English version (共催：男女共同参画室)

日時:

2013年3月6日(水)-8日（金）

場所:

名古屋大学大学院多元数理科学研究科 A館３２８セミナー室

講演者:

Nathael GOZLAN氏(University Paris-Est Marne-la-Vallée,准教授)

桑田和正(お茶の水女子大学 人間文化創成科学研究科 理学専攻 数学コース,准教授)

船野敬(京都大学 理学部数学教室,助教)

対象:

理系研究者

講演日程:

3月6日(水)

10:30-12:00 Gozlan

題目：Concentration of measure: basic properties, examples and applications.

概要：In these first two lectures, we introduce the basic notions related to the concentration of measure phenomenon. In particular we will recall and prove the Prekopa-Leindler inequality and derive from it the concentration properties of Gaussian measures. We will also recall some celebrated applications of concentration of measure such as the Dvoretzky theorem.

14:00-15:30 Gozlan

題目：Concentration of measure: basic properties, examples and applications (II).

16:00-17:30 船野

題目：Concentration, separation, and eigenvalues of Laplacian

概要：In this talk, I will discuss an application of concentration of measure to eigenvalues of Laplacian on (weighted) manifolds of non-negative Ricci curvature.

3月7日(木)

10:30-12:00 Gozlan

題目：title： From functional inequalities to concentration of measure.

概要：In this lecture we introduce classical functional inequalities related to concentration of measure : Poincaré inequality, logarithmic Sobolev inequality and Talagrand transport-entropy inequality. We prove the Bakry-Emery criterion using an optimal transport method and we state necessary and sufficient conditions for these inequalities. In particular, we will present a new necessary and sufficient condition for Talagrand’s inequality in dimension one.

14:00-15:30 Gozlan

題目：From dimension free concentration of measure to functional inequalities.

概要：This lecture is devoted to a recent characterization of dimension free concentration. We will prove using classical limit theorems of probability theory (Sanov’s theorem and the central limit theorem) that Gaussian dimension free concentration is equivalent to Talagrand’s T2 inequality. In the same vein, we will see that (exponential) dimension free concentration is equivalent to Poincare ́ inequality.

16:00-17:30 自由討論

18:00- 懇親会 ※懇親会参加予定者は3月1日(金)までに高津(takatsu'at'math.nagoya-u.ac.jp)までご連絡下さい。

3月8日(金)

10:30-12:00 Gozlan

題目：Around Otto-Villani Theorem.

概要：This lecture is devoted to a celebrated theorem by Otto and Villani stating that the logarithmic Sobolev inequality implies Talagrand’s T2 inequality. We will recall a proof based on Hamilton-Jacobi equations and give another one using concentration arguments. Then we will prove a partial converse to this theorem showing that T2 is equivalent to a log-Sobolev in- equality restricted to a class of functions. Thanks to this result we will prove the stability of dimension free concentration under bounded perturbations.

14:00-15:30 Gozlan

題目： From concentration of measure to functional inequalities under curvature assumptions.

概要： In this lecture we will present a series of results due to Wang and E. Milman showing that concentration implies functional inequalities under the assumption that the curvature of the manifold is bounded from below. We will present a recent attempt to generalize this type of results on metric spaces whose curvature is bounded from below in the sense of Lott-Sturm-Villani.

16:00-17:30 桑田

題目：Extensions of the duality in gradient estimates and Wasserstein controls

概要：It is known that there is a duality between a Lipschitz control of Wasserstein distances for Markov kernels and Bakry-Emery type gradient estimates. We extend it in two different respects. The one relaxes assumptions. As a result, gauge-Orlicz type duality holds for an arbitrary Markov kernel on any Polish extended geodesic metric space. The other deals with one parameter family of Markov kernels. It includes the result on Feng-Yu Wang's gradient estimate associated with Bakry-Emery's curvature-dimension condition with a finite dimentional upper bound.