Mathematical Sciences: Analysis and Geometry of Infinite Dimensional Manifolds

无限维流形的微分几何是量子场论研究的基础。首席研究员将把他在之前的资助下发展的关于迭代积分和狄拉克算子的Witten电流的结果扩展到超对称共形场论的设置。他将继续开发循环空间的分析工具。这些包括对数Sobolev不等式，奇异积分估计，deRham上同调和Feynman- Kac公式。最后，他将研究cones的非交换微分几何理论的应用，包括与量子场论的联系。经典场论涉及到给四维时空的每个点分配一个矢量或标量，比如电场或温度。在量子场论中，通常将某种算子或函数赋给时空中的每个点。利用内积，从这些算符中可以得到一个可观察到的量，然后可以通过反复实验来检验。在这个项目中，量子场的微妙性质将使用无限维超曲面和称为循环空间的数学家工具来研究。

Differential geometry of infinite dimensional manifolds is basic to the study of quantum field theory. The principal investigator will extend his results on iterated integrals and the Witten current for the Dirac operator developed under the previous grant to the setting of supersymmetric conformal field theory. He will continue his development of analytic tools for loop spaces. These include logarithmic Sobolev inequalities, singular integral estimates, deRham cohomology and the Feynman- Kac formula. Lastly, he will investigate applications of Connes' theory of non-commutative differential geometry including links to quantum field theory. Classical field theory involves assigning to each point of four-dimensional space-time a vector or scalar quantity such as the electrical field or temperature. In quantum field theory usually some sort of operator or function is assigned to each point in space-time. Using inner products, from these operators an observable quantity may result which then can be checked by repeated experimentation. In this project subtle properties of quantum fields will be investigated using infinite dimensional hypersurfaces and a mathematician's tool known as loop spaces.