Groups and Arithmetic

这个项目涉及两个明显截然不同的代数领域之间的深层联系。第一种是群论，它是对对称性的正式研究。第二个是代数数论，研究可以通过代数过程构造的数，比如取平方根(而不是像e和pi这样的数字，它们只能通过属于微积分的极限过程产生)。这两个领域之间的主要联系是由代数几何提供的，代数几何是对多项式方程组的几何研究。拟议工作中的一个关键主题是单行，它封装了随着变量跟随闭合环路而变化的系统所揭示的对称性。单一性问题以多种形式出现，既有纯数学的，也有应用数学的。例如，这个项目中正在研究的一些技术已经被用来确定不同类型的量子计算机可以进行哪些类型的计算。代数数论有着重要的实际应用，特别是在现代密码系统的发展中起着关键的作用。本课题的主题是群论与代数数论或算术代数几何之间的相互关系。这包括使用群论作为工具，例如，在分析有理数的伽罗瓦扩张上的L-ADDIC伽罗瓦表示的像，或阿贝尔变种的莫德尔-韦尔群。它还包括研究群，特别是离散线性群(包括有限群)，使用数论和代数几何的方法，包括圆法，等上同调和形变理论。

This project concerns the deep connections between two apparently very different areas of algebra. The first is group theory, which is the formal study of symmetry. The second is algebraic number theory, the study of numbers which can be constructed by algebraic processes, like taking square roots (as opposed to numbers like e and pi which arise only through limit processes belonging to calculus). The main link between these two fields is provided by algebraic geometry, the geometric study of systems of polynomial equations. A key theme in the proposed work is monodromy, which encapsulates the symmetries revealed by a varying system as the variable follows a closed loop. Monodromy problems arise in many guises, in both pure and applied mathematics. For instance some of the techniques under study in this project have been used to determine which kinds of computations can be carried out by different kinds of quantum computer. Algebraic number theory has found important practical applications and especially plays a key role in the development of modern cryptosystems.The theme of this project is the reciprocal relationship between group theory and algebraic number theory or arithmetic algebraic geometry. This includes using group theory as a tool, for instance in analyzing images of l-adic Galois representations, or Mordell-Weil groups of abelian varieties over Galois extensions of the rationals. It also includes studying groups, especially discrete linear groups (including finite groups), using methods from number theory and algebraic geometry, including the circle method, etale cohomology, and deformation theory.