Boundary value problems and index theory on Riemannian and Lorentzian manifolds

在先前项目“洛伦兹流形上的指标理论”的基础上，计划研究黎曼和洛伦兹环境下一阶算子的边值问题和指标理论。这包括了带边界流形上一般一阶椭圆算子的几何指标公式的推导、相对指标理论（Gromov和Lawson）、非紧非光滑边界的边值问题、高指标理论和黎曼侧的callias型算子。在洛伦兹流形上，研究了狄拉克型算子的局部指标理论，以及狄拉克算子的初边值问题和特征初值问题。

Building on the results of the previous project “Index theory on Lorentzian manifolds” it is planned to investigate boundary value problems and index theory for first order operators in both the Riemannian and the Lorentzian setting. This emcompasses the derivation of geometric index formulas for general first-order elliptic operators on manifolds with boundary, relative index theory à la Gromov and Lawson, boundary value problems for non-compact and non-smooth boundary, higher index theory and Callias-type operators on the Riemannian side. On Lorentzian manifolds we investigate local index theory for Dirac-type and more general operators, initial-boundary value problems and the characteristic initial value problem for Dirac operators.