UCSB Distinguished Lectures in the Mathematical Sciences
Vaughan Jones, November 1316, 2007
Tuesday, November 13, 2007, 3:30 p.m. State Street Room, University Center (Refreshments at 3:00 p.m.)

Vaughan Jones University of California, Berkeley 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 socalled TemperleyLieb algebra.

Wednesday, November 14, 2007, 3:30 p.m. South Hall Room 6635 
Vaughan Jones University of California, Berkeley 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 TemperleyLieb 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 
Vaughan Jones University of California, Berkeley 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. 