A first-order calculus for the Monge-Ampere quasi-metric structure and its applications to Analysis and PDEs

我们对瞬时变化率的数学理解源于牛顿和莱布尼茨在17世纪80年代发明的微积分，到现在为止，我们已经习惯了日常生活中的这一概念。例如，快速查看汽车的速度表可以让我们及时了解我们在任何给定时刻的速度，雷达或GPS系统可以告诉我们风暴接近的速度或飞机在任何给定时刻的飞行速度。 然而，在某些有趣的数学背景下，瞬时速度的概念不能被视为理所当然。这一概念需要以一种内在的方式发展，以保持我们习惯的自然属性和解释。这些性质中的一些可以以所谓的索伯列夫或变分不等式的形式表示，这些不等式将位置和速度在不同尺度上联系起来。在各种数学背景下追求瞬时速度的自然概念，以及它们的变分不等式和应用，是分析中越来越活跃和令人兴奋的研究领域。几何和测度论对象可以与凸函数相关联。这些对象的属性之间的相互作用模型的行为的解决方案，以某些椭圆和抛物偏微分方程，以及准保形的映射与凸势，几何特征的一些准度量空间，相关的潜在理论等的研究活动，由该补助金支持的目的是通过一阶微积分的基础上的变分不等式的Monge-安培准度量结构发展这样的主题。

Our mathematical understanding of instantaneous rates of change stems from the invention of calculus by Newton and Leibniz in the 1680's and, by now, we have become accustomed to that notion in our daily lives. For instance, a quick look at our car's speedometer lets us know our speed at any given moment in time and a radar or GPS system can tell us how fast a storm is approaching or an airplane is flying at any given moment in time. There are, however, certain interesting mathematical contexts in which the notion of instantaneous speed cannot be taken for granted. That notion needs to be developed in an intrinsic fashion as to retain the natural properties and interpretations we are used to. Some of those properties can be expressed in the form of the so-called Sobolev or variational inequalities which relate position and speed at different scales. The pursue of natural notions of instantaneous speed in various mathematical contexts, as well as their variational inequalities and applications, stands as an increasingly active and exciting area of research in Analysis.Geometric and measure-theoretic objects can be associated to convex functions. The interplay between properties of these objects models the behavior of solutions to certain elliptic and parabolic PDEs as well as quasi-conformality of mappings with convex potentials, geometric features of some quasi-metric spaces, associated potential theory, etc. The research activities supported by this grant aim at developing such topics by means of a first-order calculus based on variational inequalities for the Monge-Ampere quasi-metric structure.