Supervisor: Dr Abhishek Saha
Project description:One of the most important notions in modern mathematics is that of an L-function. L-functions are complex analytic functions that are key to modern number theory and central to two Millennium Prize problems, namely the Riemann Hypothesis and the Birch–Swinnerton-Dyer Conjecture. They are also intimately connected with the Langlands program, which is one of the deepest and most active areas of mathematics.
One of the ways to understand L-functions is via period formulas. A period formula expresses an L-function as an integral (period) involving certain other functions called automorphic forms. Period formulas have long been crucial to understanding arithmetic aspects of L-functions, and in recent years they have also been used to great effect to solve open problems on the more analytic side of the subject. In particular, work of Venkatesh (Fields medallist 2018) and recent works of Nelson prove new bounds on values of L-functions at s=1/2 by exploiting period formulas and using tools from representation theory and microlocal analysis.
The overarching goal of this project is to understand key period formulas in higher rank settings and make progress on some of the deepest and most substantial problems involving automorphic forms and L-functions such as the sup-norm problem, the subconvexity problem, bounds on global and local periods, and algebraicity of special values of L-functions. This project will involve tools from algebraic and analytic number theory as well as from representation theory and analysis.
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