Mathematical Sciences: Homotopy Theory and its Applications

9504989米勒代数拓扑学的几个研究方向都由这笔拨款资助。米勒教授和霍普金斯教授正在继续他们对“椭圆谱”同伦理论的研究。这包括算术和解析两个方面，与椭圆曲线的算法、指数理论和保形场理论相联系。刘教授的工作是相关的；他正在研究椭圆属的模属性的含义。Hesselholt教授正在通过一系列与拓扑Hochschild同调相关的同伦论工具来继续他对代数K-理论的分析。Garoufaidis教授的研究集中于类似于纽结的Vassiliev不变量的三维流形不变量的构造和分析。尽管这些研究项目的性质多种多样，但它们都反映了目前正在提振大部分数学学科的一股伟大浪潮。在很大程度上受到物理学的启发，许多领域正在从从“一维”角度研究可访问的现象转向使用“二维”工具进行攻击。在数论中，正是这种新方法导致了350年前的费马猜想的解决。最终的结果还有待于拓扑学的研究；但它已经对同伦理论的大部分以及著名的纽结和高维几何结构的新不变量的构造产生了更深刻的理解。***

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. ***