Algebraic Structures in Topology and Geometry

拓扑和几何中的代数结构

基本信息

  • 批准号:
    2105544
  • 负责人:
  • 金额:
    $ 24.39万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2021
  • 资助国家:
    美国
  • 起止时间:
    2021-07-01 至 2024-06-30
  • 项目状态:
    已结题

项目摘要

The goal of this project is to understand geometric space in terms of algebraic data and to develop algebraic theories that unlock effective analyses of quantitative and qualitative properties of geometric objects. The main tools come from algebraic topology, a general framework that allows one to reformulate questions about topology and geometry in terms of equivalent questions about algebraic structures such as vector spaces. Topological spaces will be encoded in terms of algebraic structures with particular operations. The PI will also use a similar algebraic framework to study string topology, a theory concerned with interactions of strings and loops in a geometric space. The study of topological and geometric spaces by means of algebraic structures is of fundamental importance in mathematics as well as in mathematical physics. The project aims to explain mathematically the sense in which the fields of topology, geometry, and algebra are equivalent, and the sense in which they are different. The computational tools and invariants that arise from studying the interplay between these fields are useful in the mathematical formulation of quantum field theory, string theory, and mirror symmetry in physics. The award provides funds for graduate students to be involved in parts of this research. The PI will build an inclusive and diverse research group and will promote initiatives directed towards groups that are currently underrepresented in mathematics research.In the first part of the project, the PI will characterize homotopy types through the algebraic concept of an E-infinity coalgebra viewed from the lens of Koszul duality theory. This viewpoint is motivated by a new observation of the PI and M. Zeinalian: the E-infinity coalgebra structure of the singular chains on a space determines the fundamental group in complete generality and this data is preserved under maps which become quasi-isomorphisms after applying the cobar functor. Once homotopy types are understood through this framework, the resulting algebraic structure will be enhanced with extra operations describing Poincaré duality at the chain level in order to characterize topological manifold structures in a homotopy type. The second part of the project is concerned with both foundational and computational questions regarding the string topology of manifolds. Some of the algebraic structures that arise in string topology, in particular the operations related to the Goresky-Hingston coproduct, are able to detect fine geometric information that go beyond the homotopy type in the non-simply connected context. String topology operations will be analyzed using the framework of Hochschild homology and Tate cohomology, as developed in previous work of the PI and Z. Wang. The PI aims to understand the full algebraic structure of string topology, its dependence on the background geometric space, and the new invariants for manifolds that arise.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.
该项目的目标是理解代数数据方面的几何空间,并开发代数理论,解锁几何对象的定量和定性特性的有效分析。主要的工具来自代数拓扑,一个通用的框架,允许一个重新制定的问题,拓扑和几何方面的等价问题,代数结构,如向量空间。拓扑空间将被编码在代数结构与特定的操作。PI也将使用类似的代数框架来研究弦拓扑,这是一种与几何空间中弦和环的相互作用有关的理论。用代数结构研究拓扑空间和几何空间在数学和数学物理中具有根本的重要性。该项目旨在从数学上解释拓扑学、几何学和代数学领域的等价性和不同性。从研究这些场之间的相互作用中产生的计算工具和不变量在量子场论、弦理论和物理学中的镜像对称的数学表述中是有用的。该奖项为研究生提供资金,以参与这项研究的一部分。PI将建立一个包容性和多样化的研究小组,并将促进针对目前在数学研究中代表性不足的群体的倡议。在项目的第一部分,PI将通过从Koszul对偶理论的透镜观察的E-无穷余代数的代数概念来表征同伦类型。这一观点是由对PI和M的新观察激发的。Zeinalian:空间上奇异链的E-无穷余代数结构确定了完全一般性的基本群,并且在应用cobar函子后成为拟同构的映射下,这些数据被保留。一旦通过这个框架理解了同伦类型,所产生的代数结构将通过在链级别描述庞加莱对偶的额外操作来增强,以便描述同伦类型中的拓扑流形结构。该项目的第二部分是关于流形弦拓扑的基础和计算问题。弦拓扑中出现的一些代数结构,特别是与Goresky-Hingston余积相关的运算,能够检测出非单连通上下文中超越同伦类型的精细几何信息。弦拓扑运算将使用Hochschild同调和Tate上同调的框架进行分析,正如PI和Z在以前的工作中所开发的那样。王. PI旨在理解弦拓扑的完整代数结构,它对背景几何空间的依赖,以及出现的流形的新不变量。该奖项反映了NSF的法定使命,并通过使用基金会的智力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(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
The simplicial coalgebra of chains determines homotopy types rationally and one prime at a time
链的单纯余代数有理地确定同伦类型并一次确定一个素数
Categorical models for path spaces
  • DOI:
    10.1016/j.aim.2023.108898
  • 发表时间:
    2022-01
  • 期刊:
  • 影响因子:
    1.7
  • 作者:
    Emilio Minichiello;M. Rivera;M. Zeinalian
  • 通讯作者:
    Emilio Minichiello;M. Rivera;M. Zeinalian
Adams' cobar construction revisited
重新审视亚当斯的科巴结构
  • DOI:
    10.4171/dm/895
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    0.9
  • 作者:
    Rivera, Manuel
  • 通讯作者:
    Rivera, Manuel
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Manuel Rivera其他文献

