The PI proposes to use geometric methods to attack problems in representation theory and number theory. Recently the PI has found a way to construct motives with exceptional Galois groups using automorphic forms and the geometric Langlands correspondence. This idea will be systematically developed in the following years and will have applications to the classical inverse Galois problem. The PI will also use moduli spaces of Hitchin type to study representations of rational Cherednik algebras, enumeration of Galois representations, and global analogs of Lusztig's character sheaves.
In mathematics, it is a general rule that the most interesting symmetries should arise geometrically. For example, the ancient classification of Platonic solids is seen nowadays as an instance of finding all 'finite symmetries' in three-dimensional space. With the revolution of algebraic geometry in the second half of the 20th century, we now have powerful geometric tools available to solve classical and new problems in number theory and group theory. The PI will use these geometric tools to attack problems in the Langlands program. In the 70s, Langlands made a series of conjectures which links arithmetic invariants (e.g., solutions to Diophantine equations) to analytic invariants (e.g., modular forms). The predictions that Langlands made, if proved, will be extremely powerful in solving classical problems (including the Fermat Problem which was solved in this way). Through this project, the PI hopes to shed light on some problems in the Langlands program and to discover more symmetries that appear in geometry.
