Algebraic Topology

代数拓扑

• 批准号: 0306429
• 负责人: Matthew Ando
• 金额: $ 34.62万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-15 至 2007-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306429&HistoricalAwards=false
• 关键词: Algebraic; Topology

DMS-0306429Matthew Ando and Randy McCarthyAndo and McCarthy will study the relationship between stable homotopytheory and mathematical physics and number theory mediated by ellipticcohomology, and the relationship between stable homotopy theory andalgebra mediated by K-theory. Ando will use elliptic cohomology toinvestigate what string theory and the theory of elliptic curves haveto say about the topology of compact manifolds. He is particularlyinterested in bringing recent work on open string theories andD-branes to bear on problems in elliptic cohomology. McCarthy willdevelop a theory of Witt structures for bimodules and a theory ofsmoothness and De Rham cohomology for commutative ring spectra. Theformer offers new insight and substantial generalizations ofcalculations of Hesselholt and Madsen, while the latter offers thepossibility of developing a crystalline Chern character and soextending to commutative ring spectra the work of Bloch.The last few years have seen an astonishing proliferation ofunexpected new structure in topology, involving new interactions withother areas of mathematics and physics. Ando and McCarthy areparticularly interested in articulating these new connections.Ando's research seeks to give explicit form to the deep and surprisingrelationship between topology, high energy physics, and number theorysignaled by elliptic cohomology. The traditional paradigm ofalgebraic topology is the use of algebra to model phenomena intopology. McCarthy's research turns this paradigm upside down, usingthe deep structure of topological spaces to tackle hard problems inalgebra.

DMS-0306429 Matthew Ando和Randy McCarthy Ando和McCarthy将研究稳定同伦理论与以椭圆上同调为中介的数学物理和数论之间的关系，以及稳定同伦理论与以K-理论为中介的代数之间的关系。 安藤将使用椭圆上同调来研究弦理论和椭圆曲线理论对紧致流形的拓扑结构有什么看法。 他特别感兴趣的是把最近的工作开弦理论和D膜承担问题的椭圆上同调。 McCarthy将发展双模的Witt结构理论和交换环谱的光滑性和De Rham上同调理论。 前者为Hesselholt和Madsen的计算提供了新的见解和实质性的推广，而后者则为发展一种晶体陈省身特征并将Bloch的工作扩展到交换环谱提供了可能性。过去几年，拓扑学中出现了令人惊讶的、意想不到的新结构，涉及到与数学和物理学其他领域的新相互作用。 Ando和McCarthy对阐明这些新的联系特别感兴趣。Ando的研究旨在明确拓扑学、高能物理学和数论之间由椭圆上同调所表示的深刻而广泛的关系。 代数拓扑学的传统范式是用代数来模拟拓扑学中的现象。 McCarthy的研究颠覆了这种范式，利用拓扑空间的深层结构来解决代数中的难题。