Mathematical Sciences: Homotopy Theory and its Applications

9504989 Miller代数拓扑的几个方向的研究是由这项拨款资助的。米勒教授和霍普金斯教授正在推进他们对“椭圆光谱”同伦理论的研究。这包括算术和解析两方面，与椭圆曲线的算术、指数理论和共形场理论有关。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同调相关的同伦理论工具进行代数k理论的分析。Garoufalidis教授的研究重点是类似于结点的Vassiliev不变量的3流形不变量的构造和分析。尽管这些研究项目的性质各不相同，但它们都反映了当前推动数学大部分发展的单一潮流。很大程度上受物理学的启发，许多领域正处于从研究从“一维”角度可获得的现象转向使用“二维”工具进行攻击的过程中。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。最终结果还有待于在拓扑学中观察；但它已经对同伦理论的大部分内容，以及结和高维几何结构的新不变量的著名构造，产生了更深入的理解。＊＊＊

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. ***