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.

该项目涉及两个明显不同的代数区域之间的深厚联系。第一个是群体理论，它是对称性的形式研究。 第二个是代数数理论，即可以通过代数过程构建的数字的研究，例如采用平方根（而不是仅通过属于计算的极限过程而产生的e和pi等数字）。这两个字段之间的主要联系是由代数几何形状（多项式方程系统的几何研究）提供的。 拟议作品中的一个关键主题是单片，它囊括了变量遵循封闭环的变量时，由变化系统揭示的对称性封装。 纯数学和应用数学中的许多伪造都出现了单片问题。 例如，该项目中研究的某些技术已用于确定可以通过不同类型的量子计算机执行哪种计算。 代数数理论发现了重要的实用应用，尤其是在现代密码系统的发展中起着关键作用。该项目的主题是群体理论与代数数理论或算术代数几何学之间的相互关系。 这包括使用组理论作为一种工具，例如分析L-ADIC GALOIS表示的图像，或在理性的Galois扩展上的Abelian品种的Mordell-Weil组。 它还包括研究组，尤其是使用数字理论和代数几何形状的方法，包括圆形方法，依托共同体学和变形理论，尤其是研究组，尤其是离散的线性群体（包括有限基团）。

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

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

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