Geometric Analysis and Spectral Theory

我从事几何分析、偏微分方程（PDE）和数学物理的交叉研究，研究天体力学、热波传播和量子力学中出现的拉普拉斯算子的谱和特征函数。拉普拉斯本征函数描述弦或鼓的振动、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、流体和等离子体流动时，也会出现相关问题。最近，我研究了光谱理论中的几个重要问题：***(a)遍历系统的高能特征函数的研究，包括分支台球的量子遍历性；双曲流形上的爱森斯坦级数（散射理论特征函数）的行为洛施密特回波效应的研究***(b)双曲表面上的共振分布，与F. Naud的一系列论文。***(c)节点和临界集（爱森斯坦级数的节点线；节点集的共形不变量，图论类似物）。***对负弯曲流形上测地线长度谱中的间隙的研究有贡献。我继续研究共形协变算子，特别地证明了0一般不是共形拉普拉斯算子的特征值。我研究了收敛Benjamini-Schramm的稀疏有界度图序列的秩多项式和Tutte多项式的行为，并证明了自然系数测度收敛于一个函数。我计划重点研究以下项目：***研究有边界流形上的共形协变算子及其在几何和偏微分方程中的相关问题；***研究双曲流形上的共振和特征函数***将秩多项式的结果推广到bollobasr - riordan多项式，以及密集图的序列，研究在统计物理中的应用，研究零的渐近分布；研究子流形上的特征函数平均值，包括全测地线子流形；***研究群环元素的谱和特征向量，重点研究表面群；***研究“随机”3流形的几何不变量和谱不变量；研究特征值函数的极值度量，以及与等距嵌入的连接；研究台球的特征函数定位；***研究长度谱；***研究黎曼流形包括曲面的“盖塔”的谱和特征函数。

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.