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I’ve added a little to it, and might write an expository paper at some point. Intriguingly, there’s a connection between what’s going on here and those Ricci flows Urs was telling us about. See page 29 of Carlos Rodriguez’ A Geometric Theory of Ignorance.

Is anything known about flows on manifolds which are locally modelled on reproducing kernel hilbert spaces?

Is anything known about flows on manifolds which are locally modelled on reproducing kernel hilbert spaces?

I am not even sure I know what this question means! In which way is your flow supposed to be modeled on a Hilbert space?

I just meant flows on infinite-dimensional Riemannian manifolds, but where the charts are maps to subspaces of a RKHS. The kind of model used in nonparametric statistics has this form.

I know next to nothing about infinite-dimensional geometry. I see there is such a thing as a Hilbert manifold, and that a Hilbert space can be given the structure of a Riemannian manifold.

Bacteria use Bayes’ Law in their evolved “inference” behavior.

physorg.com/news96301683.html

How cells deal with uncertainty

Researchers at McGill University have found that cells respond to their ever-changing environment in a way that mimics the optimal mathematical approach to doing so, also known as Bayes’ rule; an application of probability theory. Their findings are published in the April 17 issue of PNAS, the Proceedings of the National Academy of Sciences.

Biology is seeing a re-birth, said Dr. Peter Swain, an assistant professor in the Department of Physiology and a Canada Research Chair in Systems Biology, as more researchers are thinking about the cell using schemes that we know work from engineering and computer science.

The study was carried out at McGill Centre for Nonlinear Dynamics in Physiology and Medicine (CND). Eric Libby, PhD candidate at the CND and lead author on the paper, Dr. Ted Perkins, assistant professor in the School of Computer Science, and Dr. Swain simulated data on a biochemical response mechanism in a strain of E. coli bacteria.

The ideal mathematical model and the simulation meshed perfectly with Bayes rule,” remarked Swain. The bacterias collection of genes and proteins that responded to changing environmental conditions acted as a successful Bayesian inference module, which takes noisy, uncertain information and interprets what it means for the cell.

There are many known schemes for inference that exist in mathematics. This study suggests that cells may have evolved to incorporate the most efficient decision-making abilities into their biochemical pathways.

Quick, accurate cell responses to signals are necessary for survival. When we sense danger, our bodies can tell if the signal is real and trigger the production of adrenaline immediately. However, modeling the effects of a signal on one part of a cell, even in isolation from body tissues and organs, is complicated. With many drugs, we dont know how they work or exactly what they are targeting in a cell noted Swain. He explained that further study of inference modules could allow us to model more sophisticated cellular behavior, which could one day lead to computerized drug experiments and trials.

Source: McGill University

## Re: Cake Talk

It may’ve been a cake talk, but it sure wasn’t a cakewalk — either to give, or to follow. Intense stuff!