I will outline Ozsvath and Szabo's approach to investigating whether a knot has unknotting number one. I will then describe a generalisation of this to higher unknotting numbers, which is based on a refinement of Montesinos' surgery description of the branched double cover of a knot.

I will talk about a method developed by Xavier Cabr'e, Ernest Fontich, and Rafael De la Llave used to prove existence of invariant manifolds and, at the same time to find simple polynomial maps which are conjugated to the dynamics on them. This method sugests a numerical algorithm, so numerical results will be shown.

Given a finite collection of hyperplanes in a complex vector space, the complement has a rich and interesting topology. One of the central questions in the study of hyperplane arrangements is to determine to what extent the combinatorics of the arrangement can tell you about the topology of the complement. I will give an introduction to this problem, discussing the classical results as well as some of my own work.

The study of continued fractions is a classical subject in mathematics dating to the 1650's. We have been working with a relatively new collection of functions which encodes information about the continued fraction expansions of real numbers. The usefulness of these functions in proving theorems about continued fractions lies in the fact that they also form a complete system of martingale differences. In this talk we will make all of these statements precise and we will exhibit some of the main theorems which we have proved as a result of the martingale property. This is a thesis defense.

I will show how to construct determinant and pfaffian line bundles for families of Dirac operators and demonstrate how functional information is thus geometrically encoded. If time permits, I will also briefly discuss applications to physics and possible extensions of the constructions.

We explore the connections between multiparameter BMO spaces and iterated commutators with Singular integral operators such as Riesz transforms. The main tools will be dyadic operators, so called Haar shifts, projections onto cones to make up for the loss of analyticity in one variable and very general paraproducts to help with an involved bootstrapping machinery to dominate BMO by the commutators. (This part misses detail still, so there will have to be another talk in the fall).

If the fundamental group of a three manifold M acts on a tree, under certain mild conditions on the action (most importantly, that it is nontrivial) one may associate to it an essential surface in M. I will explain how to do this.

This is a thesis defense.

I will present a utility-based methodology for the valuation and hedging of mortgage-backed securities (MBS) subject to totally unpredictable prepayment risk. Incompleteness stems from the cash flow pattern of the security which is affected by its embedded prepayment option. We construct the prepayment time for a mortgage contract by its hazard rate, either deterministic or stochastic. The valuation problem for a specific mortgage pool is also considered. This is my candidacy talk.

Motivated by the construction made by Goppa on curves, we present some error-correcting codes on algebraic surfaces. A surface whose Neron-Severi group has rank 1 has a "nice" intersection property that allows us the construction of a good code. We will verify this on specific examples. Surfaces with many points and rank 1 are not easy to find. We were able, though, to find also surfaces with low rank and many points, and these gave us good codes too. Finally, we present a decoding algorithm for these codes. It is based on the realization of the code as a LDPC code, and it was inspired by the Luby-Mitzenmacher algorithm. (thesis defense)

A class of discrete auditory transforms are constructed by incorporating ear properties in the discrete Fourier transform. The transforms preserve energy, and are invertible either mathematically or perceptually. The resulting amplitude spectrum is smoother than Fourier spectrum towards high frequencies. Applications to sound (speech and music) signal processing will be shown.

For a compact set A in Euclidean space we consider the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/t^s) over all N-point subsets of A, where s>0. For a class of rectifiable sets A having finite, positive d-dimensional Hausdorff measure, we have determined the dominant term in the minimal energy and shown that such minimizing configurations have asymptotic limit distribution (as N tends to infinity with s fixed) equal to d-dimensional Hausdorff measure whenever s>d or s=d. This result is used to deduce the asymptotic behavior of "best-packing points" on A; that is, N points of A for which the minimal pairwise distance is as large as possible.

Defense