The Hawking- Penrose singularity theorems tell us that cosmological solutions of Einstein's equations generically contain a singularity. But these theorems tell us little about what happens near such a singularity. Do the gravitational fields necessarily grow without bound? Can causality break down? What about the Cosmic Censor? Work done during the past ten years--both analytical and numerical--has gotten us a lot closer to answers to these questions. We survey this work, discussing both the mathematical ideas and the physical implications. We also discuss the likely direction of future studies.

We study randomness in the automorphism group of the binary tree of depthn and its generalizations. These groups have an important role in grouptheory, and they also arise in connection with complex dynamics, fractalsand finite automata. We use branching processes to determine theasymptotic order of a random element, answering an old question of Turan.We show that three random elements generate a large subgroup with highprobability, leading to the solution of a problem of Shalev.This is joint work with M. Abert.

In Politis (2002) a method of tail index estimation for heavy-tailed time series, based on examining the growth rate of the logged sample second moment of the data, was proposed and studied. This estimator has a slow rate of convergence to the tail index, which is due to the high dependence of the summands of the statistic. To ameliorate the convergence rate, this work proposes an estimator with reduced bias, computed over subblocks of the whole data set. The resulting estimator obtains a polynomial rate of consistency for the tail index, and in simulation studies shows itself decidedly superior to competing prior art, such as the above-mentioned estimator of Politis (2002), as well as the reknowned Hill estimator.

Prof. Craw is a Recruitment candidate