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Complex Multiplication on Elliptic Curves
We describe the theory of complex multiplication on elliptic curves as it pertains to constructing abelian extensions of imaginary quadratic fields. We also briefly explore how one may construct abelian extensions of real ...
Evolutionary Dynamics in an Individual Spatial and a Mean Field Differential Equation Host-Pathogen Model
We examine a host-pathogen model in which three types of species exist: empty sites, healthy hosts, and infected hosts. In this model six different transitions can occur: empty sites can be colonized by healthy hosts, healthy ...
Coalescing random walks on n-block Markov chains
Fix a discrete-time Markov chain $(V,P)$ with finite state space $V$ and transition matrix $P$. Let $(V_n,P_n)$ be the Markov chain on n-blocks induced by $(V,P)$, which we call the n-block process associated with the base ...
Merge Times and Hitting Times of Time-inhomogeneous Markov Chains
Concepts of Markov chains mixing, the study of the convergence to stationarity, emerged three decades ago. Studying times to convergence of Markov chains is crucial as it ties to many other mathematical areas, such as Markov ...
A Stochastic Spatial Model for Tumor Growth
Evolutionary game theory can be used to study the interactions of different cell phenotypes and describe tumor population dynamics. Instead of killing tumor cells, clinical treatment could aim to change the nature of the ...