Dynamical spectral rigidity and determination for billiard systems

台球系统的动态谱刚度及其测定

• 批准号: RGPIN-2022-04188
• 负责人: DeSimoi, Jacopo
• 金额: $ 2.26万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758284
• 关键词: Dynamical; spectral; rigidity; determination; billiard

The goal of inverse problems is the reconstruction of an object from a set of coarse observations. Inverse problems constitute a surprisingly broad class of problems with far-reaching applications in virtually any field of science: high energy physics (identifying particles created in scattering events), medicine (CT scans), astrophysics (detecting elements in the photosphere of stars which are billions of light-years away from us), image processing (de-noising and de-blurring of digital images), Big Data and Machine Learning. In this proposal I will describe the analysis of an inverse problem that can be purely formulated in the context of (classical) dynamical systems. Consider the trajectories of a particle that moves freely inside a planar domain and is subject to elastic reflections upon collisions with the boundary of the domain. We call periodic those trajectories that repeat themselves after a finite amount of time: such trajectories trace closed polygons inscribed in the domain. We call Length Spectrum of the domain the set of perimeters of all such polygons. We can then formulate the following inverse dynamical problem: Dynamical Spectral Determination: is it possible to identify the domain (modulo isometries) by the knowledge of its Length Spectrum? The above question, e.g. in the very natural class of smooth domains, is still wide open, and it is unarguably considered an extremely challenging problem. Sarnak conjectured that smooth domains are locally determined by their Laplace spectrum (a question that -quoting M. Kac- is often phrased as: "Can one hear the shape of a drum?"). Due to the tight connection between the Laplace (quantum) and dynamical (classical) spectral problems established by the Wave Trace formula, we find natural to study this conjecture in the dynamical setting. As a first step, we may consider a deformational problem: we say that a domain is dynamically spectrally rigid if all smooth deformations of the domain that preserve its Length Spectrum are necessarily isometries. In the past few years, together with my collaborators, we proved dynamical spectral rigidity for symmetric convex billiards close enough to disks. Also, we proved dynamical spectral determination for a class of (symmetric) analytic open dispersing billiards (such are systems whose dynamics is reminiscent of geodesic flow on manifolds with negative curvature). In the next several years, my research team and I will leverage on the breakthrough techniques that have been developed for the above results to move towards Sarnak's conjecture. On the one hand I will set to prove (local) spectral determination results for smooth convex billiards (possibly with symmetries); on the other hand I will push my work on hyperbolic billiards towards the problem of (local) spectral determination in the smooth category. I believe that these results will be attainable in this decade, and the Discovery Grant will play a major role in their development.

逆问题的目标是根据一组粗略观察重建对象。反演问题构成了一个令人惊讶的广泛问题，在几乎所有科学领域都有深远的应用：高能物理学（识别散射事件中产生的粒子）、医学（CT 扫描）、天体物理学（检测恒星光球层中的元素，这些元素是距离我们数十亿光年）、图像处理（数字图像的去噪和去模糊）、大数据和机器学习。在本提案中，我将描述对反问题的分析，该问题可以纯粹在（经典）动力系统的背景下表述。考虑在平面域内自由移动的粒子的轨迹，并在与域边界碰撞时受到弹性反射。我们将那些在有限时间后重复自身的轨迹称为周期性轨迹：此类轨迹描绘了域中内接的闭合多边形。我们将域的长度谱称为所有此类多边形的周长集合。然后，我们可以制定以下逆动力学问题： 动态谱测定：是否可以通过其长度谱的知识来识别域（模等距）？上述问题，例如在非常自然的平滑域中​​，它仍然是开放的，并且毫无疑问被认为是一个极具挑战性的问题。萨纳克推测平滑域是由拉普拉斯谱局部决定的（引用 M. Kac 的问题通常被表述为：“人们能听到鼓的形状吗？”）。由于波迹公式建立的拉普拉斯（量子）和动力学（经典）谱问题之间的紧密联系，我们很自然地在动力学背景下研究这个猜想。作为第一步，我们可以考虑变形问题：如果保留其长度谱的域的所有平滑变形都必然是等距的，则我们说该域是动态谱刚性的。在过去的几年里，我们与我的合作者一起证明了距离圆盘足够近的对称凸台球的动态谱刚性。此外，我们还证明了一类（对称）解析开放色散台球的动态谱测定（此类系统的动力学让人想起具有负曲率的流形上的测地线流）。在接下来的几年里，我和我的研究团队将利用为上述结果开发的突破性技术，向萨纳克猜想迈进。一方面，我将证明光滑凸台球（可能具有对称性）的（局部）光谱测定结果；另一方面，我将把我在双曲台球方面的工作推向平滑类别中的（局部）谱确定问题。我相信这些成果将在这十年内实现，而探索补助金将在其发展中发挥重要作用。