Georgia Institute of Technology College of Sciences

We consider a variant of Rudin--Osher--Fatemi variational image smoothing that replaces the BV semi-norm in the penalty term with the B^1_\infty(L_1) Besov space semi-norm. The space B^1_\infty(L_1$ differs from BV in a number of ways: It is somewhat larger than BV, so functions inB^1_\infty(L_1) can exhibit more general singularities than exhibited by functions in BV, and, in contrast to BV, affine functions are assigned no penalty in B^1_\infty(L_1). We provide a discrete model that uses a result of Ditzian and Ivanov to compute reliably with moduli of smoothness; we also incorporate some ``geometrical'' considerations into this model. We then present a convergent iterative method for solving the discrete variational problem. The resulting algorithms are multiscale, in that as the amount of smoothing increases, the results are computed using differences over increasingly large pixel distances. Some computational results will be presented. This is joint work with Greg Buzzard, Antonin Chambolle, and Stacey Levine.

This series of talks will be an introduction to the use of holomorphic curves in geometry and topology. I will begin by stating several spectacular results due to Gromov, McDuff, Eliashberg and others, and then discussing why, from a topological perspective, holomorphic curves are important. I will then proceed to sketch the proofs of the previously stated theorems. If there is interest I will continue with some of the analytic and gometric details of the proof and/or discuss Floer homology (ultimately leading to Heegaard-Floer theory and contact homology).

We develop a nonparametric test to check whether the underlying continuous time process is a diffusion, i.e., whether a process can be represented by a stochastic differential equation driven only by a Brownian motion. Our testing procedure utilizes the infinitesimal operator based martingale characterization of diffusion models, under which the null hypothesis is equivalent to a martingale difference property of the transformed processes. Then a generalized spectral derivative test is applied to check the martingale property, where the drift function is estimated via kernel regression and the diffusion function is integrated out by quadratic variation and covariation. Such a testing procedure is feasible and convenient because the infinitesimal operator of the diffusion process, unlike the transition density, has a closed-form expression of the drift and diffusion functions. The proposed test is applicable to both univariate and multivariate continuous time processes and has a N(0,1) limit distribution under the diffusion hypothesis. Simulation studies show that the proposed test has good size and all-around power against non-diffusion alternatives in finite samples. We apply the test to a number of financial time series and find some evidence against the diffusion hypothesis.

In this talk we will outline proof due to Plameneveskaya and Van-Horn Morris that every virtually overtwisted contact structure on L(p,1) has a unique Stein filling. We will give a much simplified proof of this result. In addition, we will talk about classifying Stein fillings of ($L(p,q), \xi_{std})$ using only mapping class group basics.