Colloquium
3:00 p.m., Friday (March 30)
Math Annex 1100
Todd Kemp
MIT
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 longterm behaviour is intimately linked to global geometry. However, the shorttime smoothing behaviour is universal: it depends only upon the dimension of the manifold, and determines the dimension uniquely.
In noncommutative (differential) geometry, the overarching principal is to study a noncommutative algebra, pretend it is an algebra of smooth functions or differential operators on a "noncommutative 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 noncommutative manifolds. In the context of free probability (one branch of noncommutative geometry concentrating on analytic and probabilistic properties of free groups), this leads, inexorably, to the somewhat comicalsounding 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).
