Damien Pitman

Introduction. As biology becomes a more quantitative science, the lines between mathematician, biologist, and computer scientist are being blurred; and yet, in the midst of such collaboration, there are relatively few scientists able to effectively communicate between the disciplines. I would like to become such a scientist. My research into probability on graphs has interpretations and applications in each of the respective disciplines, but I am particularly interested in the field of biology, specifically evolution on fitness landscapes. The notion of fitness landscapes was introduced by a theoretical evolutionary biologist Sewall Wright in 1932 [35]. In the original formulation, fitness landscapes represent fitnesses of gene combinations. To construct a fitness landscape, one first specifies a genotype space and then assigns fitness to each gene combination in the genotype space. Since 1932, fitness landscapes have found various applications in numerous areas of both evolutionary biology and well outside of it (e.g., in biotechnology, chemistry, theoretical physics, ecology, sociology, and philosophy). Most theoretical studies of the properties of fitness landscapes have been numerical (e.g., [22]). The few analytical approaches (e.g., [23], [18], [27], [28]) have looked only at some very simple models of uncorrelated landscapes under restrictive assumptions of symmetry and very simple genotype spaces. My long term plan is to expand and generalize this previous work in a number of directions. This plan is a natural extension of my joint work [19] with my adviser, Janko Gravner and Sergey Gavrilets. We begin our work with a fitness landscape on the discrete hypercube Q n. Such a landscape is a function ω(v) that assigns fitness values 0 ≤ ω ≤ 1 to each of the 2