3.6)^{ω(abcd)} integer solutions

Generating Functional in CFT on Riemann Surfaces II: Homological Aspects

We revisit and generalize our previous algebraic construction of the chiral effective action for Conformal Field Theory on higher genus Riemann surfaces. We show that the action functional can be obtained by evaluating a certain Deligne cohomology class over the fundamental class of the underlying topological surface. This Deligne class is constructed by applying a descent procedure with respect to a \v{C}ech resolution of any covering map of a Riemann surface. Detailed calculations are presented in the two cases of an ordinary \v{C}ech cover, and of the universal covering map, which was used in our previous approach. We also establish a dictionary that allows to use the same formalism for different covering morphisms. The Deligne cohomology class we obtain depends on a point in the Earle-Eells fibration over the Teichm\"uller space, and on a smooth coboundary for the Schwarzian cocycle associated to the base-point Riemann surface. From it, we obtain a variational characterization of Hubbard's universal family of projective structures, showing that the locus of critical points for the chiral action under fiberwise variation along the Earle-Eells fibration is naturally identified with the universal projective structure.Comment: Latex, xypic, and AMS packages. 53 pages, 1 figur

Cognitive Control in Mathematics

The nature of mathematical theorizing underwent a dramatic transformation in the late 19th and early 20th centuries. Mathematicians are prone to describe this transformation by saying that mathematics became more 'conceptual' and that, consequently, we have come to enjoy more and better 'understanding' in mathematics. The purpose of my dissertation is to introduce a configuration of philosophical notions that allows us to analyze the epistemic significance of these changes. In order to arrive at such a configuration, I conduct a case study in which I compare two approaches to the solvability of polynomial equations by radicals, one characteristic of 19th century mathematics, another characteristic of 20th century mathematics. I use the pre-philosophically visible differences between the two approaches to motivate a new epistemological notion I call cognitive control. To have cognitive control over an epistemic process such as reading or writing a proof is to have epistemic guidance for the process in virtue of having an epistemic scaffolding. To have epistemic guidance at a given juncture in a process is to have a constellation of cognitive resources that allows one to represent the different possible ways of pursuing the process further; to have an epistemic scaffolding for a process is to have a suitably organized representation of the epistemically possible facts in the range of facts one has chosen to examine. I apply the notion of cognitive control to two proofs of the fact that there is no general formula for a solution by radicals for polynomial equations of degree 5, again one characteristic of 19th century mathematics, another characteristic of 20th century mathematics. I argue that we enjoy much better cognitive control over the process of reading the 20th century proof than we do over the process of reading the 19th century proof. This suggests that the epistemic significance of the said changes in the nature of mathematical theorizing consists, at least in part, in the circumstance that the conceptual resources of 20th century mathematics allow us to enjoy more and better cognitive control over the epistemic processes in mathematical research and learning