A Functional Analysis of the Hypoelliptic Laplacian

亚椭圆拉普拉斯算子的泛函分析

• 批准号: DP230100434
• 负责人: Dr Dmitriy Zanin
• 金额: $ 29.25万
• 依托单位: The University of New South Wales
• 依托单位国家: 澳大利亚
• 项目类别: Discovery Projects
• 财政年份: 2023
• 资助国家: 澳大利亚
• 起止时间: 2023-01-01 至 2025-12-31
• 项目状态: 未结题
• 来源: https://dataportal.arc.gov.au/RGS/Web/Grants/DP230100434
• 关键词: Functional; Analysis; Hypoelliptic; Laplacian

Strike a bell, a sphere, or any geometrical object, and it rings. The frequencies of ringing are the mathematical spectrum, which encodes deep secrets about the shape of the object. The spectrum of the hypoelliptic laplacian is known to carry deep truths in mathematics and physics, but it remains difficult to understand. We propose a new analytic foundation, which will replace the so far non-analytical ad hoc approach, and make accessible many new results. It is key to better understanding differential equations which lie at the boundary between quantum mechanics and the classical world. This will pave the way for Australian leadership in a new century of differential equations and geometry, and training of young mathematicians.

敲击一个钟、一个球体或任何几何物体，它就会响。振铃的频率是数学频谱，它编码了关于物体形状的深层秘密。亚椭圆拉普拉斯算子的谱在数学和物理学中被认为是有深刻意义的，但是它仍然很难被理解。我们提出了一个新的分析基础，这将取代迄今为止的非分析特设的方法，并访问许多新的结果。它是更好地理解位于量子力学和经典世界之间的微分方程的关键。这将为澳大利亚在微分方程和几何学的新世纪中的领导地位以及年轻数学家的培训铺平道路。