Greenhill did her undergraduate studies at the University of Queensland, and remained there for a master's degree, working with Anne Penfold Street there.[1] She earned her Ph.D. in 1996 at the University of Oxford, under the supervision of Peter M. Neumann. Her dissertation was From Multisets to Matrix Groups: Some Algorithms Related to the Exterior Square.[1][3]
After postdoctoral research with Martin Dyer at the University of Leeds and Nick Wormald at the University of Melbourne, Greenhill joined the University of New South Wales in 2003.[1]
She was promoted to associate professor in 2014, becoming the first female mathematician to earn such a promotion at UNSW.[4]