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Monthly Archives: September 2014
Notes of Karl-Theodor Sturm’s IHP 2014 lecture
Metric measure spaces with synthetic Ricci curvature bounds: state of the art 1. Requirements and definitions Equivalent to in Riemanian case Stable under convergence Intrinsic, synthetic. Bibliography Sturm, Lott Villani Sturm and many coauthors Ambrosio-Gigli-Savaré Erbar-Kuwada-Sturm : equivalence of Eulerian … Continue reading
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Notes of Andrea Bonfiglioli’s lecture
Maximal principles and Harnack inequalities for PDO’s in divergence form 1. Motivation CR geometry (sub-Laplacians), stochastic PDE’s. 2. Introduction 2.1. Standing assumptions Total nondegeneracy. Smooth hypoellipticity. Sometimes, we require that is hypoelliptic as well. Or even existence of a global, … Continue reading
Pierre Pansu’s slides on Differential Forms and the Hölder Equivalence Problem
Here is the completed set of slides CIRMsep14_beamer If you want to know more about the construction of horizontal submanifolds and how Gromov uses it to bound Hausdorff dimensions from below, see Pansu’s Trento notes (2005).
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Notes of Anton Thalmaiers’s lecture nr 4
1. Probabilistic content of Hörmander’s condition 1.1. Statement Theorem 1 Suppose that the Lie algebra generated by and brackets fills . Then the bilinear form on is non-degenerate 1.2. Proof Let By Blumenthal’s 0/1-law, is not random. We prove by … Continue reading
Notes of Anton Thalmaier’s lecture nr 3
1. Stochastic flows of diffeomorphisms We continue our study of SDE . Up to now, the starting point was fixed. Now we exploit the dependance on . 1.1. Random continuous paths of diffeomorphisms Let us introduce the random set of … Continue reading
Notes of Nicola Garofalo’s lecture nr 4
1. The isoperimetric problem I want to show how PDE results can be used to solve geometric problems. 1.1. The isoperimetric inequality I will prove the isoperimetric inequality in Carnot groups, It has lots of applications, see the conference in … Continue reading
Notes of Ludovic Rifford’s lecture nr 4
Open problems The Sard conjecture Regularity of geodesics Small balls 1. The Sard conjecture 1.1. Statement Theorem 1 (Morse 1939 for , Sard 1942) If is of class , and this is sharp (Whitney). Does this theorem generalize to the … Continue reading
Notes of Nicola Garofalo’s lecture nr 3
1. Fundamental solutions Exercise (related to the Hopf-Rinow): compute the sub-Riemannan metric associated to vectorfield . Observe that balls are non compact, i.e. metric is not complete. 2. Existence Theorem 1 (Folland) On a Carnot group, all sub-Laplacians have a … Continue reading
Notes of Ludovic Rifford’s lecture nr 3
1. A closer look at singular curves Today’s lecture will be full of examples. 1.1. Singularity criterion Remember that when concatenating curves, if one of them is regular, the full curve is as well. So if a curve is singular, … Continue reading
Pansu’s slides, september 1-5, 2014
Here are Pansu’s slides (version sept. 3rd, 00:13) CIRMsep14_beamer
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