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Mathematics & Statistics

Differential Geometry
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Friday, February 7, 2020


Convexity in the Heisenberg Group
Zachary Forrest
CMC 109


Convexity in the Euclidean setting plays an integral role in the study of homogeneous, proper, degenerate-elliptic partial differential operators of second-order (e.g. the Laplacian, infinity-Laplacian, \(p\)-Laplacian, and many more examples). Indeed, Euclidean convexity may be equivalently stated in terms of certain weak subsolutions of such equations. We will utilize these equivalencies in \(R^n\) as a starting point for inquiry into appropriate notions of convexity in the Heisenberg group and, time permitting, general Carnot groups. We will tour the work of Lu, Manfredi, and Stroffolini (2003) and Juutinen, Lu, Manfredi, and Stroffolini (2007), with particular emphasis on the relationships between the various notions of convexity explored in these papers.