Positivity, harmonicity, and random walks

我的建议涉及三个相互关联的主题：1)矩阵值随机漫步，遍历变换和W*因子，2)正多项式，幂级数和多面体，以及3)幂级数系数的渐近性。我的工作的影响：(1)虽然与随机漫步相关的因素的一般分类定理已经存在了几十年，但很少有结果或技术来确定特定示例的同构或非同构。这个方向的结果将是非常重要的，特别是在通过维组方法理解随机漫步方面。(2)正性问题与实代数几何问题之间的联系还处于萌芽阶段。然而，我的方法似乎很有前途。(3)渐近部分的结果不言自明。这些主题中的每一个都有很多问题，适合作为博士项目，也有一些适合作为硕士项目。在研究提案的任何部分时，学生将接触到广泛的数学知识，而不是通常在博士论文中看到的狭隘焦点。

My proposal concerns three interconnected topics: 1) matrix-valued randomwalks, ergodic transformation, & W*-factors, 2) positive polynomials,power series, & polyhedra, and 3) asymptotics of coefficients of powerseries.Impact of my work:(1) While the general classification theorems for factors relating torandom walks have been around for decades, there are very few results ortechniques for deciding isomorphism or non-isomorphism of specificexamples. Results in this direction will be very significant,particularly in understanding the random walks via the dimension groupapproach.(2) Connections between problems in positivity and in real algebraicgeometry are only in their nascent stage. However, my approach seems to be very promising.(3) The results in the asymptotics section speak for themselves.Each of these topics will have plenty of questions that will be suitableas Ph.D. projects, and some as master's level projects. In working inany of the parts of the proposal, students will be exposed to a widespectrum of mathematics, much more than the usual narrow focus seen indoctoral theses.