Mathematical Sciences: Geometry of the Fundamental Group

罗伯特·布鲁克斯将研究曲面几何和拉普拉斯本征值之间的联系。这些问题的研究工作在几年前因诸如“你能听到鼓的形状吗？”之类的问题而流行起来。如果你有完美的音高，那么理论上你可以检测到鼓的谐波，也就是拉普拉斯的特征值。所以问题是，从这些特征值的知识中，可以检测出一个曲面的几何图形有多少。Brooks的工作将集中于研究双曲流形的小特征值和紧致流形上的等谱问题。紧致流形的低本征值已经有了估计。布鲁克斯将致力于将这些推广到有限体积的情况。这是一个非平凡的推广，但肯定是一个自然的推广，因为早期估计的一个关键成分是内射性半径的概念，对于非紧致有限体积流形来说，它是零。他在等谱问题上的工作涉及到研究给定保形类中的等谱度量集的拓扑。这里的目的是将Yamabe问题的解应用到三维流形的几何中。

Robert Brooks will investigate connections between the geometry of a surface and the eigenvalues of the Laplacian. Work on these problems was popularized some years ago by questions such as "Can you hear the shape of a drum?". If you have perfect pitch then you can in theory detect the harmonics of the drum, that is the eigenvalues of the Laplacian. So the question is how much geometry of a surface can be detected from knowledge of these eigenvalues. Brooks' work will focus on the study of the small eigenvalues of hyperbolic manifolds and isospectral problems on compact manifolds. There are already estimates for the low eigenvalues of compact manifolds. Brooks will work to extend these to the finite volume case. This is a non-trivial extension but certainly a natural one since a key ingredient of earlier estimates is the the notion of injectivity radius which is zero for non-compact finite volume manifolds. His work on isospectral problems is concerned with a study of the topology of sets of isospectral metrics in a given conformal class. The intention here is to apply the solution of the Yamabe problem to the geometry of three dimensional manifolds.