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Women in Science Establishment (WiSE) presents:

Title: Hypoellipticity in infinite dimensions

Abstract: Consider a flat infinite surface to which one applies a unit of heat at a fixed point and then steps away allowing the heat to propagate. The heat flows quickly at first and then ever more slowly as it converges everywhere to zero. Contrast this now with the way heat flows on a curved surface. For a short time, the heat diffusion will be much the same, but soon the geometry of the surface will play a role. For example, if the surface has sharp corners, this can create irregularities in the heat flow. If the surface is smooth and closed, like a ball, the heat will eventually travel back around to create complex interactions with itself, converging to an equilibrium of uniformly positive heat everywhere.

It turns out that the evolution of heat in a space is exactly related to the way a particle randomly diffuses in that space subject to its geometry, in the sense that the probability that a random particle released from some initial point later finds itself inside some subset is exactly the proportion of heat in that subset from a unit of heat applied to the same initial point. As suggested above, nice properties of the geometry give rise to regularity properties of these probabilities. More particularly, in finite dimensions, "hypoellipticity" is a standard assumption required for regularity. Analogous regularity properties in infinite dimensions have allowed the development of a calculus that has become an invaluable tool in the analysis of random processes and their applications. However, in infinite dimensions it has remained elusive to demonstrate that hypoellipticity is a sufficient condition for regularity. We will discuss some infinite-dimensional model spaces where there have been positive results.

Talk will also be projected in Jones Hall, room 301

Refreshments provided - outside Jones 100 starting at 1:30 pm

Contact

Pierre Clare

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