Wavelet Methods in the Relativistic Three-Body Problem

In this paper we discuss the use of wavelet bases to solve the relativistic three-body problem. Wavelet bases can be used to transform momentum-space scattering integral equations into an approximate system of linear equations with a sparse matrix. This has the potential to reduce the size of realistic three-body calculations with minimal loss of accuracy. The wavelet method leads to a clean, interaction independent treatment of the scattering singularities which does not require any subtractions.Comment: 14 pages, 3 figures, corrected referenc

Applications of representation theory and K-theory to problems in algebraic number theory

In this thesis we consider three main problems: the Galois module structure of rings of integers in wildly ramified extensions of Q; Leopoldt's conjecture; and non-commutative Fitting ideals and the non-abelian Brumer-Stark conjecture. For each of these problems, which correspond to each main chapter, we will review and use tools from representation theory and algebraic K-theory. In the first main chapter, we will prove new results concerning the additive Galois module structure of certain wildly ramified finite non-abelian extensions of Q. In particular, when K/Q is a Galois extension with Galois group G isomorphic to A4, S4 or A5, we give necessary and sufficient conditions for the ring of integers OK to be free over its associated order in the rational group algebra Q[G]. In the second main chapter, we will work on Leopoldt's conjecture. Let p be a rational prime and let L/K be a Galois extension of number fields with Galois group G. Under certain hypotheses, we show that Leopoldt's conjecture at p for certain proper intermediate fields of L/K implies Leopoldt's conjecture at p for L; a crucial tool will be the theory of norm relations in Q[G]. We also consider relations between the Leopoldt defects at p of intermediate extensions of L/K. Finally, we will investigate new properties of (non-commutative) Fitting ideals in integral group rings, with the general idea of reducing to simpler abstract groups (such as abelian groups) that can emerge as subquotients. As an application we will provide a direct proof of the (non-abelian) Brumer-Stark conjecture in certain cases, by reducing to the abelian case as recently proved by Dasgupta and Kakde. The direct approach avoids use of technical machinery such as the equivariant Tamagawa number conjecture

The essential dimension of low dimensional tori via lattices

The essential dimension, ed_k(G), of an algebraic group G is an invariant that measures the complexity of G-torsors over fields. The correspondence between algebraic tori and finite subgroups of GL_n(Z) means one can establish bounds on ed_k(T) of an algebraic torus T by studying the action of the corresponding subgroup of GL_n(Z)

On the arithmetic of a family of degree-two K3 surfaces

Let $\mathbb{P}$ denote the weighted projective space with weights $(1,1,1,3)$ over the rationals, with coordinates $x,y,z,$ and $w$; let $\mathcal{X}$ be the generic element of the family of surfaces in $\mathbb{P}$ given by \begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$ is a K3 surface over the function field $\mathbb{Q}(t)$. In this paper, we explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with its Galois module structure, as well as derive more results on the arithmetic of $\mathcal{X}$ and other elements of the family $X$.Comment: 20 pages; v2 with some all additions and clarifications suggested by the refere