Uniform Volume Doubling and Functional Inequalities on Lie Groups

Explore the concept of uniform volume doubling and its implications for functional inequalities on Lie groups in this mathematics colloquium talk. Delve into how the volume doubling property on compact Lie groups with left-invariant Riemannian metrics can be used to prove important functional inequalities for the Laplacian, such as the Poincaré inequality and parabolic Harnack inequality. Examine the notion of uniformly doubling Lie groups and their significance in providing uniform bounds for constants in functional inequalities across all left-invariant metrics. Learn about the specific case of the special unitary group SU(2) and its uniform doubling property, including consequences for heat kernel estimates and Weyl counting functions. Discover recent progress on related results for SU(2)x\mathbb{R}^{n} and the measure contracting property (MCP) on SU(2), as presented by Masha Gordina from the University of Connecticut in this 58-minute lecture at Stony Brook University's Mathematics Department Colloquium.