Positivity, harmonicity, and random walks

我的建议涉及三个相互关联的主题：1)矩阵值随机游走、遍历变换和W*-因子，2)正多项式、幂多项式和多面体，以及3)幂级数系数的渐近性。我的工作的影响：(1)尽管与随机游走有关的因子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判定具体例子的同构或非同构。在这个方向上的结果将是非常有意义的，特别是对于通过维群方法理解随机游动。(2)正性问题和实代数几何问题之间的联系仅处于萌芽阶段。然而，我的方法似乎是非常有希望的。(3)渐近部分的结果本身就说明了问题。每个主题都有大量的问题，适合作为博士项目，也有一些适合作为硕士水平的项目。在提案的任何部分中，学生都将接触到广泛的数学知识，而不是博士论文中常见的狭隘关注。

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.