An Introduction to Subfactors in Mathematics and Physics
Introduction to von Neumann algebras. Definition and examples. Von Neumann density theorem. Factors of types I,II and III. TomitaTakesaki theory and Connes decomposition and classification. Hyperfiniteness. Vaughan F. R. JONES
Course Description
We will introduce the theory of subfactors and their interactions with topology and physics. We intend to describe the current state of classification of subfactors and attack the question of whether all subfactors have something to do with quantum field theory, including an appearance of Richard Thompson’s groups F and T.

 Introduction to von Neumann algebras. Definition and examples. Von Neumann density theorem. Factors of types I,II and III. TomitaTakesaki theory and Connes decomposition and classification. Hyperfiniteness.

Vaughan F. R. JONES
Jun. 19, 2015 1:42:17 English

 Subfactors and elementary examples. Index in the type II case. Bimodules. The tower of relative commutants and restrictions on index values. Construction of examples. Braid group representations and knot polynomials.

Vaughan F. R. JONES
Jun. 19, 2015 1:56:50 English

 Kauffman diagrams for the TemperleyLieb algebra. Planar algebras, lambda lattices – the standard invariant of a subfactor and reconstruction. Random matrices with a real number of matrices. Tensor categories, 2categories. Endomorphisms and the type III approach.

Vaughan F. R. JONES
Jun. 19, 2015 1:44:22 English

 Classification of small index subfactors. Izumi’s Cuntz algebra approach. Skein theory presentations of subfactors, the exchange relation, the YangBaxter equation and the jellyfish algorithm.

Vaughan F. R. JONES
Jun. 19, 2015 1:52:24 English

 Algebraic quantum field theory, conformal field theory and scaling limits of statistical mechanical models. Hilbert spaces from planar algebrasa toy algebraic QFT and the Thompson groups. Knots and links from the Thompson groups.

Vaughan F. R. JONES
Jun. 19, 2015 2:04:31 English
Details
 Year/Term
 2015 / Intensive, First semester
 Date
 June 11th to June 19th, 2015
 Faculty/
Graduate School  Graduate School of Science
 Language
 English
 Instructor name
 Vaughan F. R. JONES（Distinguished Visiting Professor, Kyoto University / Stevenson Distinguished Professor, Vanderbilt University）
 Place
 Room 127, Graduate School of Science Bldg No 3
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