Results 1 
2 of
2
Small time heat kernel behavior on Riemannian complexes
"... SaloffCoste Abstract. We study how bounds on the local geometry of a Riemannian polyhedral complex yield uniform local Poincaré inequalities. These inequalities have a variety of applications, including bounds on the heat kernel and a uniform local Harnack inequality. We additionally consider the e ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
SaloffCoste Abstract. We study how bounds on the local geometry of a Riemannian polyhedral complex yield uniform local Poincaré inequalities. These inequalities have a variety of applications, including bounds on the heat kernel and a uniform local Harnack inequality. We additionally consider the example of a complex, X, which has a finitely generated group of isomorphisms, G, such that X/G = Y is a complex consisting of a finite number of polytopes. We show that when this group, G, haspolynomial volume growth, there is a uniform global Poincaré inequality on
Research Statement
"... My research focuses on analysis and probability on metric measure spaces. I am particularly interested in how the local (small scale) and global (large scale) geometries affect analysis on the space. My research seeks to discover and extend these connections. If the local geometry is sufficiently ni ..."
Abstract
 Add to MetaCart
(Show Context)
My research focuses on analysis and probability on metric measure spaces. I am particularly interested in how the local (small scale) and global (large scale) geometries affect analysis on the space. My research seeks to discover and extend these connections. If the local geometry is sufficiently nice, we can explore difficult questions about abstract spaces. For carefully chosen spaces, this could form the basis of undergraduate research projects. A few interesting questions are: Isoperimetry: What is the minimum size of the boundary of a set with volume V? How does this change as we vary V? Volume Doubling: Let B(x, r) be a ball of radius r centered at x. Does there exist a constant C such that: Volume(B(x, 2r)) ≤ CVolume(B(x, r))? If not, can we find a constant, Cr, that depends only on the radius? Can we find a constant, Cx, that depends only on the center? Poincaré Inequality: Let f be an L 2 function with L 2 derivatives. Does there exist a constant C (independent of f) such that: