Groups and Arithmetic Geometry

群与算术几何

• 批准号: 2001349
• 负责人: Michael Larsen
• 金额: $ 21.6万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001349&HistoricalAwards=false
• 关键词: Groups; Arithmetic; Geometry

The principal investigator's research explores the connections between three of the central concepts in algebra: groups, fields, and varieties. Groups are symmetry types; for instance, all bilaterally symmetric animals share a symmetry group, which is different from the symmetry group of a starfish. Fields are systems of numbers which can be added, subtracted, multiplied, or divided; for instance, the usual real numbers form a field, but the rational numbers, which can be expressed as fractions, form a smaller field of particular interest to number theorists. Varieties are systems of simultaneous equations in a number of variables. The PI is trying to understand the deep connections between these three concepts. For instance, simple groups, the building blocks for all finite groups, can almost always be expressed essentially as the points of a variety over a finite field. Varieties determine fields whose symmetry groups have been of interest in mathematics for more than 200 years, when Gauss gave a compass and straightedge construction for the regular 17-gon. This project also provides research training opportunities for graduate students working with the PI on these questions.More specifically, this project involves using group theory to describe the images of Galois representations arising either from varieties or from automorphic forms. Problems of this kind can often be approached via by what might be termed the "inverse problem" in invariant theory, recognizing an algebraic group from data about its representation category. Group theory and algebraic geometry can be applied together to analyze the Galois action on Mordell-Weil groups of abelian varieties over Galois extensions. In a different direction, the project involves using algebraic geometry, alone or in combination with character-theoretic methods, to solve problems about finite simple groups. In particular, these tools can be applied to investigate word maps, defined by elements in free groups or related objects such as surface groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

首席研究员的研究探索了代数中三个中心概念之间的联系：群，域和变种。群是对称型的;例如，所有左右对称的动物都有一个对称群，这与海星的对称群不同。 域是可以加、减、乘或除的数字系统;例如，通常的真实的数形成一个域，但有理数，可以表示为分数，形成一个更小的领域，数论家特别感兴趣。 变量是由多个变量组成的联立方程组。 PI试图理解这三个概念之间的深层联系。 例如，单群，所有有限群的基石，几乎总是可以基本上表示为有限域上的簇的点。当高斯给出了规则17边形的圆规和直尺构造时，其对称群在数学中已经引起了200多年的兴趣。该项目还为与PI合作研究这些问题的研究生提供了研究培训机会。更具体地说，该项目涉及使用群论来描述从变种或自守形式产生的伽罗瓦表示的图像。 这类问题通常可以通过不变量理论中的“逆问题”来解决，即从关于代数群的表示范畴的数据中识别出代数群。 群论和代数几何可以结合起来分析交换簇在伽罗瓦扩张上的Mordell-Weil群上的伽罗瓦作用。在不同的方向，该项目涉及使用代数几何，单独或结合特征理论的方法，解决有关有限简单群的问题。特别是，这些工具可以应用于调查单词地图，由自由组或相关对象（如表面组）中的元素定义。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Abelian varieties with isogenous reductions

具有同源还原的阿贝尔簇

• DOI: 10.5802/crmath.129
• 发表时间: 2020
• 期刊: Comptes rendus
• 作者: Khare, Chandreshekhar;Larsen, Michael:
• 通讯作者: Larsen, Michael:

Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

正则半单元的字符界和 Thompson 猜想的渐近结果

• DOI: 10.1007/s00209-022-03193-3
• 发表时间: 2023
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Larsen, Michael;Taylor, Jay;Tiep, Pham Huu
• 通讯作者: Tiep, Pham Huu

Most words are geometrically almost uniform

大多数单词在几何上几乎是统一的

• DOI: 10.2140/ant.2020.14.2185
• 发表时间: 2020
• 期刊: Algebra & Number Theory
• 影响因子: 1.3
• 作者: Larsen, Michael Jeffrey

Characteristic covering numbers of finite simple groups

有限单群数的特征覆盖

• DOI: 10.1007/s00208-022-02520-7
• 发表时间: 2022
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Larsen, Michael;Shalev, Aner;Tiep, Pham Huu
• 通讯作者: Tiep, Pham Huu

The sparsity of character tables of high rank groups of Lie type

Lie型高阶群字符表的稀疏性

• DOI: 10.1090/ert/560
• 发表时间: 2021
• 期刊: Representation Theory of the American Mathematical Society
• 作者: Larsen, Michael J.;Miller, Alexander R.
• 通讯作者: Miller, Alexander R.