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# Algebra and Number Theory

### Patrick Morton

Prof. Morton works in number theory, algebra, and geometry. In number theory he has mainly studied Diophantine equations that are related to algebraic structure, including the -1 Pell equation, elliptic curves, and equations that arise from dynamical systems. Most recently he has discovered connections between subtle number theoretic properties of the Legendre polynomials and complex multiplication on supersingular elliptic curves. He has studied algebraic approaches to finding the Galois groups of equations whose roots are the periodic points of polynomial and rational maps over a field, and helped to open up the new field of algebraic and arithmetic dynamics, which studies dynamical systems from an algebraic or number-theoretic point of view. Galois theory is one of his deep interests, along with advanced Euclidean and hyperbolic geometry.

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