One of the central problems in the study of von Neumann algebras
is to find computable invariants which can distinguish nonisomorphic
algebras. In the 1980s, Dan Voiculescu developed a noncommutative probability
theory in order to understand a particular class of such von Neumann algebras.
Specifically, he defined subsets of $R^n$ called microstate spaces which
model the behavior of a generating set of a given von Neumann algebra.
Since these spaces are subsets of $R^n$, classical analytic tools such as
volume can be applied to them.

I will discuss how the application of ideas from geometric measure theory
to microstate spaces has provided insight into the general problem of
invariants and answered some longstanding questions in von Neumann
algebras.

Consider the complete regular binary tree of depth M oriented from the root to the leaves. To each node we associate a random variable and those variables are assumed to be independent. Under the null hypothesis, these random variables have the standard normal distribution while under the alternative, there is a path from the root to a leaf along which the nodes have the normal distribution with mean A and variance 1, and the standard normal distribution away from the path. We show that, as M increases, the hypotheses become separable if, and only if, A is larger than the square root of 2 ln 2. We obtain corresponding results for other graphs and other distributions. The concept of predictability profile plays a crucial role in our analysis.

Joint work with Emmanuel Candes, Hannes Helgason and Ofer Zeitouni.

Vertex algebras arose out of conformal field theory, and were
first defined mathematically by Borcherds in 1986. Since then, they have
found applications in many areas of mathematics, including representation
theory, number theory, finite group theory, and geometry. Vertex algebras
are vector spaces (generally infinite-dimensional) which are equipped with
a family of bilinear products (indexed by the integers) which in general
are neither commutative nor associative. In many ways they behave like
ordinary associative algebras, and the usual categorical and formal
algebraic notions like homomorphisms, ideals, quotients, and modules over
vertex algebras are easy to define.

In this talk, I'll define vertex algebras, give some basic examples,
indicate how to do computations, and hopefully state some interesting open
problems.

How can you tell the dimension of a manifold? One answer lies in studying
the flow of heat on the manifold. {\em Heat flow} is a smoothing process on
Riemannian manifolds, whose long-term behaviour is intimately linked to
global geometry. However, the {\em short-time} smoothing behaviour is
universal: it depends only upon the dimension of the manifold,
and determines the dimension uniquely.

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In non-commutative geometry, the over-arching principal is to study a
non-commutative algebra, pretend it is an algebra of smooth functions
or differential operators on a {\em non-commutative manifold}, and import
analytic and algebraic tools from global analysis to discover geometric
facts about this manifold.

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While using heat flow is an excessively difficult way to determine the
dimension of a manifold, it yields one approach to define dimension for
non-commutative manifolds. In the context of {\em free probability} (one
branch of non-commutative geometry concentrating on analytic properties
of free groups), this leads, inexorably, to the somewhat comical-sounding
conclusion that {\em all free groups have dimension $6$}.

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In this talk, I will outline those aspects of free probability which relate to
heat kernel analysis, and make the connection between dimension and
heat flow clear. I will also discuss recent joint work with Roland Speicher,
showing that {\em all free semigroups have dimension $4$}.