Clayton Shonkwiler (CSU) A Geometric Approach to Sampling Loop Random Flights
Equations of Lie type are ODEs on Lie groups that turn out to have their own version of a superposition formula. The usual result that a general solution to a linear system of ODEs can be written as a linear combination of homogenous and particular solutions turns out to be a special case of this idea. We'll explore the main theorem and go through some examples and possibly a proof. Applications to a special kind of connection in geometry may also be presented.
In statistical physics, the basic (and highly idealized) model of a ring polymer like bacterial DNA is a closed random flight in 3-space with equal-length steps, often called an equilateral random polygon. While random flights without the closure condition are easy to simulate and analyze, the fact that the steps in a random polygon are not independent has made it challenging to develop practical yet provably correct sampling and numerical integration techniques for polygons.
In this talk I will describe a geometric approach to the study of random polygons which overcomes these challenges. The symplectic geometry of the space of polygon conformations can be exploited to produce both Markov chain and direct sampling algorithms; in fact, this approach can be generalized to produce a sampling theory for arbitrary toric symplectic manifolds. This is joint work with Jason Cantarella, Bertrand Duplantier, and Erica Uehara.