Spectrum and Geometry

光谱与几何

• 批准号: RGPIN-2014-05385
• 负责人: Jakobson, Dmitry
• 金额: $ 2.04万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567135
• 关键词: Spectrum; Geometry

I work at the intersection of geometric analysis, PDE and mathematical physics, studying spectra and eigenfunctions of Laplace-type and Dirac-type operators. Laplacian arises in the study of celestial mechanics, heat and wave propagation, and quantum mechanics. Eigenfunctions of Laplacian describe vibrations of a string or a drum, pure states in quantum mechanics (atoms and molecules), and many other phenomena. Related problems arise when one studies acoustics (propagation of sound) and optics; data analysis ("manifold learning"); fluid flows and plasma flows in thermonuclear reactors. I have studied the behaviour of eigenfunctions for both integrable and ergodic systems. I have worked on several main questions in Spectral Theory: (a) Study of limits of eigenfunctions for ergodic systems (Quantum Ergodicity on hyperbolic surfaces; analogous results for eigenfunctions of Dirac operator and Hodge Laplacian; QE for branching billiards, an important new class of results established in 2012); as well as for integrable systems, where some of the results were extended to solutions of Schrodinger equation; (b) Spectral function (I established lower bounds and proved an accurate trace formula for surfaces of variable negative curvature); (c) Nodal and critical sets (I answered in the negative a question raised by S.T. Yau about the number of critical points of eigenfunctions; studied topology of nodal sets of random spherical harmonics; and recently found surprising connections to conformal geometry); (d) In a recent paper, I initiated the study of random wave conjectures for eigenfunctions using probabilistic methods. In the last 2 years I have also studied spectral theory of resonances. In addition, I initiated the rigorous study of averaging over geometrically natural spaces of Riemannian metrics, with applications to the study of geometric and spectral invariants of random Riemannian metrics. My long-term objectives include continuing the study of various important questions in spectral theory, with applications to geometry and PDE. My short-term objectives include continuing the work on research programs, concentrating on the following areas: (1) Semiclassical theory of discontinuous systems; (2) Measures and averaging on manifolds of metrics, including constructing "canonical" measures on conformal classes in higher dimensions, as well as the study of Random Wave conjectures; (3) Spectral theory on asymptotically hyperbolic manifolds; (4) Spectral theory of conformally covariant operators. Systems in (1) describe wave propagation through air-water interface, semiconductors, impurities in crystals, including seismic waves and elasticity. Semiclassical and ergodic theory of systems will have important practical applications. The eigenstates studied in (2) arise in the study of Loschmidt echo effect (quantum fidelity) in physics. The novel techniques described in (2) may help to answer some fundamental conjectures in Quantum Chaos raised in 1960s and 1970s; developing those techniques may also lead to progress in related problems arising in Conformal Field Theory in Physics, as well as in Quantum Gravity. Related questions arise in the study of random maps, and have applications in astrophysics, medical imaging and other fields. Questions considered in (4) have applications in conformal geometry, relativity, and certain nonlinear PDE. Problems related to the study of resonances in (3) have important application in imaging and inverse problems. Problems in semiclassical theory arise in the theory of quantum computing ("quantum dots"); in chemical and molecular physics; and in atomic physics (atomic nuclei).

我从事几何分析、偏微分方程和数学物理的交叉研究，研究拉普拉斯型和狄拉克型算子的谱和本征函数。拉普拉斯起源于天体力学、热和波传播以及量子力学的研究。拉普拉斯的本征函数描述了弦或鼓的振动，量子力学(原子和分子)中的纯态，以及许多其他现象。当人们研究声学(声音传播)和光学；数据分析(“流形学习”)；热核反应堆中的流体流动和等离子体流动时，就会出现相关的问题。我研究了可积系统和遍历系统的本征函数的行为。我研究了谱理论中的几个主要问题：(A)遍历系统的本征函数的极限的研究(双曲曲面上的量子遍历性；Dirac算子和Hodge Laplace的本征函数的类似结果；分支台球的QE，这是2012年建立的一类重要的新结果)；以及可积系统的一些结果，其中一些结果被推广到薛定谔方程的解；(B)谱函数(我建立了下界，并证明了可变负曲率曲面的精确迹公式)；(C)节点集和临界点集(我用否定的答案回答了Yau提出的关于特征函数临界点个数的问题；研究了随机球谐的节点集的拓扑；最近发现了与共形几何的惊人联系)；(D)在最近的一篇论文中，我用概率方法开始了对特征函数的随机波猜想的研究。在过去的两年里，我还学习了共振的光谱理论。此外，我还开创了黎曼度量在几何自然空间上平均的严格研究，并应用于随机黎曼度量的几何不变量和谱不变量的研究。我的长期目标包括继续研究光谱理论中的各种重要问题，以及在几何学和偏微分方程中的应用。我的短期目标包括继续研究项目的工作，集中在以下领域：(1)不连续系统的半经典理论；(2)度量流形上的度量和平均，包括在高维共形类上构造“正则”度量，以及随机波猜想的研究；(3)渐近双曲流形的谱理论；(4)共形协变算子的谱理论。(1)中的系统描述了波在空气-水界面、半导体、晶体中的杂质中的传播，包括地震波和弹性。系统的半经典和遍历理论将具有重要的实际应用。在(2)中所研究的本征态是在物理中研究洛希米特回声效应(量子保真度)时产生的。(2)中描述的新技术可能有助于回答20世纪60年代和70年代提出的量子混沌中的一些基本猜想；发展这些技术也可能导致物理学中的共形场论以及量子引力中出现的相关问题的进展。随机地图的研究中出现了相关问题，在天体物理、医学成像等领域有着广泛的应用。文(4)中讨论的问题在保角几何、相对论和某些非线性偏微分方程中都有应用。与(3)中的共振研究有关的问题在成像和反问题中有重要的应用。半经典理论中的问题出现在量子计算理论(“量子点”)、化学和分子物理以及原子物理(原子核)中。