Geometric Analysis and Spectral Theory

几何分析和谱理论

• 批准号: RGPIN-2019-03900
• 负责人: Jakobson, Dmitry
• 金额: $ 3.13万
• 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: 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=759005
• 关键词: Geometric; Analysis; Spectral; Theory

I work at the intersection of geometric analysis, partial differential equations (PDE) and mathematical physics, studying spectra and eigenfunctions of Laplace type operators, arising 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, and many other phenomena. Related problems arise when one studies acoustics, optics, data analysis, fluid and plasma flows. Recently, I worked on several important questions in spectral theory: (a) Study of high energy eigenfunctions for ergodic systems, including Quantum Ergodicity for branching billiards; behaviour of Eisenstein series (scattering theory eigenfunctions on hyperbolic manifolds); study of the Loschmidt echo effect. (b) Distribution of resonances on hyperbolic surfaces, in a series of papers with F. Naud. (c) Nodal and critical sets (nodal lines of Eisenstein series; conformal invariants from nodal sets, and graph theory analogs).  I contributed to the study of gaps in geodesic length spectra on negatively curved manifolds. I continued the study of conformally covariant operators, showing in particular that 0 generically is not an eigenvalue of the conformal Laplacian. I studied the behaviour of rank and Tutte polynomials of sequences of sparse bounded degree graphs that converge Benjamini-Schramm, and showed, that natural coefficient measures converge to a delta function. I plan to focus on the following projects: study conformally covariant operators on manifolds with boundary, and related problems in geometry and PDE; study resonances and eigenfunctions on hyperbolic manifolds; extend the results about rank polynomials to Bollobas-Riordan polynomials, and to sequences of dense graphs, study applications to statistical physics, study asymptotic distribution of zeros;  study eigenfunction averages over submanifolds, including totally geodesic submanifolds;  study of spectra and eigenvectors of elements of the group ring, focussing on surface groups; study geometric and spectral invariants on "random" 3manifolds;  study extremal metrics for eigenvalue functionals, and connections to isometric embeddings; study eigenfunction localization for billiards; study length spectra;  study spectra and eigenfunctions for "towers of covers" of Riemannian manifolds, including surfaces.

我的工作涉及几何分析、偏微分方程 (PDE) 和数学物理的交叉领域，研究天体力学、热和波传播以及描述振动的拉普拉斯本征函数的研究中出现的拉普拉斯算子的谱和本征函数。弦或鼓的结构、量子力学中的纯态以及许多其他现象在研究声学、光学、数据分析、流体和等离子体流时都会出现。谱理论：（a）遍历系统的高能本征函数的研究，包括分支台球的量子遍历性；爱森斯坦级数的行为（双曲流形上的散射理论本征函数）；（b）洛施密特回波效应的分布。双曲曲面，与 F. Naud 合作的一系列论文。 (c) 节点集和临界集（爱森斯坦级数的节点线；我对负弯曲流形上的测地线长度谱间隙的研究做出了贡献，我继续研究共形协变算子，特别表明 0 一般不是共形的特征值。我研究了收敛 Benjamini-Schramm 的稀疏有界度图序列的秩和 Tutte 多项式的行为，并表明，自然系数度量收敛于 delta 函数。我计划重点关注以下项目：研究具有边界的流形上的共变算子，以及研究双曲流形上的共振和本征函数；关于秩多项式、Bollobas-Riordan 多项式以及稠密图序列，研究在统计物理中的应用，研究零点的渐近分布；子流形上的特征函数平均值，包括完全测地线子流形；研究群环元素的谱和特征向量，重点关注表面群；研究“随机”3流形的几何和谱不变量，以及与等距嵌入的联系；研究台球的特征函数定位；研究黎曼流形（包括曲面）的“覆盖塔”的光谱和本征函数。