Weak curvature conditions and functional inequalities

We give sufficient conditions for a measured length space (X,d,m) to admit local and global Poincare inequalities. We first introduce a condition DM on (X,d,m), defined in terms of transport of measures. We show that DM, along with a doubling condition on m, implies a scale-invariant local Poincare inequality. We show that if (X,d,m) has nonnegative N-Ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant 2^N. The condition DM is preserved by measured Gromov-Hausdorff limits. We then prove a Sobolev inequality for measured length spaces with N-Ricci curvature bounded below by K>0. Finally, we prove a sharp global inequality.Comment: final versio

Riemann, Boltzmann and Kantorovich Go to a Party

Presented on April 19th, 2013 from 4:00 pm - 5:15 pm in Room 1116 of the Klaus Building on the Georgia Tech campus.Dr. Cedric Villani, director of the Henri Poincare Institute in Paris, is a French mathematician working primarily on partial differential equations and mathematical physics. He was awarded the Fields Medal in 2010 for his work on Landau damping and the Boltzmann equation. His main research interests are in kinetic theory (Boltzmann and Vlasov equations and their variants), and optimal transport and its applications, a field in which he wrote two reference books: Topics in Optimal Transportation (2003); Optimal Transport, Old and New (2008).Runtime: 62:52 minutesThis talk is the story of an encounter of three distinct fields: non-Euclidean geometry, gas dynamics and economics. Some of the most fundamental mathematical tools behind these theories appear to have a close connection, which was revealed around the turn of the 21st century, and has developed strikingly since then

Mathematics is the poetry of science

In the words of the great poet Senghor, Cedric Villani makes the bold claim that Mathematics is the Poetry of Science. Perhaps paradoxical to some, both disciplines are concerned with describing the world around us, understanding its parts, and using this knowledge to create something profound. World-renowned mathematician and Fields Medallist Cedric Villani explores this analogy in this engaging and intelligent text, and shows how mathematics, one of the world's few universal languages, holds deep similarities to the literary genre. A great lover of poetry, he insists that the two are intrinsically linked in their aim of both tackling the complexities of our reality as well as distancing us from it so that we may better appreciate its beauty. In a more light-hearted and concise approach than his more theoretical academic works, this book represents one of Villani's attempts to communicate his love of mathematics to a wider audience, drawing daring parallels between two universes that meet in their aspiration of the sublime

Lectures given at the C.I.M.E. Summer School

Leading researchers in the field of Optimal Transportation, with different views and perspectives, contribute to this Summer School volume: Monge-Ampère and Monge-Kantorovich theory, shape optimization and mass transportation are linked, among others, to applications in fluid mechanics granular material physics and statistical mechanics, emphasizing the attractiveness of the subject from both a theoretical and applied point of view. The volume is designed to become a guide to researchers willing to enter into this challenging and useful theory

A global analysis of climate-relevant aerosol properties retrieved from the network of Global Atmosphere Watch (GAW) near-surface observatories

Aerosol particles are essential constituents of the Earth's atmosphere, impacting the earth radiation balance directly by scattering and absorbing solar radiation, and indirectly by acting as cloud condensation nuclei. In contrast to most greenhouse gases, aerosol particles have short atmospheric residence times, resulting in a highly heterogeneous distribution in space and time. There is a clear need to document this variability at regional scale through observations involving, in particular, the in situ near-surface segment of the atmospheric observation system. This paper will provide the widest effort so far to document variability of climate-relevant in situ aerosol properties (namely wavelength dependent particle light scattering and absorption coefficients, particle number concentration and particle number size distribution) from all sites connected to the Global Atmosphere Watch network. High-quality data from almost 90 stations worldwide have been collected and controlled for quality and are reported for a reference year in 2017, providing a very extended and robust view of the variability of these variables worldwide. The range of variability observed worldwide for light scattering and absorption coefficients, single-scattering albedo, and particle number concentration are presented together with preliminary information on their long-term trends and comparison with model simulation for the different stations. The scope of the present paper is also to provide the necessary suite of information, including data provision procedures, quality control and analysis, data policy, and usage of the ground-based aerosol measurement network. It delivers to users of the World Data Centre on Aerosol, the required confidence in data products in the form of a fully characterized value chain, including uncertainty estimation and requirements for contributing to the global climate monitoring system.Peer reviewe