Functional Inequalities in Global Analysis and Non-Communitative Geometry

通过NSF Proposal， Dr. Kemp打算详细研究两个泛函不等式，对数Sobolev不等式和Haagerup不等式，以及它们的子代半群收缩性，超收缩性和超收缩性。这些都是在过去30年里发展起来的内容丰富的学科，它们在热核分析、概率论、流形和李群分析、几何群论和算子代数（特别是自由概率）等方面有着巨大的重要应用。这两个函数不等式在经典和非交换分析环境中都产生了重大影响；每一个都在一个领域出现，最终在另一个领域得到应用（从全局分析到非交换几何的对数索博列夫不等式，从另一个方向的哈格鲁普不等式）。Kemp博士的研究主要集中在两个不等式的一些新的和意想不到的组合方面，特别是在全纯空间（经典和非交换）的分析设置。他打算通过在全局分析（西格尔-巴格曼空间、次调和函数空间和向量束截面空间）和非交换方面（主要是自由概率论）的几个项目来显著扩展他在这一主题中的思想。特别是，通过自由概率和随机矩阵理论之间的深层联系，肯普博士希望利用计算矩阵技术来解决与这些函数不等式的极值有关的一些更难的问题，并将他的一些结果扩展到目前的组合限制之外。这个国家科学基金会提案的主要目的是研究热流和熵的数学理论的一些重要方面。在最近的数学分析中，最有趣的想法之一是认识到热量在物体中的流动方式与物体的整体几何形状密切相关。例如，想象一个白热的大头针被放在一个大而冷的平面上。热量在平面上扩散得很快，然后更慢，使整个系统最终达到相同的温度。这不是热量流动的方式，例如，来自地球表面的大陨石撞击。地球的整体几何形状开始生效；撞击产生的能量在世界各地来回传播，以一种复杂的方式与自身相互作用。撞击后不久，热扩散与大头针相似，但过了一段时间，研究热流可以为了解地球的整体形状提供线索。(它是圆的!)研究热流并由此给出几何信息的一种方法是通过熵来表示系统无序的趋势。肯普博士研究中的一个重要定理——对数索博列夫不等式（logarithmic Sobolev inequality）断言，在大范围的几何环境中，熵的速率是由系统的总能量以一种与全局几何相关的精确方式控制的。

AbstractKempThrough this NSF Proposal, Dr. Kemp intends to engage in a detailed study of two functional inequalities, the logarithmic Sobolev inequality and the Haagerup inequality, and their descendant semigroup contraction properties, hypercontractivity and ultracontractivity. These are rich subjects, developed over the past 30 years, which have an enormous body of important applications: in heat kernel analysis, probability theory, analysis on manifolds and Lie groups, geometric group theory, and operator algebras (free probability in particular). Both functional inequalities have made significant impact both in classical and non-commutative analytic settings; each debuted in one realm and eventually found application in the other (the log Sobolev inequality moving from global analysis to non-commutative geometry, the Haagerup inequality in the other direction). Dr. Kemp's research focuses on some new and unexpected combinatorial aspects of both inequalities, particularly in the setting of analysis on holomorphic space (both classical and non-commutative). He intends to significantly extend his ideas in this subject through several projects both in global analysis (in Segal-Bargmann spaces, spaces of subharmonic functions, and spaces of sections of vector bundles) and on the non-commutative side, primarily in free probability theory. In particular, through the deep connections between free probability and random matrix theory, Dr. Kemp hopes to exploit computational matrix techniques to address some of the harder questions relating to the extremals of these functional inequalities, and extend some of his results beyond their current combinatorial limitations.The principal intent of this NSF Proposal is to study some important aspects of the mathematical theory of heat flow and entropy. One of the most interesting ideas in recent mathematical analysis is the realization that the way in which heat flows in an object is intimately related to the global geometry of the object. For example, think of a white-hot pin being placed on a large, cool plane. The heat will diffuse quickly at first, then more slowly, over the plane, bringing the whole system eventually to the same temperature. This is not the way heat flows, for example, from a large meteorite impact on the surface of the Earth. The global geometry of the planet comes into effect; the energy from the impact travels around the world and back again, interacting with itself in a complicated fashion. Shortly after the impact the heat diffusion is similar to the pin's, but after a while, studying the heat flow gives clues to the global shape of the planet. (It's round!) One way to study heat flow, and therefore give geometric information, is through entropy the tendency of systems to disorder. One important theorem in Dr. Kemp's research, the logarithmic Sobolev inequality, asserts that, in a wide range of geometric settings, the rate of entropy is controlled by the total energy of the system, in a precise manner which is tied to global geometry.