Complete finite volume hyperbolic structures, when they exist, are unique in dimensions 3 and higher. However, in dimension 3 there is a rich and well-studied theory of infinite volume and incomplete hyperbolic structures. In higher dimensions little is known, but we are beginning to see glimpses of a structure that is dramatically different from that in dimension 3. This talk will give an overview of the 3-dimensional theory and discuss some recent rigidity and flexibility results in higher dimensions. Joint work with Pete Storm.

Hurwitz numbers lie at the intersection of many interesting areas: combinatorics, TQFTs, commutative algebra, and enumerative algebraic geometry. In this talk, I will give some of the basic definitions, describe a handful of these different interpretations, and end with a brief description of my thesis work. This is a preview of and preparation for my thesis defense.

I will be following a paper from Ying-Qing Wu. Let F be a closed essential surface in a hyperbolic 3-manifold M with a toroidal cusp N. The depth of F in N is the maximal distance from points of F in N to the boundary of N. We will show that if F is an essential pleated surface which is not coannular to the boundary torus of N then the depth of F in N is bounded above by a constant depending only on the genus of F. The result is then used to show that an immersed closed essential surface in M which is not coannular to the torus boundary of M will remain essential in the Dehn filled manifold M( ) after excluding C(g) curves from the torus boundary of M, where C(g) is a constant depending only on the genus g of the surface F.

The talk will overview some recent mathematical contributions related to the modelling of randomness in materials. Modeling of solid materials will be specifically addressed. Several theoretical and numerical questions around stochastic homogenization will be overviewed. The talk is based on a series of joint works with X. Blanc, PL. Lions, F. Legoll, A. Anantharamann, R. Costaouec.

This week five of your undergraduates will be giving short talks. Their titles and abstracts can be found on the Math Club website: http://math.utexas.edu/mathclub

The quadratic residue codes are a family of codes that have historically been of interest due to their strong properties - they have transmission rate close to , and a square root lower bound on their minimal distance. In this talk we will define a new family of codes, which we will call th power residue codes, that will generalize the binary quadratic residue codes using higher powered prime residues over fields of the corresponding order. Some of the important properties of these new codes will be discussed, including a lower bound on their minimum distance. In addition, we construct a family of asymptotically bad th power residue codes. This is a PhD thesis defense.