## Geometry Group 2019

Frank Morgan’s 2019 Geometry Group in SUMRY at Yale: Jack Hirsch, Kevin Li , Jackson Petty, Christopher Xue.

**Research Problems**

1. Perelman’s stunning proof of the million-dollar Poincaré Conjecture needed to consider not just manifolds but “manifolds with density” (like the density in physics you integrate to compute mass). Yet much of the basic geometry of such spaces remains unexplored. Recent results after Chambers ([8], [10]) show in various cases that if balls about the origin minimize perimeter for given volume if they are stable. Major open cases include hyperbolic space with radial density [9]. For a log-concave radial density such as e^{-1/r}, isoperimetric curves probably pass through the origin, like the isoperimetric circles for density r^{p} [4]. Double bubble in plane with density r^{p} would be interesting. See references [1-11] below.