He received his PhD in 1985 from the University of Amsterdam with Hendrik Lenstra (Elliptic Curves and Class Groups).

[3] In 1985, Schoof discovered an algorithm which enabled him to count points on elliptic curves over finite fields in polynomial time.

He obtained the best-known result extending Deligne's Theorem for finite flat group schemes to the non-commutative setting, over certain local Artinian rings.

In the past, René has also worked with Rubik's cubes by creating a common strategy in speedsolving used to set many world records known as F2L Pairs, in which the solver creates four 2-piece "pairs" with one edge and corner piece which are each "inserted" into F2L slots in the CFOP method to finish the first two layers of a 3x3x3 Rubik's cube.

This strategy is also used for all cubes of higher order (4x4x4 and up) in the Reduction, Yau, and Hoya methods if CFOP is used for their 3x3x3 stages.