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Algebra and Number Theory

A dessin d'enfant is a special type of graph embedded on a Riemann surface whose geometry encodes number theoretic information. Here is an example of a dessin d'enfant conformally drawn on a fundamental domain for the Klein quartic defined by the equation $x^3y + y^3z + z^3x = 0$.
Roughly speaking, number theory is the study of the integers. Carl Friedrich Gauss is said to have claimed: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." Number theorists are interested in topics like the distribution of prime numbers, the solutions to systems of polynomial equations with integer coefficients, the structure of symmetry groups of the roots of a polynomial, and the very deep generalizations of these topics. Many problems in number theory are simple to state but have surprising solutions that draw broadly from all areas of mathematics, and conjectures in number theory have stimulated major advances in other fields. Number theory is both beautiful in its abstraction and useful in practice: the foundation of modern cryptographic systems rely crucially on the difficulty of certain number theoretic problems.


Carl Pomerance
Number theory
Tom Shemanske
Number theory
John Voight
Number theory; Algebraic geometry; Algebraic computing


Algebra and Number Theory Seminar