Mathematical Sciences: Investigations in Spectral & Fractal Geometry: Vibrations of Fractal Drums, Spectral Zeta Functions, Analysis on Fractals, & Variational Ellip

拉皮德斯将研究具有各种边界条件的拉普拉斯本征值分布的渐近公式的锐化形式。目标是更清楚地说明边界的Minkowski维度在这种分布中的作用。关于伴随热半群的迹也有类似的公式。不仅研究区域的边界，而且研究区域本身是分形的情况。流形的几何与流形上椭圆微分算子的不变量之间的关系是当代数学的主要统一主题之一。具有分形边界的区域是如此不规则，以至于其维度不再是整数，这是最近将光谱信息与几何联系起来的一些突破的背景。这项工作在研究多孔介质和波在分形面上的散射方面具有应用价值。

Lapidus will investigate sharpened forms of the asymptotic formula for the eigenvalue distribution of Laplacians with various boundary conditions. The goal is to make yet clearer the role of the Minkowski dimension of the boundary in this distribution. There are analogous formulas for the trace of the associated heat semigroup. The case in which not only the boundary of the domain, but the domain itself, is fractal will be studied. The relationship between the geometry of a manifold and the invariants of the elliptic differential operators on it is one of the main unifying themes of contemporary mathematics. The case of a region with fractal boundary, so irregular that its dimension is no longer an integer, has been the setting for some recent breakthroughs relating spectra information to geometry. This work has applications to the study of porous media and the scattering of waves from fractal surfaces.