Groups and Arithmetic Geometry

群与算术几何

基本信息

  • 批准号:
    2001349
  • 负责人:
  • 金额:
    $ 21.6万
  • 依托单位:
  • 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

项目摘要

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核心概念:群、场和变量之间的联系。群是对称类型;例如,所有两侧对称的动物都有一个对称群,这与海星的对称群不同。字段是可以加、减、乘或除的数字系统;例如,通常的实数形成一个域,但可以表示为分数的有理数形成一个较小的域,这是数论学家特别感兴趣的。变分是由若干变量组成的联立方程组。PI正试图理解这三个概念之间的深层联系。例如,所有有限群的构成单元——简单群,基本上总是可以表示为有限域上的各种点。当高斯给出了一个指南针和正17边形的直线结构时,对称性群在数学中已经引起了200多年的兴趣。这个项目也为研究生提供研究训练的机会,让他们与PI一起研究这些问题。更具体地说,这个项目涉及使用群论来描述伽罗瓦表示的图像,这些图像要么来自变种,要么来自自同构形式。这类问题通常可以通过不变量理论中所谓的“逆问题”来解决,即从其表示范畴的数据中识别代数群。群论和代数几何可以共同应用于伽罗瓦扩展上的阿贝尔变异体的Mordell-Weil群上的伽罗瓦作用。在另一个方向上,该项目涉及使用代数几何,单独或结合特征理论方法,解决有限简单群的问题。特别是,这些工具可以用于研究由自由组中的元素或相关对象(如表面组)定义的词映射。该奖项反映了美国国家科学基金会的法定使命,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

期刊论文数量(6)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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:
  • 影响因子:
    0
  • 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:
  • 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:
  • 影响因子:
    1.3
  • 作者:
    Larsen, Michael Jeffrey
  • 通讯作者:
    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:
  • 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
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Michael Larsen其他文献

Randomized sham-controlled trial of the 6-month swallowable gas-filled intragastric balloon system for weight loss.
对为期 6 个月的可吞咽充气胃内气球系统进行减肥的随机假对照试验。
  • DOI:
  • 发表时间:
    2018
  • 期刊:
  • 影响因子:
    3.1
  • 作者:
    Shelby Sullivan;J. Swain;G. Woodman;S. Edmundowicz;T. Hassanein;V. Shayani;John Fang;M. Noar;G. Eid;Wayne J. English;N. Tariq;Michael Larsen;S. Jonnalagadda;D. Riff;J. Ponce;D. Early;E. Volckmann;A. Ibele;Matthew D. Spann;K. Krishnan;J. Bucobo;A. Pryor
  • 通讯作者:
    A. Pryor
Petechial hemorrhages of the tympanic membrane in attempted suicide by hanging: A case report
  • DOI:
    10.1016/j.jflm.2012.05.007
  • 发表时间:
    2013-02-01
  • 期刊:
  • 影响因子:
  • 作者:
    Eva Rye Rasmussen;Per Leganger Larsen;Kjeld Andersen;Michael Larsen;Klaus Qvortrup;Hans Petter Hougen
  • 通讯作者:
    Hans Petter Hougen
DIAGNOSTIC VALUE AND COST OF MRCP AND EUS FOR THE EVALUATION OF CHOLEDOCHOLITHIASIS: A RETROSPECTIVE COHORT ANALYSIS
  • DOI:
    10.1016/j.gie.2024.04.1108
  • 发表时间:
    2024-06-01
  • 期刊:
  • 影响因子:
  • 作者:
    Karena Puldon;Logan Pierce;Patrick Avila;Michael Larsen;Sun-Chuan Dai;Mustafa Arain;Abdul Kouanda
  • 通讯作者:
    Abdul Kouanda
Tensor product Markov chains and Weil representations
张量积马尔可夫链与韦伊表示
  • DOI:
    10.1016/j.jalgebra.2025.05.041
  • 发表时间:
    2025-11-15
  • 期刊:
  • 影响因子:
    0.800
  • 作者:
    Jason Fulman;Michael Larsen;Pham Huu Tiep
  • 通讯作者:
    Pham Huu Tiep
On The Correlation Of Binary M-sequences
  • DOI:
    10.1023/a:1008383811226
  • 发表时间:
    1999-01-01
  • 期刊:
  • 影响因子:
    1.200
  • 作者:
    John Friedlander;Michael Larsen;Daniel Lieman;Igor Shparlinski
  • 通讯作者:
    Igor Shparlinski

