Heat Kernel Analysis

这项研究致力于涉及有限维和无限维热方程的问题。需要解决以下问题。1.关于有限维亚椭圆型热方程的微分型不等式的存在性，包括对数Sobolev和Poincare(或质量间隙)型不等式的存在性。2.黎曼流形的路空间和圈空间上与热核测度和Wiener测度有关的微分不等式的存在性；3.紧致黎曼流形的Loop空间上的热核测度和Pinted Wiener测度之间的关系。4.类Fock空间表示从有限维群到无限维循环群的推广。5.将某些有限维近似推广到Wiener度量，以包括物理学家使用的“超对称量子力学”设置。这项研究中要研究的大多数问题都是由研究多个领域的问题引起的，这些领域是弯曲金属板中的热传播、量子力学、量子场论和概率论的某些方面。事实证明，这些看似不同的领域共享一个共同的数学描述，即“抛物型偏微分方程式”或等价的“布朗运动”的概率理论。在大多数有趣的(物理上相关的)情况下，很少可能找到要由P.I.研究的复杂偏微分方程解的显式解。然而，经常可能发现解的有趣和相关性质。抛物型偏微分方程解在等待足够长的时间后趋于稳态值是抛物型偏微分方程的一个典型现象。例如，如果金属板以不均匀的方式加热，然后不受影响，板中的热量会随着时间的推移重新分配，从而使整个板的温度达到恒定的平衡值。与“庞加莱”和“对数索波列夫”不等式有关的一个基本问题是，本研究中要研究的抛物型偏微分方程所描述的系统收敛到其平衡值的速度有多快。同样的收敛速度问题在描述相对论(即运动速度接近光速)基本粒子的量子场论的背景下有另一种解释。对于这些理论，庞加莱不等式的存在有一个令人满意的解释，即数学理论并不预测不合理的过多基本粒子。抛物线方程和相关的量子力学Shrodinger方程的粒子解释将是另一个研究重点。粒子解释在第一段提到的福克空间表示的标题下进行。

This research is devoted to questions involving heat equations in both finite and infinite dimensional contexts. The following issues are to be addressed. 1. The existence of differential type inequalities related to hypoelliptic heat equations in finite dimensions, including the existence of logarithmic Sobolev and Poincare (or mass gap) type inequalities. 2. The existence of differential inequalities related to heat kernel measure and Wiener measures on path and loop spaces of Riemannian manifolds 3. The relationship between heat kernel measures and pinned Wiener measures on Loop spaces of compact Riemannian manifolds. 4. Extensions of Fock space like representations from finite dimensional groups to infinite dimensional loop groups. 5. The generalization of certain finite dimensional approximations to Wiener measures to include the "super symmetric quantum mechanics" setting used by physicists. Most of the questions to be studied in this research are motivated by, or are outgrowths of, questions coming from studying a number of fields, namely the spread of heat in curved metal plates, quantum mechanics, quantum field theories, and certain aspects of probability theory. These seemingly disparate fields turn out to share a common mathematical description, namely "parabolic partial differential equations" or equivalently the probabilistic theory of "Brownian motion." In most interesting (physically relevant) cases it is seldom possible to find explicit solutions to the complicated partial differential equations to be studied by the P.I. Nevertheless, it is often possible to discover interesting and relevant properties of the solutions. A typical phenomenon of parabolic partial differential equations is the fact that their solutions tend towards steady state values after waiting a sufficiently long time. For example, if a metal plate is heated in a non-uniform way and then left alone, the heat in the plate will redistribute itself over time so that the temperature reaches a constant equilibrium value throughout the plate. A fundamental problem, related to "Poincare" and "logarithmic Sobolev" inequalities, is the questions of how quickly do the systems described by the parabolic partial differential equations to be studied in this research converge to their equilibrium value. This same rate of convergence question has another interpretation in the context of quantum field theories describing relativistic (i.e. moving near the speed of light) elementary particles. For these theories the existence of a Poincare inequality has the desirable interpretation that the mathematical theory does not predict an unreasonable plethora of elementary particles. The particle interpretation of parabolic and related quantum mechanical Shrodinger equations will be another point of study. The particle interpretation goes under the title of Fock space representations which were mentioned in the first paragraph.