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

数学经常在看似无关的学科之间建立深刻而令人惊讶的联系。 在过去的几十年中，谱分析中最有趣的问题之一是紧黎曼流形上谱计数函数的余项估计。 尽管这已经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有类似的对应物。 这些包括经典的格子计数问题（高斯圆问题和狄利克雷除数问题的误差估计）和著名数论函数的误差估计（包括临界线上黎曼zeta函数的均方平均）。 在另一个方向这些问题有关，通过迹公式，几何问题的计数封闭测地线（曲线最小化长度本地）与总长度小于一个给定的数字。 还有很多工作要做，这些问题和我提出的研究的一个方面，旨在进一步解决的开放式acquitures的进展，如哈代的猜想平坦2环面，在这个有趣的接口之间的微局部分析，解析数论，谱几何。另一类密切相关的问题，我建议研究的是薄annuli内的晶格点的分布。 这些问题的答案是不同的开放式结构，这取决于环的宽度和内半径。 在特殊情况下，得到薄无理椭球内格点个数的精确估计，将证明无理环面上非线性薛定谔算子的Schr hartz不等式的一些公开结果。

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.