Global motivic homotopy theory

全局动机同伦理论

基本信息

  • 批准号:
    EP/W012030/1
  • 负责人:
  • 金额:
    $ 7.41万
  • 依托单位:
  • 依托单位国家:
    英国
  • 项目类别:
    Research Grant
  • 财政年份:
    2022
  • 资助国家:
    英国
  • 起止时间:
    2022 至 无数据
  • 项目状态:
    已结题

项目摘要

This research proposal is in the areas of mathematics known as algebraic geometry and homotopy theory. Algebraic geometry studies algebraic varieties which are of principal importance. First they are relatively easy to understand since they are just defined by polynomial equations, next they usually give a rather accurate approximation to other shapes, most importantly they do appear naturally in quite a lot of subjects in theoretical physics, coding theory and computer sciences. That is why algebraic geometry - the theory of algebraic varieties is so important for the development and applications of mathematics. Homotopy theory is a considerably newer area of mathematics, being an important branch of algebraic topology, the modern development of what is popularly known as "rubber-sheet geometry", that is, the study of the properties of curves, surfaces and objects of higher dimension which are preserved under operations such as bending and stretching; in homotopy theory one allows additional modifications by "continuous deformation". Since its creation homotopy theory has become an essential component of modern mathematics. Homotopy theory has numerous applications both in and out of mathematics, including theoretical physics and computer sciences.Motivic homotopy theory is a blend of algebraic geometry and homotopy theory. Its primary object is to study algebraic varieties from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology. Motivic homotopy theory led to such striking applications as the solution of the Milnor conjecture and the Bloch-Kato conjecture, in algebraic geometry. Besides these quite spectacular applications, the fact that one can use the ideas and techniques of homotopy theory to solve problems in algebraic geometry has attracted mathematicians from both fields and has led to a wealth of new constructions and applications.The principal aim of this project is to develop global motivic homotopy theory, investigate motivic equivariant spectra and a range of associated cohomology theories of algebraic varieties. Its study will shed light on some classical problems in motivic homotopy theory. Also, we want to apply methods of global motivic homotopy theory to classical global algebraic topology. We believe that these investigations will have important computational advantages. The homotopy-theoretic and geometric outlook that we develop will also be useful in other areas of mathematics such as algebraic topology, non-commutative geometry and mathematical physics.Effective developments of the project objectives require methods of algebraic geometry, motivic homotopy theory, equivariant topology and representation theory.
这项研究计划是在数学领域被称为代数几何和同伦理论。代数几何研究代数簇是主要的重要性。首先,它们相对容易理解,因为它们只是由多项式方程定义的,其次,它们通常对其他形状给出相当准确的近似,最重要的是,它们确实自然地出现在理论物理,编码理论和计算机科学的许多学科中。这就是为什么代数几何-代数簇的理论对数学的发展和应用如此重要。同伦理论是一个相当新的数学领域,是一个重要的分支代数拓扑,现代发展什么是俗称的“橡皮片几何”,即研究的性质曲线,曲面和物体的高维下保存的操作,如弯曲和拉伸;在同伦理论之一,允许额外的修改“连续变形”。自同伦理论创立以来,它已成为现代数学的重要组成部分。同伦理论在数学中和数学之外都有许多应用,包括理论物理和计算机科学。动机同伦理论是代数几何和同伦理论的混合。它的主要目的是从同伦理论的观点来研究代数簇。这门学科的许多基本思想和技巧都起源于代数拓扑学。动机同伦理论在代数几何中有着惊人的应用,例如解决了米尔诺猜想和布洛赫-加藤猜想。除了这些相当壮观的应用,一个事实,即可以使用同伦理论的思想和技术来解决问题的代数几何吸引了数学家从这两个领域,并导致了丰富的新的建设和应用。本项目的主要目的是发展全球motivic同伦理论,研究motivic等变谱和一系列相关的上同调理论的代数簇。对它的研究将有助于解决动机同伦理论中的一些经典问题。同时,我们也希望将整体运动同伦理论的方法应用到经典的整体代数拓扑中。我们相信,这些调查将有重要的计算优势。同伦理论和几何观,我们开发的也将是有用的数学的其他领域,如代数拓扑,非交换几何和数学物理。有效的发展项目的目标需要代数几何,motivic同伦理论,等变拓扑和表示论的方法。

项目成果

期刊论文数量(4)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Semilocal Milnor K-Theory
半局部米尔诺K理论
Correspondences and stable homotopy theory
对应与稳定同伦理论
Group schemes and motivic spectra
群方案和动机谱
  • DOI:
    10.1007/s11856-023-2492-x
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    1
  • 作者:
    Garkusha G
  • 通讯作者:
    Garkusha G
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Grigory Garkusha其他文献

