This proposal is composed of problems on equidistribution in locally symmetric spaces. Specifically, the P.I. outlines questions regarding the distribution of closed geodesics, the question of arithmetic quantum unique ergodicity, and the the strong spectral gap property on locally symmetric spaces. The question of quantum unique ergodicity originates in the theory of quantum chaos, which studies the behavior of high energy states of quantum systems with underlying chaotic dynamics. Arithmetic surfaces, along with other arithmetic models, have proven a very fertile ground for testing predictions made in this theory. Recently there has been a great advancement in this field; application of new techniques from ergodic and analytic number theory resolved the (arithmetic) quantum unique ergodicity conjecture for arithmetic surfaces. However, for higher dimensional systems our knowledge is still very limited; for example, it is not even clear what the correct conjecture should be in this setting. The P.I. proposes to address this problem for certain higher dimensional symmetric spaces. On the other side of the spectrum, the notion of a strong spectral gap is related to the lowest energy state. This notion is crucial in many applications and in particular to the distribution of closed geodesics on these spaces. Following advancements in various mathematical disciplines, the existence and magnitude of the strong spectral gap is well understood in almost all cases. However, there are still a few cases missing in order to fully complete the picture, and it is the P.I.'s intention to work on closing this gap.
The problems the P.I. proposes to investigate have a long history and are still matters of active research. Although the models considered in this program are very specific and arithmetic in nature, the P.I. believes that the study of these models will lead to a better understanding of related phenomena in more general settings. In particular, results on the question of quantum unique ergodicity for arithmetic models will provide valuable insights into the behavior of other physical systems. Also, progress on the spectral gap question may lead to construction of new expanders which have uses in cryptography. Moreover, the attempts to answer these questions could lead to the development of new tools that will facilitate studying fundamental questions in number theory.
