Mathematical Sciences: Homotopy Theory and its Applications

数学科学：同伦理论及其应用

• 批准号: 9504989
• 负责人: Haynes Miller
• 金额: $ 47.62万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504989&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Homotopy; Theory; its

9504989 Miller Several directions of research in Algebraic Topology are funded by this grant. Professors Miller and Hopkins are carrying forward their investigation of the homotopy theory of "elliptic spectra." This includes both arithmetic and analytic aspects, connecting to the arithmetic of elliptic curves and to index theory and conformal field theory. Professor Liu's work is related; he is studying the implications of the modular properties of elliptic genera. Professor Hesselholt is pursuing his analysis of algebraic K-theory via a range of homotopy-theoretic tools associated with topological Hochschild homology. Professor Garoufalidis's research focuses on the construction and analysis of 3-manifold invariants analogous to the Vassiliev invariants of knots. Despite the diverse nature of these research projects, they are all reflections of a single great tide that is currently lifting large parts of Mathematics. Inspired largely by Physics, many fields are in the process of moving from a study of phenomena accessible from a "one-dimensional" perspective to an attack using "two-dimensional" tools. In Number Theory it is this new approach which led to the solution of the 350-year-old Fermat Conjecture. The final outcome remains to be seen in Topology; but already it has yielded a much deeper understanding of large parts of homotopy theory, as well as the celebrated constructions of new invariants for knots and higher-dimensional geometric structures. ***

米勒9504989 代数拓扑学的几个研究方向由该基金资助。 米勒教授和霍普金斯教授正在推进他们对“椭圆谱”同伦理论的研究。“这包括算术和分析方面，连接到椭圆曲线的算术和指数理论和共形场论。 刘教授的工作是相关的;他正在研究椭圆属的模性质的含义。 Hesselholt教授正在通过一系列与拓扑Hochschild同源性相关的同伦理论工具来进行代数K理论的分析。 教授Garoufalliev的研究重点是建设和分析的3流形不变量类似的Vassiliev不变量的结。 尽管这些研究项目的性质不同，但它们都反映了一个单一的伟大潮流，目前正在提升数学的大部分。 受物理学的启发，许多领域正在从“一维”视角的现象研究转向使用“二维”工具的攻击。 在数论中，正是这种新的方法导致了350年前的费马猜想的解决方案。 最后的结果还有待于在拓扑学中看到;但它已经产生了一个更深入的理解同伦理论的大部分，以及著名的建设新的不变量的结和高维几何结构。 ***