Singular spaces in analysis and geometry

分析和几何中的奇异空间

• 批准号: DP210103242
• 负责人: Dr Jesse Gell-Redman
• 金额: $ 27.79万
• 依托单位: The University of Melbourne
• 依托单位国家: 澳大利亚
• 项目类别: Discovery Projects
• 财政年份: 2021
• 资助国家: 澳大利亚
• 起止时间: 2021-09-30 至 2024-09-30
• 项目状态: 已结题
• 来源: https://dataportal.arc.gov.au/RGS/Web/Grants/DP210103242
• 关键词: Singular; spaces; analysis; geometry

Singularities arise naturally in many areas of mathematics, as models of symmetry, degeneracy, and asymptotic collapse. The aim of this project is to provide powerful, generlisable tools to elucidate the interplay between modes of singularity formation and solutions to the important differential equations which arise in geometric analysis. The proposed framework builds upon the established success of microlocal analysis, initiated by Melrose in the 1980's, in the generalisation of landmark theorems like the Atiyah-Singer index theorem to more general Riemannian manifolds. This project will benefit Australia by increasing its capacity in pure mathematics in this highly active research area.

奇点在数学的许多领域中自然出现，作为对称性、简并性和渐近坍缩的模型。这个项目的目的是提供强大的，generalisable工具来阐明奇异性形成和解决方案的重要微分方程，在几何分析中出现的模式之间的相互作用。建议的框架建立在建立成功的微局部分析，由梅尔罗斯在20世纪80年代，在概括的里程碑式的定理，如Atiyah-辛格指数定理更一般的黎曼流形。该项目将有利于澳大利亚提高其在这一高度活跃的研究领域的纯数学能力。