Mathematical Sciences: Inverse Spectral Problems in Riemannian Geometry

[404298] Gordon该奖项支持的研究重点是与流形几何有关的问题。它涉及确定紧致黎曼流形的拉普拉斯-贝尔特拉米算子的谱在多大程度上决定了流形的几何形状。例如，最近的例子表明，光谱本身并不能决定等距流形。进一步的工作也将在努力理解流形如何可以是等谱的，但甚至不是局部等距的。从这些例子中产生的例子为识别非光谱确定的特定局部几何不变量提供了极好的机会。与等光谱平面域相关的工作也将继续进行。这些域的构造出奇地简单；它们是某些轨道的底层空间。利用这种方法，我们将确定是否有可能构造任意有限阶的非等距平面域的等谱集。最后，对双曲流形的拉普拉斯随长度谱以及环面流形和海森堡流形上的薛定谔算子进行了研究。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，现在已经成为一个成熟的几何分析领域。我们知道，定义域的几何性质与算子谱的性质是交织在一起的。过去十年的研究结果表明，这种依赖并不像以前想象的那么简单。这项工作旨在澄清这些联系并扩大理论应用的范围。＊＊＊

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. ***