Kallin's Lemma is a technical tool useful for proving the polynomial convexity of certain unions of polynomially convex compacts (such as the union of 3-balls). We show that this result extends to holomorphically convex compacts in a Stein manifold. We will define Kallin's Lemma, introduce the necessary background from the theories of uniform algebras and Stein manifolds, and prove the generalization.

This is a PhD Comprehensive Examination.

Speaker's web page: http://www.math.uwo.ca/index.php/profile/view/208/

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

In mathematics and science, we often need to think about high (3 or more) dimensional objects, called spaces, which are hard or impossible to visualize. Besides the question of what such objects are or could be, is the problem of how can we make sense of such spaces.

The goal of this discussion is to give you an idea of how mathematicians manage to make sense of higher-dimensional spaces. We will do this by exploring the simplest spaces, and through our explorations, we will begin to see how we may tell different spaces apart.