Abstract

For 300 years, Fermat’s Last Theorem seemed to be pure arithmetic little connected even to other problems in arithmetic. But the last decades of the twentieth century saw the discovery of very special cubic curves, and the rise of the huge theoretical Langlands Program. The Langlands perspective showed those curves are so special they cannot exist, and thus proved Fermat’s Last Theorem. With many great contributors, the proof ended in a deep and widely applicable geometric result relating nice curves in rational coordinates to very nice surfaces in complex coordinates. This geometry is the only proof strategy for FLT yet known. Outstanding but not atypical of current mathematics, the proof challenges common philosophic views on abstraction and structure and raises still-open logical questions in arithmetic.