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We present an application of piecewise deterministic Markov processes to the study of the hydrological cycle. We formulate a model for the dynamical response of an arbitrary watershed to a rainfall input in the form of a Poisson process. The resulting high dimensional SDE tracks the streamflow and runoff at every stream and hillslope in the watershed. An invariant measure absolutely continuous with respect to Lebesgue measure is shown to exist, and explicitly computed in terms of the probabilisitic description of the rainfall, and the geometry of the river network.
A modern framework for studying linear differential equations with an initial value is to use semi-groups of operators. This framework generalizes the equation u'+Au = 0 from elementary differential equations in such a way that this same equation can be used to represent a partial differential equations. This talk will introduce several key concepts from the theory of semi-groups of operators, as well as provide examples in the 1 dimensional, n dimensional and infinite dimensional setting.
A map is a particular form of embedding a graph on a surface which, among other things, allows one to extend the notion of "regularity" in Platonic solids to a more diverse array of objects. In this talk, I show how to combinatorial-ize the fundamentally topological definition of a map. I proceed to give definitions and examples of two types of symmetry in maps. Finally, I give a generalization of combinatorial maps to higher dimensions and present several results from my Master's work.