Groups and Arithmetic

群与算术

• 批准号: 1702152
• 负责人: Michael Larsen
• 金额: $ 17.7万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702152&HistoricalAwards=false
• 关键词: Groups; Arithmetic

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.

这个项目涉及代数的两个明显不同的领域之间的深层联系。第一个是群论，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字，比如平方根（与像e和pi这样的数字相反，这些数字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，多项式方程组的几何研究。 在拟议的工作中的一个关键主题是monodromy，它封装的对称性揭示了一个不断变化的系统作为变量遵循一个闭环。 在纯数学和应用数学中，单值问题以多种形式出现。 例如，该项目中正在研究的一些技术已被用于确定不同类型的量子计算机可以执行哪些类型的计算。 代数数论有着重要的实际应用，特别是在现代密码体制的发展中起着关键的作用，本项目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l-进伽罗瓦表示的图像，或在有理数的伽罗瓦扩展上的阿贝尔变种的莫德尔-韦伊群。 它还包括研究群体，特别是离散线性群体（包括有限群体），使用数论和代数几何的方法，包括圆方法，etale上同调和变形理论。

项目成果

Words, Hausdorff dimension and randomly free groups

词、Hausdorff 维数和随机自由群

• DOI: 10.1007/s00208-017-1635-y
• 发表时间: 2018
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Larsen, Michael;Shalev, Aner
• 通讯作者: Shalev, Aner

A note on Lie algebra cohomology

关于李代数上同调的注解

• DOI: 10.2140/ant.2021.15.773
• 发表时间: 2021
• 期刊: Algebra & Number Theory
• 影响因子: 1.3
• 作者: Larsen, Michael J.;Lunts, Valery A.
• 通讯作者: Lunts, Valery A.

Waring’s problem for unipotent algebraic groups

单能代数群的韦林问题

• DOI: 10.5802/aif.3283
• 发表时间: 2019
• 期刊: Annales de l'Institut Fourier
• 作者: Larsen, Michael;Nguyen, Dong Quan
• 通讯作者: Nguyen, Dong Quan

Flatness of the Commutator Map Over $\textrm{SL}_n$

$ extrm{SL}_n$ 上换向器图的平坦度

• DOI: 10.1093/imrn/rnz285
• 发表时间: 2019
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Larsen, Michael;Lu, Zhipeng
• 通讯作者: Lu, Zhipeng

Many Zeros of Many Characters of GL(n,q)

GL(n,q) 的多个字符的多个零

• DOI: 10.1093/imrn/rnaa160
• 发表时间: 2020
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Gallagher, Patrick X;Larsen, Michael J;Miller, Alexander R
• 通讯作者: Miller, Alexander R