Quadratic Forms, Division Algebras, and Elliptic Curves

本课题详细研究了一类非交换代数的结构。从技术上讲，它们在除法代数中作为最大阶出现，但它们也可以理解为由有限数量的元素和有限数量的关系产生。例如，通常的多项式是由变量（一级）产生的，关系表明变量具有交换乘法（即xy = yx）。事实上，这个项目的一些激励例子，3度的Sklyanin代数，具有与通常的三变量多项式非常相似的性质（除了非对易性），因此给出了所谓的“量子平面”。Sklyanin代数是由Sklyanin在他关于Yang-Baxter方程和量子逆问题的工作中引入的，它们的理论在过去十年中利用非交换几何的工具得到了广泛的发展。这个项目融合了数论、代数几何和k理论的现代技术。这个项目中的数学问题最初是由没有视觉几何图片的物理问题引起的。该项目将开发将这些“非交换”几何问题转化为更容易理解和更经典的“交换”几何问题的方法。长久以来，几何问题都被编码在一种叫做代数的数学结构中。代数可以用方程来描述。描述重要几何代数的方程通常只在“理论上”为人所知，但本项目中使用的技术可以明确地确定这些方程。此外，这些方程可以直接与用于定义原始代数的曲线相关。使用这些信息，对许多非交换（“量子”）问题的分析可以简化为产生可视化的交换问题，从而帮助生成答案。

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.