Mathematics

On no-k-equal spaces

Yuliy Baryshnikov (UIUC Math/ECE)

Abstract: No-k-equal spaces are the spaces of (ordered) points with no k of them occupying the same position in a topological space X (for k=2 they are, of course, the "usual" configuration spaces). They became popular as a testing ground for lower bounds in TCS, but are interesting on their own right (if X is a vector space, the no-k-equal spaces are subspace arrangement manifolds, with a lot of machinery to understand them). In this talk I will survey what we know about the Betti numbers of these configuration spaces in simplest situations. In particular, we'll see that the exponential generating function for the Poincare polynomials for the no-k-equal spaces on the interval is \[ \sum_n P_n(t)\frac{z^n}{n!} =\frac{e^z}{1 +(-t)^{k-2}(e^z q_k(z) - 1)}. \] Some problems will be posed.

Coboundaries and ergodic sums.

Joe Rosenblatt (Illinois)

Abstract: The behavior of the norms of ergodic sums can be used to characterize coboundaries. But the behavior of the norms of ergodic sums can be fairly chaotic. Moreover, for a given function, which classes of transformations have that function as a coboundary is a complex issue. These types of things will be considered in some detail for general ergodic transformations of a probability space.

Uniqueness of group measure space decomposition for Popa's HT factors (after Adrian Ioana): Part I

Bogdan Udrea (UIUC Math)

Abstract: This is a first in a series of talks on deformation/rigidity theory. In particular, we focus first on Ioana's results on the HT factors introduced by Popa. Students may receive credit by registering for the seminar, which meets 5-6:30pm on Mondays.