A Functional Analysis of the Hypoelliptic Laplacian

敲打一个钟、一个球体或任何几何物体，它就会响起。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。众所周知，亚椭圆拉普拉斯的光谱蕴含着数学和物理学中的深刻真理，但它仍然很难理解。我们提出了一种新的分析基础，它将取代迄今非分析的特别方法，并使许多新的结果变得可用。这是更好地理解位于量子力学和经典世界之间的微分方程式的关键。这将为澳大利亚在微分方程式和几何的新世纪中的领导地位以及年轻数学家的培训铺平道路。

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.