Mathematical Sciences: Inverse Spectral Problems in Riemannian Geometry

9404298戈登，这个奖项支持的研究集中在与流形几何有关的问题上。它涉及确定紧致黎曼流形的Laplace-Beltrami算子的谱决定流形几何的程度。例如，最近的例子表明，单靠频谱并不能决定流形是否达到等距。进一步的工作还将努力理解流形如何可以是等谱的，但甚至不是局部等距的。从这些例子中发展出来的例子提供了极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。这类区域的构造出奇地简单；它们是作为某些奥比沃尔德的底层空间而出现的。利用这种方法，我们将确定是否可以构造任意有限阶的非等距平面域的等谱集。最后，我们将研究双曲流形的拉普拉斯-长度谱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区域和流形，现在是几何分析的一个既定领域。人们知道，域的几何性质与算子的谱性质交织在一起。过去十年的结果表明，这种依赖并不像人们曾经认为的那么简单。这项工作旨在澄清两者之间的联系，并拓宽该理论的应用范围。***

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. ***