Analytic aspects of optimal transportation

项目的主要目标是研究最优交通理论中与最优映射或最优计划的各种限制有关的新问题的分析方面。所考虑的问题对数学和应用的许多领域具有重要意义。预期将取得下列结果：附加限制输运问题的Kantorovich对偶及其与遍历理论的关系研究。考虑鞅输运问题，一般线性限制，和一些其他类型的限制。由遍历分解生成的交通规划结构的研究。构造一个非交换的Monge-Kantorovich理论。能源测量空间及相关交通问题研究。无限维空间上最优运输理论的动力学问题研究。最优方案的变换。最优交通规划空间上非线性泛函分布绝对连续的条件。基于kantorovich度规的概率测度空间梯度流的最优控制。研究真实版本的Kähler-Einstein方程。一个可能的应用是得到凸体等周常数的最佳可能渐近界。计划研究Kähler-Einstein方程的先验估计和各种几何特征，特别是解的三阶导数的估计。在Wienerspace中图像测度与Wiener测度重合的无限维实数Kähler-Einstein方程的研究。计划证明在无限维情况下Kähler-Einstein方程解的存在性，其中最优输运由某一测度的对数梯度给出。最优运输势诱导的带有度量和黑森度量的流形研究。主要的预期应用是对等周常数和Sobolev常数的新估计，以及Kantorovich距离（输运不等式）的边界。有许多（多于两个）边际的运输问题。我们计划获得具有一维边际的运输问题的解的精确描述和成本函数是仿射函数的最小值。我们还计划将项目1中提到的一些结果推广到更多的边际。用最优运输方法得到新的凸体几何不等式。

The chief goal of the project is investigation of analytic aspects of new problems in thetheory of optimal transportation related to diverse restrictions on optimal mappings oron optimal plans. The considered problem have significant importance for a number ofareas of mathematics and applications. It is envisaged to obtain the following results.1. A study of the Kantorovich duality for transport problems with additional restrictionsand its relations to ergodic theory. Consideration of martingale transport problems,general linear restrictions, and some other types of restrictions. Investigation of the structureof transport plans generated by ergodic decompositions.2. Constructing a noncommutative Monge-Kantorovich theory.3. A study of energy measure spaces and related transportation problems.4. Investigation of dynamical problems of the theory of optimal transportation oninfinite-dimensional spaces. Transformations of optimal plans. Conditions for the absolutecontinuity of the distributions of nonlinear functionals on spaces with optimaltransportation plans.5. Optimal control of gradient flows in the space of probability measures with theKantorovich metric.6. Investigation of the real version of the Kähler-Einstein equation. One of possibleapplications is obtaining best possible asymptotic bounds for isoperimetric constants ofconvex bodies. It is planned to study a priori estimates and various geometric characteristicsof the Kähler-Einstein equation, in particular, estimates for the third order derivativesof solutions.7. Investigation of the infinite-dimensional real Kähler-Einstein equation on the Wienerspace, where the image measure coincides with the Wiener measure. It is planned to provethe existence of a solution to the Kähler-Einstein equation in the infinite-dimensional case,where the optimal transport is given by the logarithmic gradient of a certain measure.8. Investigation of manifolds equipped with measures and Hessian metrics inducedby potentials of optimal transports. The main expected applications are new estimatesof isoperimetric constants and Sobolev constants, bounds for the Kantorovich distance(transport inequalities).9. The transport problem with many (more than two) marginals. We plan to obtain aprecise description of solutions to the transport problem with one-dimensional marginalsand the cost function that is the minimum of affine functions. We also plan to generalizesome results mentioned in item 1 to a larger number of marginals.10. Obtaining new geometric inequalities for convex bodies by means of optimal transportation.