The role of dietary arachidonic acid and docosahexaenoic acid in preventing the phenotype observed with highly unsaturated fatty acid deficiency
膳食花生四烯酸和二十二碳六烯酸在预防高度不饱和脂肪酸缺乏的表型中的作用
  • DOI:
  • 发表时间:
    2010
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Manuel Rivera
  • 通讯作者:
    Manuel Rivera
Perceptions of service attributes in a religious theme site: an importance–satisfaction analysis
对宗教主题网站服务属性的看法:重要性-满意度分析
  • DOI:
    10.1080/17438730902822939
  • 发表时间:
    2009
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Manuel Rivera;Amir Shani;D. Severt
  • 通讯作者:
    D. Severt
Human agency shaping tourism competitiveness and quality of life in developing economies
  • DOI:
    10.1016/j.tmp.2017.03.002
  • 发表时间:
    2017-04-01
  • 期刊:
  • 影响因子:
  • 作者:
    Marketa Kubickova;Robertico Croes;Manuel Rivera
  • 通讯作者:
    Manuel Rivera
PERCUTANEOUS EPICARDIAL MAPPING AND ABLATION OF VENTRICULAR TACHYCARDIA: A SYSTEMATIC REVIEW OF SAFETY OUTCOMES
  • DOI:
    10.1016/s0735-1097(17)33735-x
  • 发表时间:
    2017-03-21
  • 期刊:
  • 影响因子:
  • 作者:
    Rhanderson Cardoso;Manuel Rivera;Harold Rivner;Rodrigo Mendirichaga;Andre D'Avila
  • 通讯作者:
    Andre D'Avila
THYROID DYSFUNCTION AS A PREDICTOR OF ADVERSE CARDIOVASCULAR OUTCOMES IN HEART FAILURE: A META-ANALYSIS
  • DOI:
    10.1016/s0735-1097(19)31488-3
  • 发表时间:
    2019-03-12
  • 期刊:
  • 影响因子:
  • 作者:
    Amanda Fernandes;Gilson Fernandes;Leonardo Knijnik;Manuel Rivera;Rosario Colombo;Amit Badiye;Sandra Chaparro
  • 通讯作者:
    Sandra Chaparro

Manuel Rivera的其他文献

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{{ truncateString('Manuel Rivera', 18)}}的其他基金

Algebraic Structures in String Topology
弦拓扑中的代数结构
  • 批准号:
    2405405
  • 财政年份:
    2024
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Standard Grant
Conference: Algebraic Structures in Topology 2024
会议:拓扑中的代数结构 2024
  • 批准号:
    2348092
  • 财政年份:
    2024
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Standard Grant
Algebraic Structures in Topology Conference, San Juan, Puerto Rico
拓扑中的代数结构会议,波多黎各圣胡安
  • 批准号:
    2200130
  • 财政年份:
    2022
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Standard Grant

相似海外基金

Algebraic Structures in String Topology
弦拓扑中的代数结构
  • 批准号:
    2405405
  • 财政年份:
    2024
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Standard Grant
Conference: Algebraic Structures in Topology 2024
会议:拓扑中的代数结构 2024
  • 批准号:
    2348092
  • 财政年份:
    2024
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Standard Grant
Low-dimensional topology and algebraic structures
低维拓扑和代数结构
  • 批准号:
    22K03311
  • 财政年份:
    2022
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Algebraic Structures in Topology Conference, San Juan, Puerto Rico
拓扑中的代数结构会议,波多黎各圣胡安
  • 批准号:
    2200130
  • 财政年份:
    2022
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Standard Grant
Synthesising directed structures in Computer Science using Directed Algebraic Topology
使用有向代数拓扑合成计算机科学中的有向结构
  • 批准号:
    19K20215
  • 财政年份:
    2019
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Grant-in-Aid for Early-Career Scientists
Algebraic and category-theoretic structures in low-dimensional topology
低维拓扑中的代数和范畴论结构
  • 批准号:
    18H01119
  • 财政年份:
    2018
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
Generalized complex structures, 4 dimensional differential topology, noncommutative algebraic geometry and derived category
广义复结构、4维微分拓扑、非交换代数几何和派生范畴
  • 批准号:
    16K13755
  • 财政年份:
    2016
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Grant-in-Aid for Challenging Exploratory Research
Low-dimensional topology and algebraic and category-theoretic structures
低维拓扑以及代数和范畴论结构
  • 批准号:
    15K04873
  • 财政年份:
    2015
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
New Algebraic Structures in Topology
拓扑中的新代数结构
  • 批准号:
    1510417
  • 财政年份:
    2015
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Continuing Grant
The algebraic structures behind higher homotopies in symplectic topology.
辛拓扑中更高同伦背后的代数结构。
  • 批准号:
    1105837
  • 财政年份:
    2011
  • 资助金额:
    $ 24.39万
  • 项目类别:
    Standard Grant
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