Systems of diagram categories and K-theory. II
  • DOI:
    10.1007/s00209-004-0726-0
  • 发表时间:
    2004-10-15
  • 期刊:
  • 影响因子:
    1.000
  • 作者:
    Grigory Garkusha
  • 通讯作者:
    Grigory Garkusha

Grigory Garkusha的其他文献

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{{ truncateString('Grigory Garkusha', 18)}}的其他基金

Enriched motivic homotopy theory
丰富的动机同伦理论
  • 批准号:
    EP/J013064/1
  • 财政年份:
    2012
  • 资助金额:
    $ 7.41万
  • 项目类别:
    Research Grant
Symplectically oriented cohomology theories of algebraic varieties
代数簇的面向辛的上同调理论
  • 批准号:
    EP/H021566/1
  • 财政年份:
    2010
  • 资助金额:
    $ 7.41万
  • 项目类别:
    Research Grant

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环面空间的上同调与motivic稳定同伦
  • 批准号:
    12271183
  • 批准年份:
    2022
  • 资助金额:
    45 万元
  • 项目类别:
    面上项目
代数群作用下复射影簇的Lawson同调与morphic上同调
  • 批准号:
    12126309
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    2021
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    10.0 万元
  • 项目类别:
    数学天元基金项目
代数群作用下复射影簇的Lawson同调与morphic上同调
  • 批准号:
    12126354
  • 批准年份:
    2021
  • 资助金额:
    20.0 万元
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    数学天元基金项目
同伦群的结构与计算
  • 批准号:
    11801082
  • 批准年份:
    2018
  • 资助金额:
    25.0 万元
  • 项目类别:
    青年科学基金项目
Motivic稳定同伦与环面拓扑中R-S谱序列的研究
  • 批准号:
    11871284
  • 批准年份:
    2018
  • 资助金额:
    53.0 万元
  • 项目类别:
    面上项目
Wall crossing现象和内禀Higgs态
  • 批准号:
    11305125
  • 批准年份:
    2013
  • 资助金额:
    22.0 万元
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相似海外基金

Computations in Classical and Motivic Stable Homotopy Theory
经典和动机稳定同伦理论的计算
  • 批准号:
    2427220
  • 财政年份:
    2024
  • 资助金额:
    $ 7.41万
  • 项目类别:
    Standard Grant
Conference: Motivic and non-commutative aspects of enumerative geometry, Homotopy theory, K-theory, and trace methods
会议:计数几何的本构和非交换方面、同伦理论、K 理论和迹方法
  • 批准号:
    2328867
  • 财政年份:
    2023
  • 资助金额:
    $ 7.41万
  • 项目类别:
    Standard Grant
RUI: Motivic, Operadic, and Combinatorial Homotopy Theory
RUI:动机、操作和组合同伦理论
  • 批准号:
    2204365
  • 财政年份:
    2022
  • 资助金额:
    $ 7.41万
  • 项目类别:
    Continuing Grant
Computations in Classical and Motivic Stable Homotopy Theory
经典和动机稳定同伦理论的计算
  • 批准号:
    2204357
  • 财政年份:
    2022
  • 资助金额:
    $ 7.41万
  • 项目类别:
    Standard Grant
Analyzing algebraic varieties from the point of view of motivic homotopy theory
从动机同伦论的角度分析代数簇
  • 批准号:
    2101898
  • 财政年份:
    2021
  • 资助金额:
    $ 7.41万
  • 项目类别:
    Standard Grant
Motivic Homotopy Theory and Applications to Enumerative Geometry
本征同伦理论及其在枚举几何中的应用
  • 批准号:
    2103838
  • 财政年份:
    2021
  • 资助金额:
    $ 7.41万
  • 项目类别:
    Continuing Grant
Classical, Motivic and Equivariant Stable Homotopy Groups of Spheres.
球面的经典、动机和等变稳定同伦群。
  • 批准号:
    2105462
  • 财政年份:
    2021
  • 资助金额:
    $ 7.41万
  • 项目类别:
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Motivic Homotopy Theory, Stable Homotopy Groups of Spheres and the Kervaire Invariant
动机同伦理论、球面稳定同伦群和 Kervaire 不变量
  • 批准号:
    2043485
  • 财政年份:
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  • 资助金额:
    $ 7.41万
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Motivic Cohomology, Motivic Homotopy Theory and K-theory
动机上同调、动机同伦理论和 K 理论
  • 批准号:
    2001417
  • 财政年份:
    2020
  • 资助金额:
    $ 7.41万
  • 项目类别:
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Equivariant and Motivic Deformations of Stable Homotopy Theory
稳定同伦理论的等变和动机变形
  • 批准号:
    2005476
  • 财政年份:
    2020
  • 资助金额:
    $ 7.41万
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