Eigenfunctions in analysis, geometry and PDE

我研究的光谱理论问题起源于数学物理（拉普拉斯型和狄拉克型算子的光谱和特征函数）。我经常研究高能领域的问题，如谱统计，以及高能特征态的渐近性质：均匀分布，L^p$范数，节点和临界集，半经典渐近。这些问题存在于分析、偏微分方程、微分几何、动力系统和数学物理的交叉点。我的主要兴趣之一是量子混沌领域。这门学科起源于使用随机矩阵理论来模拟原子核的光谱。后来人们认识到，类似的现象也发生在数学和物理的许多其他领域。我打算继续研究部分双曲动力系统的遍历理论与矩阵值算子（如Hodge Laplacian和Dirac算子）的特征函数的高能行为之间的关系。我也打算研究高能谱和特征函数的系统，其中波的传播涉及分支（或射线分裂）；例如，在弹性和声学中就出现了这样的系统。这类系统的半经典分析中的许多自然问题仍未得到探索，需要在遍历理论中建立新的结果。

I work on questions in spectral theory that have their origins in mathematical physics (spectra and eigenfunctions of Laplace-type and Dirac-type operators). I often study questions in the high-energy regime such as spectral statistics, as well as asymptotic properties of of high energy eigenstates: uniform distribution, $L^p$ norms, nodal and critical sets, semiclassical asymptotics. These questions lie at the intersection of analysis, partial differential equations, differential geometry, dynamical systems and mathematical physics. One of my main interests is the area of quantum chaos. The subject originated with the use of random matrix theory to model the spectra of atomic nuclei. Later it was recognized that similar phenomena occur in many other areas of mathematics and physics. I propose to continue my study of relationship between ergodic theory of partially hyperbolic dynamical systems, and high energy behavior of eigenfunctions of matrix-valued operators, such as Hodge Laplacian and Dirac operators. I also intend to study high energy spectra and eigenfunctions for systems where wave propagation involves branching (or ray-splitting); such systems arise, for example, in elasticity and acoustics. Many natural questions in semiclassical analysis of such systems remain unexplored, and require establishing new results in ergodic theory.