Quadratic Forms, Division Algebras, and Elliptic Curves

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DMS 9970374This project studies the structure of certain noncommutative algebras in detail. Technically, they arise as maximal orders in division algebras, but they can also be understood as being generated by a finite number of elements together with a finite number of relations. For example, the usual polynomials are generated by the variables (in degree one) with the relations saying that the variables have commutative multiplication (i.e. xy = yx). In fact, some of the motivating examples for this project, the Sklyanin algebras in degree 3, have properties very similar to the usual polynomials in three variables (except being noncommutative) and therefore give what is called a "quantum plane". The Sklyanin algebras were introduced by Sklyanin in his work on the Yang-Baxter Equation and the quantum inverse problem, and their theory has been extensively developed during the past decade using the tools of noncommutative geometry.This project blends modern techniques from number theory, algebraic geometry, and K-theory. The mathematical problems in this project were motivated originally by physics problems for which there are no visual geometric pictures. This project will develop ways to transform these "noncommutative" geometry problems into better understood and more classical "commutitive" geometric problems. Geometric problems have long been encoded in a mathematical construction called an algebra. An algebra can be described by equations. The equations describing important geometric algebras are often known only "in theory", but the techniques used in this project can determine the equations explicitly. Moreover, these equations can then be directly related to the curves which were used in defining the original algebras. Using this information, the analysis of many noncommutative ("quantum") questions can be reduced to commutative questions producing the visualizations that help produce answers.