Title: Configuration spaces of disks in an infinite strip
Speaker: Matt Kahle (OSU)
Abstract: (This is joint work with Bob MacPherson.) Configurations spaces of points, in $\mathbb{R}^2$ for example, are well studied in algebraic topology. We understand the homology, homotopy groups, etc., extremely well. But if the points have some finite thickness, and if the region of the plane is restricted, then much less is known. However this situation is quite natural from the point of view of physics---such configuration spaces describe the phase space or energy landscape of a hard spheres gas. As a case study, we investigate the configuration space of $n$ disks of unit radius in an infinite strip of width $w$. We are especially interested in the asymptotic behavior of the $k$th Betti number as $n \to \infty$, and we find two regimes. For some choices of $(w,k)$, the $k$th Betti number grows polynomially fast, and for some it grows exponentially fast. The line separating these two regimes might be considered a kind of phase transition. Our main results are asymptotic estimates for the Betti numbers in every case, correct up to a constant factor.
Seminar URL: https://research.math.osu.edu/topology/