Oliver defended his thesis **Toroidal automorphic forms for function fields** in 2008. In the 1970's, Don Zagier defined a new space of automorphic forms for GL(2), so-called *toroidal automorphic forms*, defined by the vanishing of their constant Fourier coefficients along all tori. The Fourier coefficient of an Eisenstein series is the zeta function of the quadratic extension that corresponds to the torus, which links toroidal forms to zeros of zeta functions. This thesis is the first published study of this space of toroidal automorphic forms for global function fields. Oliver proves that the space is finite-dimensional. He also shows that the dimension is not twice the genus of the curve (as one would expect from the zeros of the corresponding zeta function), but rather, relates to the class number of the field and the number of cusp forms with vanishing central L-value. He makes a detailed study of the case of elliptic function fields. The tools are a mixture of classical adelic methods and methods more akin to geometric Langlands. On the way, he develops a generalisation of the tree of SL(2) to more general "Graphs of Hecke operators". The research was funded by NWO through my VIDI-project on "Nonarchimedean geometry and automorphic forms". Oliver went on to do a post-doc at MPIM Bonn, became assistant of Markus Reineke in Wuppertal, was Feodor Lynnen Fellow at CUNY and temporary professor at Frankfurt, before joining IMPA (Rio de Janeiro) as tenured researcher.

**Published results related to PhD research***Toroidal automorphic forms for some function fields*(with Gunther Cornelissen). Journal of Number Theory**129**(6), 1456-1463, 2009.*Toroidal automorphic forms, Waldspurger periods and double Dirichlet series*(with Gunther Cornelissen). Multiple Dirichlet series, L-functions and automorphic forms, 131-146, Progr. Math.,**300**, Birkhauser/Springer, New York, 2012.*Automorphic forms for elliptic function fields.*Math. Z.**272**, no. 3-4, 885-911, 2012.*Graphs of Hecke operators.*Algebra Number Theory**7**, no. 1, 19-61, 2013.*Toroidal automorphic forms for function fields.*Israel J. Math.**194**, no. 2, 555-596, 2013.