This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The PI (John Millson) proposes three main lines of research all within the general framework of reductive algebraic groups and geometry. In the first main line (with Jens Funke) the PI continues his project of using the theta correspondence and differential geometry to construct Siegel (resp. Hermitian) modular forms that are generating functions for the intersection numbers of certain (special) cycles in locally symmetric spaces of orthogonal (resp. unitary) groups. The second main line deals with the ring R of projective invariants of n ordered points on the projective line. This ring was much studied by the invariant theorists in the late eighteenth and early twentieth centuries. In 1894, A. Kempe found that R was generated by certain invariants of lowest degree. In the last year, working with Ben Howard, Andrew Snowden and Ravi Vakil, the PI computed the relations between the Kempe generators. They are quadratic, binomial and have a simple graphical description. The third main line is a continuation of the project dealing with the generalized triangle inequalities and related (saturation) problems from algebra. This project was the subject of the PI's most recent NSF grant, an FRG grant with Prakash Belkale, Thomas Haines, Misha Kapovich and Shrawan Kumar. This project is joint with Thomas Haines and Misha Kapovich.
The PI (John Millson) proposes three main lines of research all within the general framework of reductive algebraic groups and geometry. The first part of the proposal should have applications to number theory along the lines described by S. Kudla in his Beijing ICM talk and also to string theory (according to a preprint of Ai-Ko Lu).. The second part of the proposal is motivated in part because the problem completes work of mathematicians working in the late nineteenth and early twentieth centuries solving a one hundred year old problem. The third part deals with basic problems in representation theory e.g. decomposing tensor products and branching formulas which if solved would be of use in many disciplines.All the above projects are in collaboration with other mathematicians from within the USA or abroad. The PI presently has collaborations with eight mathematicians continuing a history of extensive collaboration (over forty joint papers).