In particular, he used new integral geometric methods to establish fundamental existence theorems for differential equations on symmetric spaces as well as some new results on the representations of their isometry groups.

Helgason became a professor of mathematics at the Massachusetts Institute of Technology in 1965,[1] and he retired from the faculty in 2014.

[2] Helgason received the Børge Jessen Diploma Award of the Danish Mathematical Society in 1982, and the Grand Knight's Cross (Stórriddarakross) of the Icelandic Order of the Falcon in 1991.

On 31 May 1996, Helgason received an honorary doctorate from the Faculty of Science and Technology at Uppsala University, Sweden.

[8] He was made an honorary member of the Icelandic Mathematical Society when he turned 70, and a symposium was held in his honor.