8 May 2013, 16:15
Abbeanum, lecture hall 4

The geometry of random polygons

Clayton Shonkwiler, PhD (Dept. of Mathematics, University of Georgia, USA)

Here is a natural question in statistical physics: what is the expected shape of a ring polymer with n monomers in solution? For example, what is the expected radius of gyration or total curvature? What is the likelihood of knotting? Numerical experiments are essential in this field, but pose some interesting geometric challenges since the space of closed n-gons in 3-space is a nonlinear submanifold of the larger space of open n-gons.

I will describe a natural probability measure on n-gons of total length 2 which is pushed forward from the standard measure on the Stiefel manifold of 2-frames in complex n-space using methods from algebraic geometry. We can directly sample the Stiefel manifold in O(n) time, which gives us a fast, direct sampling algorithm for closed n-gons via the pushforward map. We can also explicitly compute the expected radius of gyration and expected total curvature and even recover some topological information. This talk describes joint work primarily with Jason Cantarella (University of Georgia) and Tetsuo Deguchi (Ochanomizu University).