Arithmetic Applications of the Trace Formula

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The PI proposes several research projects in number theory which involve the trace formula at the core of proof. Largely the projects can be divided into three parts. First, the PI proposes to study the cohomology of Shimura varieties with a view towards the classical and mod p/p-adic Langlands program. Second, the PI is concerned with arithmetic statistics for in finite families of automorphic representations and their L-functions. Third, the PI is involved in a joint project to classify the automorphic representations of unitary groups with arithmetic applications in mind. Some projects will be carried out in collaboratiton with Kaletha, Minguez, Scholze, Templier and White. The proposed projects stem from basic questions that ancient Greeks were trying to answer, such as studying properties of prime numbers and finding systematic solutions to polynomial equations in integers or rational numbers. The trace formula may be viewed as a twentieth century invention to tackle such fundamental problems effectively. Automorphic forms and the Langlands program become increasingly important in theoretical physics and would possibly shed light on our understanding of the universe. Developments in number theory are particularly important in the new era making extensive use of internet and electronic devices in view of its wide applications to cryptography, error-correcting codes and internet security, just to name a few.