Activities This Week

A discrete set $Y$ in $R^d$ is called a dense forest if for every positive $\epsilon$, $Y$ is epsilon close to all line segments of length $V(\epsilon)$, for some function $V(\epsilon)$. We will discuss the intuition of this definition and the motivation for having such sets. Then I will present three constructions for dense forests by Bishop-Peres, S.-Weiss, and by Alon, that use basic Diophantine approximations, homogeneous dynamics, and the Lovasz local lemma, respectively. The focus will be on our result (jointly with Barak Weiss) for which I hope to give all the details of the construction. All the definitions and the background will be given in the talk.

This is joint work with Govind Menon, Sheehan Olver and Thomas Trogdon. The speaker will present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm

with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time. Two-component universality is observed for the fluctuations of the halting time, i.e., the

histogram for the halting times, centered by the sample average and scaled by the sample variance, collapses to a universal curve, independent of the input data distribution, as the dimension increases. Thus, up to two components,

the sample average and the sample variance, the statistics for the halting time are universally prescribed. The case studies include six standard numerical algorithms, as well as a model of neural computation and decision making.