## Geometry Group 2019

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]. See references [1-10] below.

2. A regular hexagon is the least-perimeter tile of given area for the Euclidean plane. What about the hyperbolic plane? Since there is no scaling, this is a different problem for every given area.

3. Use the Surface Evolver to relax the conjectured least-area *n*-hedral 3D tiles [11].

**References**

[1] Frank Morgan, Manifolds with density, Notices Amer. Math. Soc. 52 (2005), 853-858, http://www.ams.org/notices/200508/fea-morgan.pdf

[2] Ivan Corwin, Neil Hoffman, Stephanie Hurder, Vojislav Sesum, Ya Xu (2004 Geometry Group), Differential geometry of manifolds with density, Rose-Hulman Und. Math. J. 7 (1) (2006), http://www.rose-hulman.edu/mathjournal/v7n1.php

[3] Colin Carroll, Adam Jacob, Conor Quinn, Robin Walters (2006 Geometry Group), The isoperimetric problem on planes with density, Bull. Austral. Math. Soc. 78 (2008), 177-197.

[4] Jonathan Dahlberg, Alexander Dubbs, Edward Newkirk, Hung Tran (2008 Geometry Group), Isoperimetric regions in the plane with density r^{p}, NY J. Math. 16 (2010), 31-51, http://nyjm.albany.edu/j/2010/16-4.html

[5] Alexander Díaz, Nate Harman, Sean Howe, David Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589–619; http://arxiv.org/abs/1012.0450 (2010); see blog posts http://sites.williams.edu/Morgan/2009/06/11/sobolev-type-inequality/ and http://sites.williams.edu/Morgan/2009/07/18/sectors-with-density-in-granada/.

[6] Frank Morgan, Geometric Measure Theory, Academic Press, 5th ed., 2016, Chapters 18 and 15.

[7] Frank Morgan, The log-convex density conjecture.

[8] Gregory R. Chambers, Proof of the log convex density conjecture, J. Eur. Math. Soc., to appear; arxiv.org.

[9] Leo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, The log convex density conjecture in hyperbolic space, Rose-Hulman Und. Math. J. 18 (2017), http://scholar.rose-hulman.edu/rhumj/vol18/iss1/9/; arxiv.org (2016).

[10] Leo Di Giosia, Jay Habib, Lea Kenigsberg, Dylanger Pittman, Weitao Zhu, Balls isoperimetric in **R**^{n} with volume and perimeter densities r^{m} and r^{k}, arxiv.org (2016).

[11] Paul Gallagher, Whan Ghang, David Hu, Zane Martin, Maggie Miller, Byron Perpetua, and Steven Waruhiu, Surface-area-minimizing *n*-hedral tiles, Rose-Hulman Und. Math. J. 15(1) (2014).

Part of Summer Undergraduate Mathematics Research at Yale, June 12—August 19, 2019. We may sometimes have activities evenings or weekends. July 13-27 we’ll go on a retreat to Ocean City, New Jersey, living and working together free from the usual distractions; the four of you will be provided with two bedrooms to share (each with two beds, or other options). We may also attend and speak at the Young Mathematicians’ Conference, usually in early August.