I will discuss two closely related problems.
(1) Take integer lattice points in a large ball of radius R and project them onto the unit sphere S centered at x. It is well known that the sequence of projected points becomes uniformly distributed on S as R becomes large. I will show that, for every fixed x, the statistical correlation functions of this sequence have limiting distributions with some remarkable properties. The proof uses Ratner's classification of ergodic measures invariant under unipotent flows.
(2) The periodic Lorentz gas describes a point particle moving in a periodic array of spherical scatterers of radius r. I'll explain why the probability for a particle to hit the first scatterer after a fixed time T has a limiting distribution in the small scatterer limt r->0 and discuss some of its properties. This particular question was raised by Sinai in the early 1980s.
This is joint work with A. Strombergsson.

Inferences on the proportion of non-null effects in large-scale multiple comparisons

Natural color images captured by digital cameras often exhibit noticeable noise, particularly when the pictures are taken under low lighting or artificial lighting conditions. Traditional denoising techniques, which are often tested for removing artificial noise in monochromatic images, often do not work well for noisy color images.

In this talk, we present an overview of some of the traditional methods for denoising. We discuss a new strategy, which we call the cross-channel principle, that can be applied for very effective denoising of color images. In particular we show how this principle can be applied to the total variation denoising scheme and an ENO type denoising scheme.

Roughing it up: Disentangling Continuous and Jump Components in Measuring, Modeling and Forecasting Asset Return Volatility

On March 11, 1944, the famous Eremitani's Church in Padua (Italy) was destroyed in an Allied bombing together with the inestimable frescoes by Andrea Mantegna et al. contained in the Ovetari Chapel. In the last 60 years, several attempts have been made to restore the fresco fragments by traditional methods, but without much success. We have developed a fast, robust, and efficient pattern recognition algorithm in order to map the original position and orientation of the fragments, based on comparisons with an old gray level image of the fresco prior to the damage. This innovative technique allowed for the partial reconstruction of the frescoes. Unfortunately, the surface covered by the fragments is only 77 m^2, while the original area was of several hundreds. This means that we can currently reconstruct only a fraction (less than 8%) of this inestimable artwork. In particular the original color of the blanks is not known. This begs the question of whether it is possible to estimate mathematically the original colors of the frescoes by making use of the potential information given by the available fragments and the gray level of the pictures taken before the damage. Can one estimate how faithful such restoration is?

In this talk we retrace the development of the recovery of the frescoes as an inspiring and challenging real-life problem for the development of new mathematical methods. We introduce two models for the recovery of vector valued functions from incomplete data, with applications to the fresco recolorization problem. The models are based on the minimization of a functional which is formed by the discrepancy with respect to the data and additional regularization constraints. The latter refer to joint sparsity measures with respect to frame expansions for the first functional and functional total variation for the second. We establish the relations between these two models. As a byproduct we develop the basis of a theory of fidelity in color recovery, which is a crucial issue in art restoration.

We use the martingale method to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the way, we obtain bounds on certain martingale differences associated with the random sequences, which may be of independent interest. As an application of our result, we also derive a concentration inequality for inhomogeneous Markov chains, and establish an extremal property associated with their martingale difference bounds. This work complements certain concentration inequalities obtained by Marton and Samson, while also providing a different proof of some known results. Paper written with Kavita Ramanan. http://arxiv.org/abs/math.PR/0609835