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

数学常常在看似无关的学科之间建立起深刻而惊人的联系。在过去的几十年里，谱分析中最有趣的问题之一是估计紧致黎曼流形上的谱计数函数的余项。尽管这已经引起了许多数学家的注意，但得到证明的一般结果却很少。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有类似的对应。这些问题包括经典的点阵计数问题（高斯圆问题和狄利克雷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（包括临界线上黎曼ζ函数的均方平均值）。在另一个方向上，通过迹公式，这些问题与计算总长度小于给定数的封闭测地线（局部最小长度的曲线）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是在微局部分析、解析数论和光谱几何之间这个有趣的接口上，进一步推进解决开放猜想的进展，比如哈代的平面2环面猜想。我要研究的另一类密切相关的问题是薄环内晶格点的分布。这些问题的答案是根据环的宽度和内半径的不同而进行不同的开放猜想。在一种特殊情况下，获得一个薄的无理数椭球内晶格点数目的精确估计，将证明无理数环面上非线性Schrödinger算子关于Strichartz不等式的一些开放猜想。

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.