Geometric Phenomena in Homotopy Theory

DMS-0072300 Igor Kriz本项目的目的是研究数学和数学物理之间的某些联系。代数学涉及的领域是代数拓扑学，它采用代数方法来研究拓扑空间（形状）在连续变形下保持不变（不变）的性质。在这些性质或不变量中，某些线性性质的不变量（称为广义上同调理论）扮演着重要的角色。本项目的主要目的是研究这种上同调理论与物理学、几何学和代数几何学（代数方程解的空间）的某些领域的解释和相互作用。具体来说，研究者集中在两个领域。第一个目标是找到椭圆上同调的几何（或物理）模型，证明椭圆上同调可以由一类全纯共形场论的无限圈空间理论恢复。另一个领域是研究Voevodsky提出的代数变量同伦范畴中的Morava K-理论。

DMS-0072300Igor KrizThe aim of the present project is to study certain connections between mathematics and mathematical physics. The area of mathematicsconcerned is Algebraic topology, which employs methods of algebrato investigate properties of topological spaces (shapes) which remain unchanged (invariant) under continuous deformation. Among such properties or invariants, a distinguished role is played by certain invariants of linear nature, known as generalized cohomology theories. The main goal of this project is studying interpretations and interactions of such cohomology theories with certain areas of physics, geometry and algebraic geometry (spaces of solutions of algebraic equations).Concretely, the investigator focuses on two areas. The first objectis finding a geometric (or physical) model of elliptic cohomology.It is conjectured that elliptic cohomology can be recovered byinfinite loop space theory from a category of holomorphicconformal field theories. The other area concerns the study ofMorava K-theories in a homotopy category of algebraic varietiesintroduced by Voevodsky.