Tuesday, October 1, 2019
10/01/2019 - 4:00pm
Abstract: Consider a pressureless gas interacting through an attractive-repulsive potential given as a dierence of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and
If the attraction is not assumed to be strong, we show these congurations are at least local energy minimizers in the relevant d metric from optimal transportation, as are all of the other uncountably many unbalanced congurations with the same support.
An ingredient in the proof which may have independent interest is the establishment of a simple isodiametric variance bound which generalizes Popoviciu’s inequality from one to higher dimensions and characterizes regular simplices: it shows that among probability measures on Rn whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass 1/(n + 1) to each vertex of a (unit-diameter) regular simplex.
Based on preprint with Tongseok Lim at https://arxiv.org/abs/1907.13593
10/01/2019 - 4:15pm
A higher spin structure is a structure on a Riemann surface that assigns a ``winding number’’ to each isotopy class of simple closed curve. Higher spin structures appear in many problems involving families of Riemann surfaces, such as the study of smooth plane curves or strata of Abelian differentials. In this talk I will discuss the basic theory of higher spin structures, the subgroup of the mapping class group preserving such a structure, and some recent advances in the theory that lead to developments in algebraic geometry and the theory of translation surfaces. Portions of this work are joint with Aaron Calderon.