Functional Inequalities in Global Analysis and Non-Communitative Geometry

肯普博士通过NSF的这个提案，打算对两个函数不等式，对数Sobolev不等式和Haagerup不等式，以及它们的后代半群压缩性质，超压缩性和超压缩性进行详细的研究。这些都是丰富的学科，在过去的30年中发展，其中有一个巨大的身体的重要应用：在热核分析，概率论，分析流形和李群，几何群论，算子代数（特别是自由概率）。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;每个不等式都在一个领域首次亮相，并最终在另一个领域找到了应用（对数Sobolev不等式从全局分析转向非交换几何，Haagerup不等式在另一个方向）。肯普博士的研究集中在一些新的和意想不到的组合方面的两个不等式，特别是在设置的分析全纯空间（古典和非交换）。他打算显着扩大他的想法在这个问题上通过几个项目都在全球分析（在西格尔-巴格曼空间，空间的次调和函数，空间的部分向量丛）和非交换方面，主要是在自由概率论。 特别是，通过自由概率和随机矩阵理论之间的深刻联系，肯普博士希望利用计算矩阵技术来解决一些与这些函数不等式的极值有关的更困难的问题，并将他的一些结果扩展到超出其当前的组合限制。这个NSF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流动的方式与物体的整体几何形状密切相关。例如，想象一个白热化的针被放置在一个大的，凉爽的飞机上。热量首先会迅速扩散，然后慢慢地扩散到整个平面上，最终使整个系统达到相同的温度。这不是热量流动的方式，例如，来自地球表面的大型陨石撞击。地球的全球几何学开始生效;撞击产生的能量在世界各地旅行并再次返回，以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，研究热流会为行星的整体形状提供线索。(It的圆！）研究热流并由此给出几何信息的一种方法是通过熵来研究系统的无序趋势。肯普博士研究中的一个重要定理，对数索伯列夫不等式，断言在广泛的几何设置中，熵的速率由系统的总能量控制，以一种精确的方式与全局几何相联系。

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.