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Ricci curvature for metric-measure spaces via optimal transport
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of theExpand
Notes on Perelman's papers
These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds".
Particle models and noncommutative geometry
Abstract We write three particle models in terms of noncommutative gauge theory: the Glashow-Weinberg-Salam model, the Peccei-Quinn model and the standard model.
Flat vector bundles, direct images and higher real analytic torsion
We prove a Riemann-Roch-Grothendieck-type theorem concerning the direct image of a flat vector bundle under a submersion of smooth manifolds. We refine this theorem to the level of differentialExpand
It is well known that there are relationships between the heat flow, acting on differential forms on a closed oriented manifold M, and the topology of M. From Hodge theory, one can recover the BettiExpand
Some geometric properties of the Bakry-Émery-Ricci tensor
Abstract The Bakry-Émery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive orExpand
Some Geometric Calculations on Wasserstein Space
We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.
Weak curvature conditions and functional inequalities
We give sufficient conditions for a measured length space (X, d,ν) to admit local and global Poincare inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X, d,ν) ,dExpand
Optimal transport and Perelman’s reduced volume
We show that a certain entropy-like function is convex, under an optimal transport problem that is adapted to Ricci flow. We use this to reprove the monotonicity of Perelman’s reduced volume.
We define topological and analytic indices in R/Z Ktheory and show that they are equal.