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The classical isoperimetric inequality in Euclidean space is equiva-
lent to the statement that spheres provide unique enclosures of least

perimeter for any given volume. In this talk we sketch some of the ideas

of the proof that this inequality also holds in complete simply con-
nected spaces of nonpositive curvature, known as Cartan-Hadamard

manifolds. Immediate applications include sharp extensions of the
Sobolev and Faber-Krahn inequalities to Cartan-Hadamard manifolds.

The essential step in the proof is to show that the total positive Gauss-
Kronecker curvature of any closed hypersurface embedded in a Cartan-
Hadamard manifold Mn

, n ≥ 2, is bounded below by the volume of the

unit sphere in Euclidean space Rn

. Our starting point is a comparison
formula for the total curvature of level sets in Riemannian manifolds.
This is joint work with Mohammad Ghomi.

Abstract: The Ribe program is the research program concerned with generalizing local properties of Banach spaces to biLipschitz invariant properties of metric spaces. Among such generalizations that have been found is the notion of Markov p-convexity, proven by Mendel-Naor to generalize uniform p-convexity. One of the first important spaces for which this invariant has been calculated is the Heisenberg group, proven by Li to be Markov p-convex for every p at least 4 and not Markov p-convex for any p less than 4. In this talk, we'll start with background on the Ribe program and applications to metric space embedding theory, and then introduce the filiform groups - a class of Carnot groups containing the Heisenberg group - and explain how to use embeddings of graphs to compute their Markov convexities.

Join us for a talk by former local Logic Group member Dave Ripley (Monash)!

https://logic.uconn.edu/calendar/

In this talk I will discuss joint work with Pierre Albin in which we study the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the eta-invariant and the determinant of the Laplacian. Time permitting, I will discuss connections with the Rumin complex, and the associated limit of Analytic Torsion.

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