Optimal shapes in geometry and physics: Isoperimetry in modern analysis

这个项目将在弯曲的空间中找到最佳的等周形状：优化几何或解析量的形状，例如由给定区域的表面包围的体积，或给定区域的鼓的共振频率。最优形状产生了被广泛应用于微分方程式、几何分析、统计物理、概率论和量子计算的工具。通过这项工作，我们将在弯曲空间的几何和其中的算子的物理之间建立联系。该项目的显著好处包括增加基础数学知识，建设澳大利亚世界级几何分析社区的能力，以及与国际合作伙伴建立牢固的联系。

This project will find the best isoperimetric shapes in curved spaces: shapes that optimise geometric or analytic quantities, such as the volume enclosed by a surface of a given area, or the resonant frequency of a drum of given area. The optimal shapes lead to tools that are widely used in differential equations, geometric analysis, statistical physics, probability theory, and quantum computing. Through this work, we will forge connections between the geometry of curved spaces, and the physics of operators therein. The significant benefits of this project include increasing fundamental mathematical knowledge, building capacity in Australia’s world-class geometric analysis community, and strong links with international partners.