Determinants, analytic torsion, and functional analytic models for Dirac operators

DMS-0072551 Matthias LeschDirac类算子是黎曼几何中自然产生的对称一阶椭圆微分算子。他们对理论物理很感兴趣。在有边界的紧致流形上给出这样一个算子，就有可能刻画导致自伴椭圆问题的边界条件。这些所谓的适定边值问题是由R.T.Seeley提出的。最近，这位研究人员与J.Bruning合作，从功能分析的角度给出了一份报告。这是当前项目的起点，在该项目中，研究人员希望研究与带边界流形上的狄拉克算子相关的谱不变量。更具体地说，我们想要给出任意扁丛存在时解析扭转的粘合公式的一个解析证明。作为结果，我们得到了在最一般的背景下，包括带边界的流形，关于解析挠率和拓扑挠率之间关系的Cheeger-Muller定理的另一种方法。与解析挠率有关的一个不变量是Zeta正则化行列式。我们想把这个行列式作为边界条件的函数来研究，推广了Burghelea-Friedlander-Kappeler和Scott-Wojciechowsk.第三个子项目，与J.Bruning一起，研究了Dirac型算子边值问题的泛函分析方法.最终目的是为流形上的Dirac算子找到一个简单的泛函分析模型，该模型可以得到热迹展开式、指数定理和谱流定理等基本结果。Dirac算子是黎曼几何中自然产生的对称一阶椭圆型微分算子。他们对理论物理很感兴趣。在算符的谱(物理系统的‘可测量的’量)和基本几何之间有有趣的关系。一般来说，算符的谱很难计算。然而，可以提取与几何密切相关的某些光谱不变量。如果基本几何是一个有边界的流形，那么首先必须施加边界条件才能得到一个自伴问题，从而得到一个明确定义的谱。在当前的项目中，作者想要研究这类边值问题的某些谱不变量(解析挠、行列式)。一个特殊的问题是不变量对边界条件的选择的依赖性。

DMS-0072551Matthias LeschDirac type operators are symmetric first order elliptic differential operators arising naturally in Riemannian geometry. They are of considerable interest in theoretical physics. Given such an operator on a compact manifold with boundary it is possible to characterize those boundary conditions which lead to a self--adjoint elliptic problem. These so--called well--posed boundary value problems were introduced by R. T. Seeley. A recent account from a functional analytic perspective was given by the researcher in collaboration with J. Br\"uning. This is the starting point of the current project in which the researcher wants to investigate spectral invariants associated to Dirac operators on manifolds with boundary. More specifically,we want to present an analytic proof of the gluing formula for the analytic torsion in the presence of an arbitrary flat bundle. As a consequence one obtains an alternative approach to Cheeger--Muller type theorems on the relation between analytical and topological torsion in the mostgeneral setting, including manifolds with boundary. An invariant related to the analytic torsion is the zeta-regularized determinant. We want to study this determinant as a functionof the boundary condition, generalizing earlier work of Burghelea-Friedlander--Kappeler and Scott-Wojciechowski.The third subproject, jointly with J. Bruening, deals with a functional analytic approach to boundary value problems for Dirac type operators. The ultimate goal is to find a simple functional analytic model for a Dirac type operator on a manifold with boundary which allows to derive basic resultslike heat trace expansions, index theorems and the spectralflow theorem. Such a model would hopefully lead to such theorems for manifolds with corners.Dirac type operators are symmetric first order elliptic differential operators arising naturally in Riemannian geometry. They are of considerable interest in theoretical physics. There are interesting relations between the spectrum of the operator (the 'measurable' quantities of the physical system) and the underlying geometry. In general the spectrum of the operator is hard to compute. However, it is possible to extract certain spectral invariants which are intimately related to the geometry. If the underlying geometry is a manifold with boundary then, at first, one has to impose boundary conditions in order to get a self--adjoint problem and hence a well--defined spectrum. In the current project the author wants to investigate certain spectral invariants (analytic torsion, determinant) for such boundary value problems. A special issue is the dependence of the invariants on the choice of the boundary condition.