Mathematical Sciences: Spectral Theory of Riemannian Manifolds

主要研究者将研究完备黎曼流形上拉普拉斯算子的本征函数。当空间是紧的时，他将提供这些特征函数的节点集的Hausdorff测度的上界和下界。在非紧的情况下，他将得到这些特征函数衰减的下界。拉普拉斯算子的本征函数是钟声和乐器的谐波的数学表示。特殊形状的钟可以有错综复杂的各种谐波；通常是第三或第七次谐波占主导地位。在更高的维度上，这些钟的形状由黎曼流形数学表示。如果这些钟声扩展到无限大，黎曼流形将不得不是非紧凑的。首席研究员将研究这种谐波的衰减或衰减。

The principal investigator will study the eigenfunctions of the Laplacian operator on complete Riemannian manifolds. When the space is compact, he will provide upper and lower bounds for the Hausdorff measure of the nodal set of these eigenfunctions. In the noncompact situation, he will derive lower bounds for the decay of these eigenfunctions. Eigenfunctions of the Laplacian operator are the mathematical representations of harmonics of bells and musical instruments. Peculiar shaped bells can have an intricate assortment of harmonics; often the third or seventh harmonics dominate. In higher dimensions, the shapes of these bells are represented mathematically by Riemannian manifolds. Were these bells to flare out to infinity, the Riemannian manifold would have to be noncompact. The principal investigator will study the decay or damping out of such harmonics.