Free probability theory and random matrices

几年前，我开始了一个项目，用自由概率论的工具来理解随机矩阵的波动。最近，我分离出了一个新的概念，称为“二阶自由”，它是许多与随机矩阵有关的现象的正确的底层结构，并有望给该领域带来革命性的变化。NSERC通过领导支持计划奖来支持这种方法，我在未来几年的研究将集中在带来收获上。虽然我的大多数短期目标更多地放在随机矩阵方面(即，利用我在算子代数和自由概率论方面的专业知识来理解随机矩阵及其波动)，但我也想追求一个更具投机性的项目，即随机矩阵的波动与冯·诺伊曼代数的子因子理论之间令人兴奋的联系的可能性(后者导致琼斯发现了新的纽结不变量)。一个令人费解的联系来自于这样一个事实，即琼斯在平面代数上的工作使用了与我在随机矩阵的组合方法中所展示的相同类型的组合对象。我想调查这是表面上的相似之处，还是有更深层次的联系。在后一种情况下，人们可以期待对子因素的全新视角，这将极大地改变这一领域。这个项目的一个重要特点是，它解决了非常深刻的问题，而且可能相当困难，但人们也可以分离出更容易的子问题，这些子问题的理解对于实现最终目标至关重要，但这些问题是在没有更深的理论背景的情况下就可以接触到的，因此是研究生甚至是优秀的本科生的理想话题。这将提供一个独特的机会，通过可接触的具体问题与最令人兴奋的数学前沿研究之一的年轻人对抗。该项目将以每年培养2名本科生暑期学生和3名研究生为特色。

A few years ago I started a project to understand fluctuations of random matrices with tools from free probability theory. Recently, I have isolated a new concept, called "second order freeness", which is the right underlying structure for many phenomena concerning random matrices and which promises to revolutionize the field. NSERC supported this approach with a Leadership Support Initiative Award and my research in the next years will concentrate on bringing in the harvest. Whereas most of my short term goals lie more on the random matrix side (i.e., understanding random matrices and their fluctuations by using my expertise from operator algebras and free probability theory), I also want to pursue as a long term goal a more speculative project, namely the exciting possibility of a link between fluctuations of random matrices and subfactor theory of von Neumann algebras (the latter resulted in Jones's discovery of new knot invariants). A puzzling link comes from the fact that the work of Jones on planar algebras uses the same kind of combinatorial object as show up in my combinatorial approach to random matrices. I want to investigate whether this is a superficial similarity or whether there is some deeper connection. In the latter case, one can expect a totally new perspective on subfactors, which will change the field dramatically. One of the important features of this project is that it attacks very deep and presumably quite hard problems, but that one can also separate much easier subproblems whose understanding is crucial for achieving the final goal, but which are accessible without a deeper theoretical background, and thus ideal topics for graduate, and even strong undergraduate, students. This will present a unique opportunity to confront young people via accessible concrete problems with one of the most exciting cutting edge research in mathematics. This project will feature the training of 2 undergraduate summer students and 3 graduate students per year.