Michael Larsen的其他文献

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{{ truncateString('Michael Larsen', 18)}}的其他基金

Groups and Arithmetic
群与算术
  • 批准号:
    2401098
  • 财政年份:
    2024
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Continuing Grant
RUI: Dynamic Guanidine-based Polymer Networks
RUI:动态胍基聚合物网络
  • 批准号:
    2105149
  • 财政年份:
    2021
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Continuing Grant
Collaborative Research to Explore the Spatial/Temporal Statistical-Physical Structures of Rain in the Vertical Plane
探索垂直平面降雨时空统计物理结构的合作研究
  • 批准号:
    2001490
  • 财政年份:
    2020
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Standard Grant
Developing a Life Sciences Workforce with Strong Quantitative Skills
培养具有强大定量技能的生命科学员工队伍
  • 批准号:
    1742241
  • 财政年份:
    2018
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Standard Grant
Collaborative Research: The Relationship of the Spatial/Temporal Variability of Rain to Scaling
合作研究:降雨的时空变化与尺度的关系
  • 批准号:
    1823334
  • 财政年份:
    2018
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Standard Grant
Groups and Arithmetic
群与算术
  • 批准号:
    1702152
  • 财政年份:
    2017
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Standard Grant
Collaborative Research: The Meteorological Variability of the Two Dimensional/Temporal Structures of Drop Size Distributions and Rain
合作研究:雨滴尺寸分布和降雨的二维/时间结构的气象变化
  • 批准号:
    1532977
  • 财政年份:
    2015
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Continuing Grant
Arithmetic, Groups, and Monodromy
算术、群和单数
  • 批准号:
    1401419
  • 财政年份:
    2014
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Standard Grant
Collaborative Research: Characterization of the Two-dimensional/Temporal Mosaic of Drop Size Distributions and Spatial Variability (Structure) in Rain
合作研究:雨中液滴尺寸分布和空间变化(结构)的二维/时间镶嵌特征
  • 批准号:
    1230240
  • 财政年份:
    2012
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Continuing Grant
Groups, Arithmetic, and Monodromy
群、算术和单数
  • 批准号:
    1101424
  • 财政年份:
    2011
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Continuing Grant

相似海外基金

Conference on Arithmetic Geometry and Algebraic Groups
算术几何与代数群会议
  • 批准号:
    2305231
  • 财政年份:
    2023
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Standard Grant
Geometry, Arithmetic, and Groups.
几何、算术和群。
  • 批准号:
    2204684
  • 财政年份:
    2022
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Standard Grant
Higher-dimensionalization of arithmetic geometry concerning arithmetic fundamental groups
关于算术基本群的算术几何的高维化
  • 批准号:
    20H01796
  • 财政年份:
    2020
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
The geometry of character variety and classification of arithmetic Kleinian groups
字符变换的几何与算术克莱尼群的分类
  • 批准号:
    20K03612
  • 财政年份:
    2020
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Model theory of absolute Galois groups with a view towards arithmetic geometry
算术几何视角下的绝对伽罗瓦群模型论
  • 批准号:
    2099876
  • 财政年份:
    2018
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Studentship
Thin Groups in Geometry and Arithmetic
几何和算术中的薄群
  • 批准号:
    1802119
  • 财政年份:
    2018
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Continuing Grant
Fundamental groups and applications to arithmetic geometry
基本群及其在算术几何中的应用
  • 批准号:
    1789793
  • 财政年份:
    2016
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Studentship
Various problems in arithmetic geometry concerning arithmetic fundamental groups and their interrelationships
算术几何中有关算术基本群及其相互关系的各种问题
  • 批准号:
    15H03609
  • 财政年份:
    2015
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
Geometry and Cohomology of Arithmetic and Related Groups
算术及相关群的几何和上同调
  • 批准号:
    1509182
  • 财政年份:
    2015
  • 资助金额:
    $ 21.6万
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    Standard Grant
Algebraic groups, arithmetic subgroups and geometry
代数群、算术子群和几何
  • 批准号:
    1401380
  • 财政年份:
    2014
  • 资助金额:
    $ 21.6万
  • 项目类别:
    Continuing Grant
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