At Berkeley, Simons faced his first setback - Chern was on sabbatical his first year. Working with another advisor, he developed a "little theorem" which reminded his advisor of a famous open problem. The advisor warned him against attempting it, citing failed attempts by distinguished mathematicians.
Naturally, this only motivated Simons more. He tackled the problem and solved it, completing his PhD work. This pattern of defying conventional wisdom and solving problems others deemed too difficult would define his career.
During his time at the Institute for Defense Analyses (a classified NSA facility), he worked on minimal surfaces in higher dimensions. He successfully proved results up to dimension seven, though his approach failed in dimension eight. Years later, other mathematicians proved his suspected counterexample was indeed correct, demonstrating the fundamental limitations of the problem.