简介: |
2012清华大学陈省身讲座
报告人：Vaughan F. R. Jones，美国范德堡大学
时间：2012年12月19-26日
地点：清华大学
General lecture: What is a von Neumann algebra?
Time: 19:30 – 20:30pm, Dec. 19 (Wed.)
(Tea at 19:00 – 19:30pm)
Place: Reception Hall, The Main Building, Tsinghua University
Abstract:
Von Neumann was motivated by many things in his introduction, with Murray, of "Rings of Operators", now called von Neumann algebras: unitary group representations, abstract algebra, projective geometry, operator theory, ergodic theory and most of all, the mathematical structure of quantum physics. I will describe how all these threads are united by the seductively simple notion of a closed *-algebra of operators on Hilbert space. I will also sketch the development of the subject by von Neumann, and beyond, including the notion of subfactor.
Lecture two: Von Neumann algebras and quantum physics
Time: 14:00 – 15:00pm, Dec. 21 (Fri.)
(Tea at 15:00 – 15:30pm)
Place: Lecture Hall, Floor 3, Jin Chun Yuan West Building, Tsinghua University
Abstract:
The strong and weak operator topologies have direct physical meaning so that the closedness of a von Neumann algebra is as natural to quantum physics as the completeness of the reals is tp classical physics. Haag and Kastler postulated a model-independent framework for quantum field theory in which loalised observables form von Neumann algebras. This structure has turned out to be extraordinarily rich and has in turn enriched the theory of von Neumann algebras.
Lecture three: Von Neumann algebras and topology
Time: 14:00 – 15:00pm, Dec. 24 (Mon.)
(Tea at 15:00 – 15:30pm)
Place: Lecture Hall, Floor 3, Jin Chun Yuan West Building, Tsinghua University
Abstract:
Of the many interactions between von Neumann algebras and topology I will discuss two-the first being the study of manifolds via the group von Neumann algebra of their fundamental group, for instance Atiyah's L^2 index theorem, and second the development of 2+1 dimensional topological quantum field theory beginning with the Jones polynomial which originated in sufbactors.
Lecture four: Von Neumann algebras and random matrices.
Time: 14:00 – 15:00pm, Dec. 26 (Wed.)
(Tea at 15:00 – 15:30pm)
Place: Lecture Hall, Floor 3, Jin Chun Yuan West Building, Tsinghua University
Abstract:
The large N limit of a single self-adjoint random NxN matrix is particularly well understood even in the presence of a quite arbitrary potential. But for more than one random matrix the theory is less satisfactory. This is because of the non-commutative nature of the beast. The work of Voiculescu shows that even in the purely Gaussian case we land inevitably in the world of type II factors, and indeed we land right on top of some deep unsolved problems in type II factors. I will describe this and more recent work of Guionnet, Shlyakhenko and myself where we create matrix models with a real (non-integer) number of random matrices.
报告人简介
Vaughan F. R. Jones爵士是一位新西兰数学家。他是冯•诺伊曼代数和纽结多项式方面的知名学者。1990年，Jones爵士被授予了菲尔兹奖。Jones爵士现就职于美国范德堡大学，任数学教授。此前，他任职于美国加州大学伯克利分校，并且是奥克兰大学的杰出校友教授。
Jones爵士出生在新西兰的吉斯本，在剑桥市长大。他在奥克兰大学获得了学士和硕士学位。之后，他赴瑞士留学深造，并且于1979年在瑞士日内瓦大学获得博士学位。他的博士论文题目为超有限II_1因子的有限群作用，他的导师是André Haefliger教授。1980年他移居到美国，先后在加州大学洛杉矶分校（1980-1981），宾夕法尼亚大学（1981-1985），以及加州大学伯克利分校任数学教授（1985年以后）。
1984年Jones爵士发现了冯•诺伊曼代数和几何拓扑之间令人震惊的联系。在此基础上，他发现了一个新的纽结不变量，这是他最为著名的工作。在这之前60多年间，拓扑学家和相关领域的学者虽然有很多成就，但是他们都没有发现这个不变量，因而这个发现是令人震惊的。这个发现解决了一系列的经典纽结理论问题，为纽结理论和低维拓扑注入了新鲜的血液。他发现的这个不变量被命名为“琼斯多项式”，已经成为纽结理论和低维拓扑中的最基本概念之一。随后的几年里，人们发现这个纽结不变量事实上和当代数学和物理不同的领域都有着深刻的联系。这些领域（除几何拓扑之外）包括统计力学的可解模型，量子群，Dynkin图和李代数的表示等等。而联系这些领域的纽带则源自Jones爵士在冯•诺伊曼代数的成就：II_1型因子的指标定理。
Jones爵士在1990年获得菲尔兹奖，在1991年被新西兰皇家学会授予了卢瑟福奖。在1990年，他当选为英国皇家学会院士，并分别在1993和1999年当选美国国家科学院及美国艺术和科学院院士。此外，Jones爵士在2002年被授予新西兰杰出贡献骑士勋章。
Sir Vaughan Frederick Randal Jones is a New Zealand mathematician, known mostly for his work on von Neumann algebras and knot polynomials. He was awarded a Fields Medal in 1990. Jones is currently on the faculty of Vanderbilt University as a distinguished professor of mathematics. He previously served as a professor at the University of California, Berkeley and a Distinguished Alumni Professor at the University of Auckland.
Jones was born in Gisborne, New Zealand and brought up in Cambridge. His undergraduate studies were at the University of Auckland, from where he obtained a B.Sc. in 1972 and a M.Sc. in 1973. For his graduate studies, he went to Switzerland, where he completed his Ph.D. at the University of Geneva in 1979. His thesis, titled Actions of finite groups on the hyperfinite II_1 factor, was written under the supervision of André Haefliger. In 1980, he moved to the United States, where he taught at the University of California, Los Angeles (1980–1981) and the University of Pennsylvania (1981–1985), before being appointed as Professor of Mathematics at the University of California, Berkeley.
In 1984 Jones discovered an astonishing relationship between von Neumann algebras and geometric topology. As a result, he found a new polynomial invariant for knots and links in 3-space. His invariant had been missed completely by topologists, in spite of intense activity in closely related areas during the preceding 60years, and it was a complete surprise. His invariant solved of a number of the classical problems of knot theory and is now a fundamental concept in geometric topology, called the Jones polynomial. As time went on, it became clear that his discovery had to do in a bewildering variety of ways with widely separated areas of mathematics and physics and so on. These included (in addition to knots and links) that part of statistical mechanics having to do with exactly solvable models, the very new area of quantum groups, and also Dynkin diagrams and the representation theory of simple Lie algebras. The central connecting link in all this mathematics was a tower of nested algebras which Jones had discovered some years earlier in the course of proving a theorem which is known as the "Index Theorem".
Jones was awarded the Rutherford Medal by the Royal Society of New Zealand in 1991, and the Fields Medal in 1990. Also in 1990 he was elected a Fellow of the Royal Society, and elected to American Academy of Arts & Sciences and US National Academy of Sciences in 1993 and 1999. After that, Jones was awarded Distinguished Companionship of the New Zealand Order of Merit in 2002.
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