Integral Geometry and Geometric Probability
Summer 2007 REU
Led by Prof. Joseph Fu
The first problem in geometric probability is the famous Buffon needle problem: what is the probability that a needle, dropped on a plane ruled by parallel lines an inch apart, crosses one of the lines? This is a very special case of the more general question: given two 2-dimensional shapes, what is the average shape of their intersection if one of them is spun around and tossed randomly onto the other? The same question can be asked for higher dimensional shapes, and if one applies certain measurements, called valuations, to the shape of the intersection, there is a very satisying answer in the form of the "kinematic formulas". It is known that the same kind of answer must exist even if the spinning is restricted to a subgroup of the full rotation group, but the actual formulas remain a mystery! These questions can be asked for finite group actions on finite sets as well. The kinematic formulas turn out to be a repackaged form of a natural multiplication on the set of valuations. The resulting algebra is poorly understood except in the simplest case, but seems to have a lot of combinatorial structure. We will study it in some finite cases and in the geometric case of the unitary group, with lectures and detours to fill in background as needed.
Participants:
Kelly Aman, Seongyoon Cheong, William Hudelson, Kenneth Knox, Caitlyn Phillips, and Jonathan Weed.
