Geometric Phenomena in Homotopy Theory

本项目的目的是研究数学和数学物理之间的某些联系。数学涉及的领域是代数拓扑，它采用代数方法研究拓扑空间（形状）在连续变形下保持不变（不变性）的性质。在这些性质或不变量中，某些线性性质的不变量起着显著的作用，被称为广义上同调理论。该项目的主要目标是研究这种上同调理论与物理、几何和代数几何（代数方程的解空间）的某些领域的解释和相互作用。具体来说，研究者主要关注两个方面。第一个目标是找到椭圆上同调的几何（或物理）模型。从全纯共形场论的范畴出发，提出了用无限环空间理论可以恢复椭圆上同调的猜想。另一个领域涉及Voevodsky引入的代数变种同伦范畴中的morava k理论的研究。

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.