In elementary school one learns how to solve quadratic equations using the quadratic formula; it involves taking a certain square root. One can also find solutions for degree three polynomials and degree four polynomials by taking radicals, but in general by the Abel-Ruffini theorem there is no formula for the roots of polynomial of degree five or higher in terms of radicals. This theorem, first proven in the 1800's, is an indication of a profound difference between so-called solvable polynomials (those that can be solved by radicals) and nonsolvable polynomials (those that cannot be solved by radicals). The latter are much harder to understand. Existing techniques in the so-called Langlands program are often limited to the solvable setting. Part of this proposal is dedicated to providing new techniques in the Langlands program that will work in nonsolvable settings.
Though only proven in special (albeit important) cases the Langlands functoriality conjectures have become a cornerstone of modern mathematics. The conjectures had their genesis in the area of automorphic forms, which lies at the intersection of number theory, representation theory, and harmonic analysis on Lie groups. The theory developed to understand and prove cases of the conjectures has become crucial to understanding these subjects, and has reached beyond them, leading to important applications in other areas such as topology, algebraic geometry and mathematical physics. This proposal aims to develop tools to resolve important cases of the conjectures and their analogues in other contexts. The proposed research has two parts. First, it will investigate Langlands' so-called ``Beyond Endoscopy'' idea in settings designed to establish descent and base change of automorphic representations along nonsolvable Galois extensions. Second, it will develop relative analogues of the theory of twisted endoscopy. This will lead to a better understanding of special cycles on Shimura varieties with a view towards establishing cases of the Tate and Beilinson-Bloch conjectures.
