Mathematical Sciences: Determinants and Other Spectral Invariants for Elliptic and Toeplitz Operators on Manifolds

PI：Okikiolu DMS-9506057 Okikiolu将研究具有边界的流形上某些椭圆算子的Zeta正则行列式和边界上的某些Toeplitz行列式之间的联系。她证明了一个椭圆算子的Campbell-Hausdorff公式和一个相关的迹公式，并发展了新的符号计算方法。她现在将这些结果推广到有边界的流形上。Okikiolu还证明了二维和三维球面上强Szego极限定理的一个类似结果，并将进一步推广这一结果。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

PI: Okikiolu DMS-9506057 Okikiolu will investigate the connection between zeta-regularized determinants of certain elliptic operators on a manifold with boundary and certain Toeplitz determinants on the boundary. She has proved a Campbell-Hausdorff formula for elliptic operators and a related trace formula, and developed new methods for symbolic computation. She will now extend these results to manifolds with boundary. Okikiolu also has proved an analogue of the strong Szego limit theorem for spheres of dimension two and three, and will further generalize this result. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.