Coalescent processes and population models

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The PI studies several problems related to coalescentprocesses and population models. Coalescent processesare stochastic processes that model a system of particleswhich start out separated and merge into clusters as timegoes forward. These processes can be used to describethe genealogy of a population because if one takes asample from a population and follows the ancestral linesbackwards in time, the ancestral lines will coalesce.When a beneficial mutation occurs in a population andspreads rapidly, many ancestral lines will merge atalmost the same time, as they will all be traced back tothe individual that had the beneficial mutation. Onegoal is to use results from the theory of coalescentprocesses with multiple mergers to get further insightinto tests that are used to detect beneficial mutations.A second project is to determine the distribution of thetime that it takes for one individual in a population toexperience k mutations. The PI will also study a modelof coalescence in which the total mass of the systemincreases over time. A coalescent model in which newparticles appear at rate one and clusters merge at a rateproportional to the product of the masses has previouslybeen studied, but a qualitatively different phasetransition is conjectured to arise in an alternativemodel in which clusters merge at a rate proportional tothe sum of the masses.Stochastic models of coalescence have a wide range ofapplications in other fields of science such as biology,physical chemistry, and astronomy. Biologists interestedin understanding evolution are concerned with the mergingof the ancestral lines of a sample from a population. Itshould be possible to use the mathematical theory ofcoalescence to gain further insight into how beneficialmutations impact this process. The project of determiningthe amount of time for one individual in a population toexperience several mutations is motivated by simple modelsof cancer, in which it is assumed that a cell becomescancerous only after several harmful mutations take place.The study of coalescent processes in which the mass of thesystem increases over time is motivated by recent interestin randomly growing networks.