I am going to discuss the evolution of compact convex curves in two-dimensional space forms under an inverse curvature type flow, first proposed by Brendle, Guan and Li.

Consider families of circular means where the radii are restricted to a given subset of a compact interval. One is interested in the $L^p$ improving estimates for the associated maximal operators and related objects.

We will introduce the probabilistic notion of strong convergence, discuss the basics of rough path theory, discover how we can acquire pathwise estimates of RSSDEs, and learn how these pieces fit together to establish convergence.

This seminar will present an algorithm for decomposing immersions satisfying a weaker inequality into a finite number of handles using a surgically modified (high codimension) mean curvature flow.