"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."
This proposal is aimed at breaking the convexity bounds for $L$-functions on high rank groups. The principal investigator has already made the first progress on breaking the convexity bounds for self dual L-functions on GL(3). The problems related to breaking the convexity bounds for L-functions on high rank groups in various aspects are discussed in the proposal. Concrete conjectures are made, possible difficulties are discussed, new techniques and tools are mentioned. These projects are closely related to one of the most fundamental open problems in number theory - the Lindeloff hypothesis for degree n L-functions. The subconvexity bounds are the first steps toward the Lindeloff hypothesis.
The proposal is concerned with the study of the size of certain number theoretical functions (infinite sums with one complex parameter) which are constructed on some geometric spaces. Bounding such functions are important not only because of its theoretical value but also because of its vast applications. For example, one application would be a stronger quantum unique ergodicity conjecture which tells us the convergence of certain measures on certain geometric spaces as well as the rate of convergence of these measures. This is a very important problem in mathematical physics. Other applications will be found in the future.
