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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611608
• 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).

我工作在几何分析，偏微分方程和数学物理的交叉点，研究拉普拉斯型和狄拉克型算子的谱和本征函数。拉普拉斯算子出现在天体力学、热和波传播以及量子力学的研究中。拉普拉斯算子的本征函数描述了弦或鼓的振动，量子力学中的纯态（原子和分子）以及许多其他现象。当人们研究声学（声音的传播）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)研究遍历系统的本征函数极限（双曲曲面上的量子遍历性; Dirac算子和Hodge Laplacian的本征函数的类似结果;分支台球的QE，2012年建立的一类重要新结果）;以及可积系统，其中 部分结果推广到了薛定谔方程的解; (b)谱函数（建立了变负曲率曲面的下界，证明了变负曲率曲面的精确迹公式）; (c)节点集和临界集（我对S. T.丘关于本征函数临界点的数量;研究了随机球谐函数节点集的拓扑结构;最近发现 与共形几何的惊人联系）; (d)在最近的一篇论文中，我开始了用概率方法研究本征函数的随机波动。 在过去的两年里，我也研究了共振的光谱理论。此外，我发起了严格的研究平均几何自然空间的黎曼度量，与应用研究的几何和谱不变量的随机黎曼度量。 我的长期目标包括继续研究光谱理论中的各种重要问题，并将其应用于几何和偏微分方程。我的短期目标包括继续研究工作 方案，集中于以下领域： (1)不连续系统的半经典理论 (2)度量流形上的测度和平均，包括在高维共形类上构造“规范”测度，以及随机波结构的研究; (3)渐近双曲流形上的谱理论 (4)共形协变算子的谱理论。 （1）中的系统描述了通过空气-水界面、半导体、晶体中的杂质的波传播，包括地震波和弹性。系统的半经典和遍历理论将有重要的实际应用。（2）中研究的本征态是在研究物理学中的洛什基回波效应（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的一些基本问题;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。随机映射研究中出现了相关问题，并在天体物理学、医学成像等领域有应用。在（4）中考虑的问题在共形几何、相对论和某些非线性偏微分方程中有应用。与研究共振有关的问题在成像和逆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”）、化学和分子物理学以及原子物理学（原子核）中。