Analysis of Spectral Invariants on Manifolds

提议0302647 p。摘要黎曼流形中存在着拉普拉斯-贝尔塔米算子等自然几何算子，我们可以研究流形的几何与这些算子的谱之间的关系。（关于这个主题有一个著名的问题是“你能听到鼓的形状吗？”）这导致了几何算子的谱不变量的研究，它是流形的几何不变量。几何椭圆算子的不变量构成了一类谱不变量，其中特别包括拉普拉斯算子的行列式和某些积分局域不变量，如Yamabe泛函。众所周知，在给定体积和保形类的度量中，Yamabe泛函被极化为常数标量曲率的度量，对于曲面上的拉普拉斯行列式也是如此。这就提出了许多问题，比如拉普拉斯算子的行列式在高维空间中是如何表现的，不同类型的拉普拉斯算子会发生什么，以及其他的不变量是如何表现的。在这些问题上已经有了一些进展，其动机是既要理解特定zetainvariant的行为，又要研究zetainvariant在几何中的可能应用。Okikiolu建议研究与拉普拉斯不变量和其他zeta不变量的行列式的存在性，唯一性和关键度量的行为有关的一些问题，包括有关全局上界或下界，梯度流，共形类的行为，模型问题的行为以及zeta不变量基本分析理论的发展的问题。此外，Okikiolu建议研究流形上Toeplitzoperators的谱不变量的相关问题，并在一个稍微不同的方向上研究从量子场论推广Verlinde公式的问题。理解空间的几何性质和空间上的自然几何微分算子的谱之间的关系是许多科学分支中出现的一个问题。单个特征值很难分析，通常空间的几何形状更清楚地反映在特征值的某些加权平均值上，例如不变量。特别是，流形上拉普拉斯的行列式已经被研究并应用于数学和物理的几个领域，包括拓扑学、量子场论、弦理论、代数几何和保形几何。这里提出的研究应该会导致更完整的几何和物理应用的zeta不变量的数学理论。

Proposal 0302647P.I.: Kate Okikiolu (UCSD)Analysis of Spectral Invariants on Manifolds: AbstractAssociated to a Riemannian manifold, there exist natural geometricoperators such as the Laplace-Beltami operator, and one can study therelationship between the geometry of the manifold andthe spectra of these operators. (A famous question on this theme is``can you hear the shape of a drum?".) This leads to the study of spectral invariants of geometric operators, whichare geometric invariants of the manifold. Thezeta invariants of geometric elliptic operators form a family ofsuch spectral invariants which includes in particular thedeterminant of the Laplacian and certain integrated localinvariants such as the Yamabe functional. It is well known that among metrics of a givenvolume and conformal class, the Yamabe functional is extremized ata metric of constant scalar curvature, and the same is true forthe determinant of the Laplacian on surfaces. This raises manyquestions regarding how the determinant of the Laplacian behavesin higher dimensions, what happens for different types ofLaplacian, and how other zeta invariants behave. There hasalready been progress on several of these questions, themotivation being both to understand the behavior of specific zetainvariants and to investigate possible applications of zetainvariants to geometry. Okikiolu proposes to study a number ofissues related to the existence, uniqueness and behavior ofcritical metrics for the determinant of the Laplacian and otherzeta invariants, including questions concerning global upper orlower bounds, gradient flow, behavior across conformal classes,behavior of model problems, and development of the basic analytictheory of zeta invariants. In addition, Okikiolu proposes to workon related questions concerning spectral invariants of Toeplitzoperators on manifolds, and in a somewhat different direction, onthe problem of extending the Verlinde formulas from quantum fieldtheory. Understanding the relationship between thegeometry of a space and the spectra of natural geometricdifferential operators on the space is a problem which arises in anumber of branches of science. Individual eigenvalues are hard toanalyze and often the geometry of a space is more clearlyreflected by certain weighted averages of the eigenvalues such aszeta invariants. In particular, determinants of Laplacians onmanifolds have been studied and applied in several fields ofmathematics and physics including topology, quantum field theory,string theory, algebraic geometry, and conformal geometry. Theresearch proposed here should lead to a more complete mathematicaltheory of zeta invariants for geometrical and physicalapplications.