Activities This Week

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.

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. The most intuitive example is to imagine blowing up a sphere so that eventually it becomes so large, it looks like a plane. This is a transition from spherical geometry to Euclidean geometry.

We will study limits of the Cartan subgroup in $SL(n,R)$. A limit group is the limit under a sequence of conjugations of the Cartan subgroup in $SL(n,R)$. We will show using the hyperreal numbers that in $SL(3,R)$ there are 5 limit groups, each determined by a degenerate triangle.

In the second part of the talk, we will show that for $n \geq 7$, there are infinitely many nonconjugate limit groups of the Cartan subgroup.