Standard information-theoretic results show that data over small, typically binary, alphabets can be compressed to Shannon's entropy limit. Yet most practical sources, such as text, audio, or video, have essentially infinite support. Compressing such sources requires estimating probabilities of unlikely, even unseen, events, a problem considered by Laplace. Of existing estimators, an ingenious if cryptic one derived by Good and Turing while deciphering the Enigma code works best yet not optimally. Hardy and Ramanujan's celebrated results on the number of integer partitions yield an asymptotically optimal estimator that compresses arbitrary-alphabet data patterns to their entropy. The same approach generalizes Fisher's seminal work estimating the number of butterfly species and its extension authenticating a poem purportedly written by The Bard. The talk covers these topics and is self contained.

Joint work with Prasad Santhanam, Krishna Viswanathan, and Junan Zhang.

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