Algebraic Topology

Randy McCarthy(Pi)Matthew Ando(co-Pi)代数拓扑学的一个基本思想是通过附加在变形下不变的代数结构来研究拓扑空间。一个好的结构往往会与数学的其他领域产生深刻的联系。Ando和McCarthy建议研究涉及三种结构的问题：椭圆上同调、代数K-理论和函子演算。Ando提出了几个涉及椭圆上同调的项目，它提供了拓扑学、代数几何和弦理论之间的关系。例如，他基于theta函数的代数理论，提出了一种新的方法来证明椭圆族的紧性定理。麦卡锡计划进一步推进他已经成功的计划，即使用函数式演算来更好地理解代数K-理论。例如，他希望函子演算中的观点能够澄清尼曼关于三角范畴的K理论。Ando和McCarthy还计划一个联合项目，研究函子演算在Voevodsky等人的新方案同伦理论中的作用。

DMS-0071482Randy McCarthy (PI)Matthew Ando (co-PI)A fundamental idea in algebraic topology is to study topological spaces by attaching to them algebraic structures which are invariant under deformations. A good structure will often produce profound relationships with other areas of mathematics. Ando and McCarthy propose to investigate problems involving three suchstructures, elliptic cohomology, algebraic K-theory, and the calculus of functors. Ando proposes several projects involving elliptic cohomology, which provides a relationship between topology, algebraic geometry, and string theory. For example, he proposes a new approach to therigidity theorems for elliptic genera, based on the algebraic theory of theta functions. McCarthy plans to further pursue his already successful program of using the calculus of functors to betterunderstand algebraic K-theory. For example, he hopes that ideas from the calculus of functors will clarify Neeman's K-theory of triangluated categories. Ando and McCarthy also plan a joint project toinvestigate the role of the calculus of functors in the new homotopy theory of schemes of Voevodsky and others.