Georgia Institute of Technology College of Sciences

James Curry will finish the discussion of arXiv:0911.3437

Exact Topic TBA. Talk will be a general survery of branched covers, possibly including covers from the algebraic geometry perspective. In addition we will look at branched coveres in higher dimensions, in the contact world, and my current research interests. This talk will be a general survery, so very little background is assumed.

We consider the statistical deconvolution problem where one observes $n$ replications from the model $Y=X+\epsilon$, where $X$ is the unobserved random signal of interest and where $\epsilon$ is an independent random error with distribution $\varphi$. Under weak assumptions on the decay of the Fourier transform of $\varphi$ we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators $f_n$ for the density $f$ of $X$, where $f: \mathbb R \to \mathbb R$ is assumed to be bounded. We then derive lower bounds for the minimax sup-norm risk over Besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds. We further show that linear estimators adapt to the unknown smoothness of $f$ if the Fourier transform of $\varphi$ decays exponentially, and that a corresponding result holds true for the hard thresholding wavelet estimator if $\varphi$ decays polynomially. We also analyze the case where $f$ is a 'supersmooth'/analytic density. We finally show how our results and recent techniques from Rademacher processes can be applied to construct global nonasymptotic confidence bands for the density $f$.

In this talk we will present some recent results about the matrix representation of the multiplication operator in terms of a basis of either orthogonal polynomials (OPUC) or orthogonal Laurent polynomials (OLPUC) with respect to a nontrivial probability measure supported on the unit circle. These are the so called GGT and CMV matrices.When spectral linear transformations of the measure are introduced, we will find the GGT and CMV matrices associated with the new sequences of OPUC and OLPUC, respectively. A connection with the QR factorization of such matrices will be stated. A conjecture about the generator system of such spectral transformations will be discussed.Finally, the Lax pair for the GGT and CMV matrices associated with some special time-depending deformations of the measure will be analyzed. In particular, we will study the Schur flow, which is characterized by a complex semidiscrete modified KdV equation and where a discrete analogue of the Miura transformation appears. Some open problems for time-depending deformations related to spectral linear transformations will be stated.This is a joint work with K. Castillo (Universidad Carlos III de Madrid) and L. Garza (Universidad Autonoma de Tamaulipas, Mexico).