Spectrum and Geometry

我从事几何分析、偏微分方程和数学物理的交叉研究，研究拉普拉斯算子和狄拉克算子的谱和特征函数。拉普拉斯出现在天体力学、热波传播和量子力学的研究中。拉普拉斯本征函数描述弦或鼓的振动、量子力学中的纯态（原子和分子）以及许多其他现象。在研究声学（声音的传播）和光学时，会出现相关的问题；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。我研究了可积系统和遍历系统的特征函数的行为。我主要研究了谱理论中的几个主要问题：(a)遍历系统本征函数极限的研究（双曲曲面上的量子遍历性；Dirac算子和Hodge拉普拉斯算子本征函数的类似结果；分支台球的QE，这是2012年建立的一个重要的新结果）；以及可积系统，其中一些结果被推广到薛定谔方程的解；(b)谱函数（建立了变负曲率曲面的下界，证明了一个精确的迹线公式）；(c)节点和临界集（我否定地回答了邱绍德关于特征函数的临界点数目的问题；研究了随机球谐的节点集拓扑；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，我开始使用概率方法研究特征函数的随机波猜想。在过去的两年里，我还研究了共振的光谱理论。此外，我还开创了黎曼度量几何自然空间平均的严谨研究，并将其应用于随机黎曼度量的几何和谱不变量的研究。我的长期目标包括继续研究光谱理论中的各种重要问题，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)度量流形上的测度与平均，包括高维共形类上的“正则”测度的构造，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)中的系统描述了波通过空气-水界面、半导体、晶体中的杂质的传播，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。(2)所研究的本征态是在研究物理学中的洛施密特回波效应（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中的一些基本猜想；发展这些技术也可能导致在物理学共形场论以及量子引力中出现的相关问题的进展。在随机图的研究中出现了相关的问题，并在天体物理学、医学成像和其他领域有应用。(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都有应用。与(3)中的共振研究相关的问题在成像和逆问题中有重要的应用。量子计算理论（“量子点”）中出现了半经典理论中的问题；在化学和分子物理；在原子物理学中（原子核）。

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).