Georgia Institute of Technology College of Sciences

The main goal is to characterize the dilatation of an outer automorphisms of free groups. It is known that for any automorphism, its dilatation is a weak Perron number. The converse was recently shown by Thurston; for every weak Perron number, there is an automorphism represented by a train track map. We will discuss Thurston's theorem and its proof.

We will prove a duality between locally compact Hausdorff spaces and the C*-algebra of continuous complex-valued functions on that space. Formally, this is the equivalence of the opposite category of commutative C*-algebras and the category of locally compact Hausdorff spaces.

TBA

The main purpose of this talk is to better understand how to use branched covers to construct 3-manifolds. We will start with branched covers of 2-manifolds, carefully working through examples and learning the technology. Using these methods in combination with open book decompositions we will show how to construct 3-manifolds by branching over link and knots in S^{3}. Particular emphasis will be placed on using the map to get a "coloring" of the branched locus and how this combinatorial data is useful both for explicit constructions and for the general theory.