Spectral asymptotics on compact manifolds and related problems in analytical number theory

数学经常在看似不相关的学科之间建立起深刻而令人惊讶的联系。在过去的几十年里，谱分析中最有趣的问题之一就是估计紧致黎曼流形上的谱计数函数的余项。尽管这引起了许多数学家的注意，但很少有一般性的结果被证明。有趣的是，在一些重要的情况下，这些分析问题在解析数论中也有类似的情况。这些问题包括经典的格点计数问题(Gauss圆问题和Dirichlet除数问题中的误差估计)和众所周知的数论函数的误差估计(包括临界线上Riemann Zeta函数的均方平均)。在另一个方向上，这些问题通过迹公式与计算总长度小于给定数的闭测地线(局部最小化长度的曲线)的几何问题有关。在这些问题上还有很多工作要做，我提出的研究的一个方面旨在进一步解决开放猜想的进展，例如Hardy关于平坦2-环面的猜想，在微局域分析、解析数论和谱几何之间的这个有趣的界面上。我建议研究的另一类密切相关的问题是薄环内格点的分布。根据环的宽度和内径的不同，这些问题的答案是不同的开放猜想。在一种特殊情况下，得到薄的无理椭球内格点个数的精确估计将证明一些关于无理环面上的非线性薛定谔算子的Strichartz不等式的公开猜想。

Mathematics often makes deep and surprising connections between seemingly unrelated subjects. One of the most interesting problems in spectral analysis of the past few decades has been estimating the remainder term of the spectral counting function on compact Riemannian manifolds. Even though this has occupied the attention of many mathematicians, very few generic results have been proven. Interestingly, in some important cases these analysis problems have analogous counterparts in analytic number theory. These include classic lattice counting problems (the error estimates in the Gauss circle problem and the Dirichlet divisor problem) and the error estimates for well known number theory functions (including the mean square average of the Riemann zeta function on the critical line). In another direction these problems are related, via the trace formulae, to the geometric problem of counting closed geodesics (curves minimizing lengths locally) with total length less than a given number. Much work remains to be done on these problems and one aspect of my proposed research aims at furthering the progress toward resolving the open conjectures, such as Hardy's conjecture for flat 2-tori, at this interesting interface between microlocal analysis, analytic number theory, and spectral geometry. Another class of closely related problems I propose to study is the distribution of lattice points within thin annuli. The answer to these problems are different open conjectures depending on the width of the annulus and the inner radius. In a special case, obtaining the sharp estimates of the number of the lattice points inside a thin irrational ellipsoid would prove some open conjectures about Strichartz' inequalities for the non-linear Schrödinger operator on irrational tori.