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

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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.