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Wesley Pegden, Assistant ProfessorPh.D., Rutgers University E-mail: wes@math.cmu.eduOffice: Wean Hall 7105Phone: 412-268-9782## Research:My research is in the broad area of Discrete Mathematics, including probabilistic combinatorics, combinatorial game theory, graph theory, and discrete geometry. My recent focus has been on the Abelian sandpile, a simple deterministic diffusion process on the lattice which produces striking fractal limits. My paper with Charles Smart on the convergence of the Abelian sandpile gives a new paradigm—based on the set of quadratic growths attainable by integer superharmonic functions on the integer lattice—for understanding the sandpile's limiting process. Recent work with Lionel Levine and Charles Smart has used this paradigm to analyze the fractal geometry of the sandpile process, through a computationally established Apollonian property of such quadratic growths. In very recent work, we prove this Apollonian property underlying the sandpile process, through a recursive construction of integer superharmonic functions which assigns discrete tilings of the plane to circles in the Apollonian band circle packing. There is hope that our construction may ease progress on problems unrelated to the sandpile, including number-theoretic conjectures related to Apollonian circle packings. |