Geometric Analysis and Spectral Theory

我的工作在几何分析，偏微分方程（PDE）和数学物理的交叉点，研究光谱和本征函数的拉普拉斯型运营商，产生在研究天体力学，热和波的传播，量子力学。拉普拉斯算子的本征函数描述了弦或鼓的振动，量子力学中的纯态以及许多其他现象。当人们研究声学、光学、数据分析、流体和等离子体流动时，就会出现相关的问题。最近，我工作的几个重要问题，在光谱理论：（一）研究高能量本征函数的遍历系统，包括量子遍历分支台球;行为爱森斯坦系列（散射理论本征函数的双曲流形）;研究的Loschirst回声效应。 (b)在与F.诺(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，以及图论类似物）。 我贡献了负弯曲流形上的测地长度谱的间隙的研究。我继续研究共形协变算子，特别表明，0一般不是共形拉普拉斯算子的本征值。我研究了行为的秩和Tutte多项式的序列稀疏有界度图收敛Benjamini-Schramm，并表明，自然系数措施收敛到一个δ函数。我计划集中在以下几个方面：研究带边界流形上的共形协变算子及其几何和偏微分方程的相关问题;研究双曲流形上的共振和本征函数;将秩多项式的结果推广到Bollobas-Riordan多项式和稠密图序列，研究其在统计物理中的应用，研究零点的渐近分布;研究子流形上的特征函数平均，包括全测地子流形; 研究群环元素的谱和本征向量，重点是曲面群;研究“随机”流形上的几何和谱不变量; 研究本征值泛函的极值度量，以及等距嵌入的连接;研究台球的本征函数局部化;研究长度谱;研究黎曼流形的“覆盖塔”的谱和本征函数，包括表面。

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.