Grant: Discovery Grants Program - Individual

The first topic to be considered is the essential dimension of moduli stacks of bundles. The dimension of a geometrical object is an important invariant. Stacks are an important generalisation of ordinary spaces that arise naturally when considering parameter spaces for objects. For example the most natural parameter ""space"" for curves is in fact not a space but a stack. One may view stacks as a modern theory of symmetries. For stacks there are two notions of dimension. Classical dimension, can be computed via methods from deformation theory that were developed many decades ago. The second notion of dimension, essential dimension, turns out to be a far more subtle invariant. It is the center of much intensive work and interesting conjectures. In our research we are interested in studying the essential dimension of certain stacks parametrising bundles over algebraic curves. The second topic in this work is to study the category of Nori finite parabolic bundles in characteristic zero on the projective line minus three points. In mathematics, the collection of symmetries of an object is called a group. One can consider the collection of symmetries of numbers with certain properties, or more precisely the absolute Galois group of the rational numbers. This group is large, interesting and unfortunately a concrete description of it is not within our grasp.The collection of algebraic covers of the line punctured at 0 and 1 has a very interesting action of this group. We would like to try and shed some light on this action. Covers correspond to subgroups of the free group on two letters and can be studied by the universal cover of the punctured line. The universal cover is the ancestor of all covers. The problem with the universal cover is that because it is topological in nature it is difficult to describe the Galois action on its finite quotients. We proceed by introducing a proxy for the universal cover called the category of Nori finite parabolic bundles on the line. The Galois group action on this side is more transparent but this category has yet to be described. Describing this category is the most immediate goal of this research.