5 Year Impact Statement - 2017

The 2017 special year covered several related topics.  The topics that seem to have been the most active in the intervening time    are those related to analysis of automorphic forms and periods. Here are a few of these developments that come readily to my mind:

During the special year, Simon Marshall announced a new higher rank subconvexity result for the Gross-Prasad family, with the "small" form fixed and the "large" form varying in the depth aspects. This breakthrough seems to have truly opened a door, for a series of striking higher rank results have followed, in particular Nelson's resolution of subconvexity for standard L-functions on GL_n. 

The learning seminar during the special year studied Knop's work on spherical varieties. I believe this has catalyzed a greater, and more systematic, use of spherical varieties when studying periods of automorphic forms. The  joint work of Pollack, Wan and Zydor (all postdocs in the special year), "On the residue method for period integrals," gives a good example of this trend. Another important development here has been the work of Jonathan Wang and Yiannis Sakellaridis. They revisited Sakellaridis' foundational work on Plancherel measure for spherical varieties from a geometrical viewpoint, both proving new theorems and finding rich algebraic structures.    

The special year was also transformative for my own work, because a chance conversation between myself, David Ben-Zvi and Yiannis Sakellaridis sparked an joint project, still on going, which reformulates the theory of periods in terms of duality of certain Hamiltonian spaces. This gives a new and powerful context in which to study automorphic periods, and brings this study into contact with the physics literature, where it is related to the study of duality of boundary conditions in quantum field theory.