Dimension in global analysis and free probability.

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

In non-commutative (differential) 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 "non-commutative manifold", and import analytic and algebraic tools from global analysis to discover geometric facts about this manifold.

While using heat flow to determine the dimension of a manifold is serious overkill, it yields one approach to define dimension for non-commutative manifolds. In the context of free probability (one branch of non-commutative geometry concentrating on analytic and probabilistic properties of free groups), this leads, inexorably, to the somewhat comical-sounding conclusion that all free groups have dimension between 4 and 6.

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 my recent work showing that all free semigroups have dimension 4.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).