Tuesday, November 13, 2007, 3:30 p.m.

State Street Room, University Center

(Refreshments at 3:00 p.m.)

Abstract: Planar algebras give operations on graded vector spaces quite analogous to multiplication of polynomials in several variables. We will begin with the most evolved definition of planar algebras, at least those of relevance to the study of von Neumann algebras. Adopting the operadic point of view (though not all the formalism!) we see that the meaning of the identity for a planar algebra is a tangle without input discs. This gives the already rather rich structure of the so-called Temperley-Lieb algebra.

Wednesday, November 14, 2007, 3:30 p.m.

South Hall Room 6635

Abstract: If constant tangles are ones with no input discs, linear tangles are those with one input disc. The most complete analysis of these (for the Temperley-Lieb algebra) was done by Graham and Lehrer. We will give a framework for the complete understanding of linear tangles from a subfactor point of view and contrast the TQFT ideas of Kevin Walker in this regard and point out a connection with cyclic homology.

Thursday, November 15, 2007, 3:30 p.m.

South Hall Room 6635

Abstract: Constant and linear tangles are relatively tame structures but when one considers tangles with two input discs all hell breaks loose as these, together with constant and linear tangles, generate the whole operad. But by asking the right questions it is possible to approach the systematic study of quadratic tangles and obtain significant results. Of considerable interest are the tangles which give bilinear forms and associative algebra structures. It is possible to calculate all bilinear forms on labelled quadratic tangles in terms of relatively few structure constants and this leads to some powerful constraints and an alternative derivation of some results of Haagerup. Also a trace on some of the graded algebras given by quadratic tangles is suggested by random matrices and has been exploited in recent work in collaboration with Guionnet and Shlyakhtenko.