On the approximate parametrization problem of algebraic curves.

The problem of parameterizing approximately algebraic curves and surfaces is an active research field, with many implications in practical applications. The problem can be treated locally or globally. We formally state the problem, in its global version for the case of algebraic curves (planar or spatial), and we report on some algorithms approaching it, as well as on the associated error distance analysis

Restricted Hausdorff spectra of qq-adic automorphisms

We solve two well-known open problems on the Hausdorff dimension of branch groups. Firstly, we completely determine the self-similar Hausdorff spectrum of the group of $q$-adic automorphisms where $q$ is a prime power, answering a question of Grigorchuk. Indeed, we take a further step and completely determine its Hausdorff spectra restricted to the most important subclasses of self-similar groups, providing examples differing drastically from the previously known ones in the literature. Our proof relies on a new explicit formula for the computation of the Hausdorff dimension of closed self-similar groups and a generalization of iterated permutational wreath products. Secondly, we provide the first examples of just infinite branch pro-$p$ groups with trivial Hausdorff dimension, disproving a well-known conjecture of Boston which has been open for over two decades.Comment: 31 page

A Geometrical Construction of Rational Boundary States in Linear Sigma Models

Starting from the geometrical construction of special Lagrangian submanifolds of a toric variety, we identify a certain subclass of A-type D-branes in the linear sigma model for a Calabi-Yau manifold and its mirror with the A- and B-type Recknagel-Schomerus boundary states of the Gepner model, by reproducing topological properties such as their labeling, intersection, and the relationships that exist in the homology lattice of the D-branes. In the non-linear sigma model phase these special Lagrangians reproduce an old construction of 3-cycles relevant for computing periods of the Calabi-Yau, and provide insight into other results in the literature on special Lagrangian submanifolds on compact Calabi-Yau manifolds. The geometrical construction of rational boundary states suggests several ways in which new Gepner model boundary states may be constructed.Comment: 45 pages, 8 Postscript figures, LaTeX2e. v2: the construction reproduces a larger set of CFT boundary states; clarified discussion of instanton contributions and moduli; other minor improvements; references added . v3: version accepted for publication in Nuclear Physics B (minor changes

Tight Probability Bounds for Hausdorff Random Variables with Applications to Optimal Cancer Radiotherapy Treatment Design

The goal of this thesis is to examine tight computable bounds on a probability measure generated by its moments. We study measures supported on a real line and propose an extension of the classical moment problem to the so-called rational moments. Specifically, we examine semidefinite and linear optimization formulations for solving the univariate rational Hausdorff moment problem given a vector of moments. We further investigate shifted moments to reduce the distance between probability bounds, and propose a numerical method to better position such shifts. In addition, when only a few raw and shifted moments are known, we derive novel extensions of the Markov and Chebyshev bounds. Motivated by the problem of optimal radiotherapy treatment design, we present a novel and first in its class cutting plane method to be included within the mixed integer branch and cut scheme. Implementing these cuts results in as much as a 40% runtime reduction