Given an algebraic curve with marked points and integer weights on the markings, we can consider the corresponding divisor. This defines an Abel--Jacobi map from the moduli space of marked curves of compact type to the universal Jacobian variety. Pulling back relations from the universal Jacobian for various values of the weights gives a plethora of tautological relations on the space of curves of compact type. These relations include those discovered by Faber, Getzler, Belorousski and Pandharipande, and Tavakol, and this method was also used by Hain to compute the double ramification cycle. I will talk about the extension of the Abel--Jacobi map to a larger moduli space of marked curves, the corresponding extended formula for the double ramification cycle, and Pixton's conjectures about the extensions of these relations to the entire Deligne-Mumford compactification.
We will discuss the motion a tagged gas particle in a gravitational field. Our starting point will be a Markov approximation to a Lorentz gas model with variable density. We investigate how the density of the ambient gas impacts the recurrence or transience of the tagged particle and we will show that there are multiple scaling regimens leading to nontrivial diffusive limits. This talk is based on joint work with Krzysztof Burdzy.
