Free probability theory and random matrices

自由概率论和随机矩阵

• 批准号: 238386-2006
• 负责人: Speicher, Roland
• 金额: $ 2.91万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=393901
• 关键词: Free; probability; theory; random; matrices

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.

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，我分离出一个新的概念，称为“二阶自由度”，这是正确的基础结构的许多现象有关的随机矩阵，并承诺革命性的领域。NSERC通过领导支持倡议奖支持这种方法，我在未来几年的研究将集中在收获上。而我的大多数短期目标更多地是在随机矩阵方面（即，理解随机矩阵和他们的波动，利用我的专业知识，从算子代数和自由概率论），我还想追求作为一个长期目标，一个更具投机性的项目，即令人兴奋的可能性之间的联系波动的随机矩阵和子因子理论的冯诺依曼代数（后者导致琼斯的发现新的结不变量）。一个令人费解的联系来自这样一个事实，即工作琼斯对平面代数使用相同的组合对象显示在我的组合方法随机矩阵。我想研究一下这是表面上的相似还是有更深层次的联系。在后一种情况下，人们可以期待一个全新的角度来看子因素，这将大大改变该领域。这个项目的一个重要特点是，它攻击非常深，可能相当困难的问题，但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，但无需更深的理论背景就可以访问，因此是研究生，甚至是强大的本科生的理想主题。这将提供一个独特的机会，让年轻人通过数学中最令人兴奋的前沿研究之一来面对具体的问题。该项目每年将培训2名本科暑期学生和3名研究生。