The proposal aims to apply methods from algebraic geometry to study representations of algebraic groups. The PI and his collaborators will further develop his theory of global Springer representations, which involves the geometry of Hitchin integrable systems, and apply it to representations of p-adic groups. The proposal also proposes to use geometric methods of Laumon, Ngo, etc. to prove orbital integral identities (Fundamental Lemmas) in relative trace formulae and to construct explicit automorphic forms (sheaves) for function fields and their Hecke eigenvalues (local systems), especially those with wild ramifications. e.g., Kloosterman sheaves for general reductive groups. Finally the proposed project also plans to study Koszul duality patterns for sheaves on generalized flag varieties.
This project naturally sits at the intersection of algebraic geometry, representation theory, number theory, and mathematical physics. In these subjects as well as in understanding our physical world, symmetry is a central theme. Group theory is a uniform way to study such symmetries, and representation theory is trying to classify the actions of groups on vector spaces. As often happens, the most interesting representations come from geometric objects with symmetries, which in turn appear in physics. This is why geometric methods are so powerful in solving representation-theoretic problems. Through this project, the PI hopes to shed light on hard problems in representation theory and number theory, and to discover more symmetries that appear in geometry.
I use geometric tools to study problems about numbers and collections of symmetries (called "groups"). In number theory and group theory, many problems can be solved once the hidden symmetry is discovered. 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 a 3-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 past thirty years have witnessed enormous success in this direction, resulting in at least five Fields medals for the solution of number-theoretic problems using geometry. My research focuses on geometric problems in the Langlands program, a central subject in modern mathematics. 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. For example, the 350-year-old Fermat Problem was solved by Wiles in 1993 as a consequence of a special case of the Langlands program. My recent research under the support of NSF provides yet another application of the Langlands program to solving classical problems. The attempt of solving polynomial equations by radicals lead to Galois theory in the late 19th and early 20th century. Galois theory attaches a finite group to every polynomial with integer coefficients. The structure of this group tells in particular whether the roots of the polynomial can be solved by radicals. A central problem in Galois theory is to determine which groups appear as Galois groups of polynomials. This is called the inverse Galois problem. Through my research, I discovered that there is a link between certain "rigid" objects in the Langlands program and the inverse Galois program which was not known before. I used this link to solve certain open cases of the inverse Galois problem. For example I showed that the groups E8(p) and F4(p) are Galois groups for large enough primes p. The groups E8(p) and F4(p) are among the most complicated groups that cannot be disassembled into smaller ones. Other results from my research are more technical to state. Together with Jochen Heinloth and Bao-Chau Ngo, we have discovered new types of exponential sums that generalize Kloosterman sums, an important player in analytic number theory. I have also initiated a theory called Global Springer Theory, which combines techniques from algebraic geometry and representation theory to study representations of double affine Hecke algebras. This topic is still being actively investigated. I have also published several papers on the geometry of affine flag varieties, which is a key geometric object in modern representation theory. I have presented my work in many conferences and seminars. I gave a plenary talk in the International Congress of Chinese Mathematicians in 2013 on the inverse Galois problem. I also organized several conferences on or related to geometric representation theory. For example, with Zongzhu Lin we organized an AMS special session on geometric representation theory in March 2012 at Kansas State University. I also helped organize summer programs on arithmetic geometry at Beijing International Center for Mathematical Research in 2011 and 2012.
