Top problem taken on by US Institute (6)
I believe the answer is:
summit
'top' is the definition.
('summit' can be a synonym of 'top')
'problem taken on by us institute' is the wordplay.
'problem' becomes 'sum' (arithmetic problem).
'taken on by' is a charade indicator (letters next to each other).
'us institute' becomes 'mit' (Massachusetts Institute of Technology).
'sum'+'mit'='SUMMIT'
(Other definitions for summit that I've seen before include "Highest point (of hill)" , "Crown" , "Apex" , "Topmost point" , "Talks" .)