Mathematical Sciences: Inverse Spectral Problems in Riemannian Geometry

小行星9404298 该奖项支持的研究重点是与流形几何相关的问题。 它涉及到确定在何种程度上的频谱的拉普拉斯-贝尔特拉米算子的紧凑黎曼流形决定几何的流形。 最近的例子表明，例如，光谱本身并不确定流形的等距。 进一步的工作也将努力了解如何流形可以isospectral但甚至没有局部等距。 从这些例子中产生的例子提供了很好的机会来识别特定的局部几何不变量，这些不变量不是光谱确定的。 与等谱平面域有关的工作也将继续进行。 这种域的构造出奇地简单;它们是作为某些轨道折叠的底层空间而出现的。 使用这种方法，将做的工作，以确定是否有可能构建任何有限阶的非等距平面域的等谱集。 最后，研究了双曲流形上的拉普拉斯谱与长度谱的关系，以及环面和海森堡流形上的薛定谔算子。 通过研究定义在这些空间上的经典微分算子来分析域和流形，现在是几何分析的一个既定领域。 我们知道，域的几何性质与算子谱的性质交织在一起。 过去十年的结果表明，这种依赖性并不像人们曾经认为的那么简单。 这项工作旨在澄清的联系，并扩大该理论的应用范围。 ***

9404298 Gordon The research supported by this award focuses on questions related to the geometry of manifolds. It concerned with the determination of the extent to which the spectrum of the Laplace- Beltrami operator of a compact Riemannian manifold determines the geometry of the manifold. Recent examples show, for instance, that the spectrum alone does not determine manifolds up to isometry. Further work will also be done in efforts to understand how manifolds can be isospectral yet not even locally isometric. Examples growing out of these examples provide excellent opportunities to identify specific local geometric invariants which are not spectrally determined. Work related to isospectral plane domains will also continue. Constructions of such domains are surprisingly simple; they arise as underlying spaces of certain orbifolds. Using this method, work will be done to establish whether or not it is possible to construct isospectral sets of non-isometric plane domains of any finite order. Finally, studies on Laplace versus length spectra of hyperbolic manifolds and on the Schrodinger operator on tori and Heisenberg manifolds will be carried out. Analysis of domains and manifolds through the study of classical differential operators defined on these spaces is now an established field of geometric analysis. One knows that geometric properties of the domains are intertwined with properties of the spectrum of the operators. Results of the last decade show that the dependence is not as simple as once thought. This work seeks to clarify the connections and broaden the scope of the theory's applications. ***