1. Automorphic forms and their Fourier coefficients Autmorphic forms are functions on adelic groups invariant under translations by a discrete subgroup and can be seen as a generalization of modular forms where we replace the underlying group $SL_2$ by any reductive group $G$ and appear in many applications.
Because of the invariance, the automorphic form is "periodic" with respect to unipotent subgroups and can therefore be expanded in a Fourier series. The Fourier coefficients for many choices of unipotent subgroups are often difficult to compute directly by integration, but often contain sought after arithmetic information. I and my collaborators have developed a reduction algorithm which writes difficult to compute Fourier coefficients in terms of other types of Fourier coefficients which are either known or easier to compute.
An important application is the automorphic forms appearing in scattering amplitudes in string theory where the Fourier coefficients contain information about the number of quantum states of D-instantons and black holes.
This project is joint work with Dmitry Gourevitch, Axel Kleinschmidt, Daniel Persson and Siddhartha Sahi.
2. p-adic Whittaker functions The Fourier coefficients of automorphic forms have local $p$-adic constituents that are so-called Whittaker functions. More specifically, in certain cases the Fourier coefficients become an Euler product over primes $p$ of Whittaker functions, but in general the composition of Whittaker functions into global Fourier coefficients is more complicated.
Whittaker functions by themselves are crucial tools in the study of a $p$-adic representation $\pi$ by embedding it into a convenient function space and the image of this embedding is called a Whittaker model. In a collaboration with Ben Brubaker, Valentin Buciumas and Daniel Bump, we construct a basis of Whittaker functions for these models and have shown that the values of these Whittaker functions are exactly the partition functions of certain solvable lattice models from statistical mechanics. These lattice models can be pictured as different paths traversing a two-dimensional grid according to certain rules with configuration weighted by the different vertex configurations. Using these lattice models we can then prove identities for the original Whittaker functions. The connections between different lattice models and Whittaker functions is very rich and goes beyond the identification with the partition function. For example, we have shown that there is a bijection between the data describing a particular Whittaker function value and the data describing the boundary condition for the lattice model.
3. Counting automorphic representations This is a new project I am working on together with Shai Evra and Mathilde Gerbelli-Gauthier where we give bounds on the multiplicities of both local and automorphic representations using the Langlands program and the Arthur classification of representations. The aim is to prove the Sarnak-Xue density hypothesis which can often serve as a replacement for the (Naive) Ramanujan conjectures in applications.