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We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan-Gross-Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding L-functions. Ratner’s results on measure classification provide an important input to the proof. Our local results include asymptotic expansions for certain special functions arising from representations of higher-rank Lie groups, such as the relative characters defined by matrix coefficient integrals as in the Ichino-Ikeda conjecture.

This dissertation splits into two distinct halves. The first half is devoted to the study of the holography of higher spin gauge theory in AdS_3. We present a conjecture that the holographic dual of W_N minimal model in a 't Hooft-like large N limit is an unusual ``semi-local" higher spin gauge theory on AdS_3×S^1. At each point on the S^1 lives a copy of three-dimensional Vasiliev theory, that contains an infinite tower of higher spin gauge fields coupled to a single massive complex scalar propagating in AdS_3. The Vasiliev theories at different points on the S^1 are correlated only through the AdS_3 boundary conditions on the massive scalars. All but one single tower of higher spin symmetries are broken by the boundary conditions. This conjecture is checked by comparing tree-level two- and three-point functions, and also one-loop partition functions on both side of the duality. The second half focuses on the holography of higher spin gauge theory in AdS_4. We demonstrate that a supersymmetric and parity violating version of Vasiliev's higher spin gauge theory in AdS_4 admits boundary conditions that preserve N=0,1,2,3,4 or 6 supersymmetries. In particular, we argue that the Vasiliev theory with U(M) Chan-Paton and N=6 boundary condition is holographically dual to the 2+1 dimensional U(N)_k×U(M)_{−k} ABJ theory in the limit of large N,k and finite M. In this system all bulk higher spin fields transform in the adjoint of the U(M) gauge group, whose bulk t'Hooft coupling is M/N. Our picture suggests that the supersymmetric Vasiliev theory can be obtained as a limit of type IIA string theory in AdS_4×CP^3, and that the non-Abelian Vasiliev theory at strong bulk 't Hooft coupling smoothly turn into a string field theory. The fundamental string is a singlet bound state of Vasiliev's higher spin particles held together by U(M) gauge interactions.

This paper mainly focuses on the optimized methods of the sprinkling irrigation for greenery patches, by maximally equalizing the amount of water sprayed on a certain area. Various models are being discussed, where the main mathematical tool is analytic geometry, employed to research the possible effects of different proposals. Firstly, the simplest models are built based on a totally ideal situation. Assuming that sprinkling spouts are spinning over plain lawn with a set of specified radii, install them in arrangements of simple geometric figures. Areas of overlapping and blank parts are being calculated and the most reasonable arrangement of all that are studied is selected. Secondly, real factors are taken into consideration separately as follows: 1. The disequilibrium of the water that drops in a line from the sprinkling center is transformed into a functional expression, whose graphs are drawn to show the water distributed over the area; 2. The plane models are changed into solid ones on the assumption that the sprinkling spouts are placed on slopes. Analytic geometry methods are employed to describe the range of sprayed water on the oblique surface. Through calculation and analysis, models can be adjusted to specific situations. Finally, the boundary problems and landscape effects are involved.

The log-optimal strategy in gambling and asset allocation problem attempts to maximize the expectation of the logarithmic rate of return but not the gross wealth itself. This strategy has been shown to have long termsuperiority overother strategies in theory. Another remarkable virtue is it allows updating the real-time market information quickly which could increase of the logarithmic rate of return. We will prove these properties ourselves in a simple way. We further consider how to do asset allocation by applying the log-optimal strategy in practice. The distribution function of the returns is usually unknown and need to be estimated from the history in a real world. The estimation error will affect the asset allocation directly, but such effect mayresult in a "butterfly effect" which could bring an investment disaster. Therefore, it is an important issue to answer whether there is logoptimal strategy resisted "butterfly effect". We develop a new log-optimal strategy based on the allocation utility which is a quadratic approximation to the expectation of the logarithmic rate of return. We show that the upper bound of the allocation utility deviation can be controlled by the l_1-norm of portfolio and l_∞-norm of the bias of variance matrix estimator. This turns out that our proposed strategy is robust. We apply our strategy for NYSE data. The results showed that our strategy has high performance in return.