Projective geometry, invariants and momentum

射影几何、不变量和动量

基本信息

  • 批准号:
    282475916
  • 负责人:
  • 金额:
    --
  • 依托单位:
  • 依托单位国家:
    德国
  • 项目类别:
    Research Grants
  • 财政年份:
    2016
  • 资助国家:
    德国
  • 起止时间:
    2015-12-31 至 2018-12-31
  • 项目状态:
    已结题

项目摘要

The subject of study are geometric spaces with many symmetries. In particular, I study embeddings of such spaces and linear representations of the symmetry groups. It is well-known that the class of semisimple complex Lie groups is exactly the class of linear automorphism groups of homogeneous projective varieties. The latter are called flag varieties. The irreducible representations of a semisimple group on finite dimensional vector spaces are in one-to-one correspondence with the projective embeddings of its flag varieties. The group acts on the ring of polynomials on the vector space. The invariant polynomials form a subring - the fundamental object in invariant theory. The invariant ring is related to a momentum map, defined with respect to a maximal compact subgroup of the symmetry group. The interplay between representations, invariants, momentum and projective geometry of flag varieties defines the subject area of the first part of this project. In particular, there is a remarkable relation between linear spans in the representation space and convex hulls in the momentum image. This naturally leads to the notion of secant varieties to flag varieties. These are classical geometric objects, but have not been systematically studied in the context of momentum maps. I plan to pursue such a study. Another central topic in representation theory is the branching problem. Given an embedding of groups, one asks about decompositions of representations of the ambient group with respect to the subgroup. This can be interpreted in the context of Geometric Invariant Theory, generalizing the above classical setting. The branching problem has two aspects: local, where one studies saturation coefficients, and global, where one seeks a single object encoding the complete branching laws for a given embedding. Objects containing global asymptotic information on multiplicities have recently been introduced by Seppänen, in the form of Okounkov bodies for Hilbert quotients. Many of their properties are not known. An investigation on Hilbert quotients and the local-global interaction in branching laws is intended as an essential part of the project. The second part of the project is about real semisimple Lie groups. In this context momentum maps are replaced by gradient maps. I intend to study gradient maps on representation spaces, in relation to convexity and sphericity. Gradient maps also apply to the geometry of orbits of real forms in complex flag varieties. In particular, I focus on bisectors in Hermitian symmetric spaces. Those are related to discrete groups. I am interested in a particular type of hypergeometric groups constructed recently by Brav and Thomas.
研究的对象是具有许多对称性的几何空间。特别是,我研究嵌入这样的空间和线性表示的对称群。众所周知,半单复李群类恰是齐次投射簇的线性自同构群类。后者被称为旗品种。有限维向量空间上半单群的不可约表示与其旗簇的投射嵌入一一对应。群作用于向量空间上的多项式环。不变多项式形成一个子环-不变理论的基本对象。不变环与动量映射有关,该动量映射是关于对称群的最大紧子群定义的。旗帜的表现形式、不变量、动量和射影几何之间的相互作用定义了本项目第一部分的主题领域。特别是,有一个显着的关系之间的线性跨度的表示空间和凸壳的动量图像。这自然导致正割簇的概念到旗簇。这些都是经典的几何对象,但尚未在动量映射的背景下进行系统的研究。我打算进行这样的研究。表示论的另一个中心议题是分支问题。给定一个群的嵌入,人们会问关于子群的周围群的表示的分解。这可以在几何不变理论的背景下解释,推广了上述经典设置。分支问题有两个方面:局部的,其中一个研究饱和系数,和全球性的,其中一个寻求一个单一的对象编码的完整的分支法律为一个给定的嵌入。包含多重性的全局渐近信息的对象最近由Seppänen以奥昆科夫体(英语:Okounkov bodies)的形式介绍给了希尔伯特对称性。它们的许多性质尚不清楚。对分支律中的希尔伯特导数和局部-全局相互作用的研究是该项目的重要组成部分。 该项目的第二部分是关于真实的半单李群。在这种情况下,动量映射被梯度映射取代。我打算研究表示空间上的梯度映射与凸性和球形的关系。梯度图也适用于复杂的旗帜品种的真实的形式的轨道几何。特别是,我专注于厄米对称空间中的平分线。这些都与离散的群体有关。我感兴趣的是最近由Brav和托马斯构造的一种特殊类型的超几何群。

项目成果

期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Quantum marginals from pure doubly excited states
纯双激发态的量子边际
{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

数据更新时间:{{ journalArticles.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ monograph.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ sciAawards.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ conferencePapers.updateTime }}

{{ item.title }}
  • 作者:
    {{ item.author }}

数据更新时间:{{ patent.updateTime }}

Dr. Valdemar Tsanov其他文献

Dr. Valdemar Tsanov的其他文献

{{ item.title }}
{{ item.translation_title }}
  • DOI:
    {{ item.doi }}
  • 发表时间:
    {{ item.publish_year }}
  • 期刊:
  • 影响因子:
    {{ item.factor }}
  • 作者:
    {{ item.authors }}
  • 通讯作者:
    {{ item.author }}

相似国自然基金

2019年度国际理论物理中心-ICTP School on Geometry and Gravity (smr 3311)
  • 批准号:
    11981240404
  • 批准年份:
    2019
  • 资助金额:
    1.5 万元
  • 项目类别:
    国际(地区)合作与交流项目
新型IIIB、IVB 族元素手性CGC金属有机化合物(Constrained-Geometry Complexes)的合成及反应性研究
  • 批准号:
    20602003
  • 批准年份:
    2006
  • 资助金额:
    26.0 万元
  • 项目类别:
    青年科学基金项目

相似海外基金

Motivic invariants and birational geometry of simple normal crossing degenerations
简单正态交叉退化的动机不变量和双有理几何
  • 批准号:
    EP/Z000955/1
  • 财政年份:
    2024
  • 资助金额:
    --
  • 项目类别:
    Research Grant
Conference: Tensor Invariants in Geometry and Complexity Theory
会议:几何和复杂性理论中的张量不变量
  • 批准号:
    2344680
  • 财政年份:
    2024
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
A study of invariants of singularities in birational geometry via arc spaces
基于弧空间的双有理几何奇点不变量研究
  • 批准号:
    23K12958
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Grant-in-Aid for Early-Career Scientists
Fusion of enumerative and algebraic geometry and exploration of quasi-geometric invariants
枚举几何与代数几何的融合以及准几何不变量的探索
  • 批准号:
    23K17298
  • 财政年份:
    2023
  • 资助金额:
    --
  • 项目类别:
    Grant-in-Aid for Challenging Research (Pioneering)
Geometry from Donaldson-Thomas invariants
唐纳森-托马斯不变量的几何
  • 批准号:
    EP/V010719/1
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Research Grant
Floer Invariants, Cobordisms, and Contact Geometry
Floer 不变量、配边和接触几何
  • 批准号:
    2238131
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Surfaces and Geometry and Topology of Quantum Link Invariants
量子链接不变量的表面、几何和拓扑
  • 批准号:
    2244923
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Continuing Grant
Invariants in Several Complex Variables and Complex Geometry
多个复变量和复几何中的不变量
  • 批准号:
    2154368
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Birational Geometry: Invariants, Reconstruction, and Deformation Problems
双有理几何:不变量、重构和变形问题
  • 批准号:
    2201195
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
Categorical Invariants in Non-commutative Geometry
非交换几何中的分类不变量
  • 批准号:
    2202365
  • 财政年份:
    2022
  • 资助金额:
    --
  • 项目类别:
    Standard Grant
{{ showInfoDetail.title }}

作者:{{ showInfoDetail.author }}

知道了