Algebraic geometry
Not long before his death he developed new tools in Algebraic geometry and used them to obtain results in Discrete geometry, proving Purdy's Conjecture. Micha Sharir organized, extended and published Elekes's posthumous notes on these methods. Then Nets Katz and Larry Guth used them to solve (apart from a factor of (log n) 1/2 ) the Erdős distinct distances problem, posed in 1946.