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 正则化行列式与边界上的某些 Toeplitz 行列式之间的联系。 她证明了椭圆算子的坎贝尔-豪斯多夫公式和相关的迹公式，并开发了符号计算的新方法。 她现在将这些结果扩展到有边界的流形。 Okikiolu 还证明了二维和三维球体的强 Szego 极限定理的类似物，并将进一步推广这一结果。 偏微分方程构成了物理世界数学建模的基础。 数学分析的作用与其说是创建方程，不如说是提供有关解决方案的定性和定量信息。 这可能包括有关独特性、平滑性和成长性问题的答案。 此外，分析通常会开发近似解的方法并估计这些近似的准确性。

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.