Mathematical Sciences: Analysis and Geometry of Infinite Dimensional Manifolds

无穷维流形的微分几何是 量子场论的基础 校长 研究者将他的结果推广到迭代积分， 狄拉克算子的维滕流是在 超对称共形场设定的前期授权 理论 他将继续开发分析工具， 循环空间。 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Connes的应用 含链环的非交换微分几何理论 量子场论 经典场论涉及分配给每个点， 四维时空一个矢量或标量，如 电场或温度。 在量子场论中 通常会为每个对象分配某种运算符或函数， 时空中的一点 使用内积，从这些运算符 可以得到一个可观测的量，然后可以通过以下方式进行检查： 反复实验。 在这个项目中， 量子场将使用无限维 超曲面和数学家的工具称为循环空间。

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.