In 1966, Almgren showed that any immersed minimal surface in
S^3 of genus 0 is totally geodesic, hence congruent to the equator. In
1970, Blaine Lawson constructed many examples of minimal surfaces in S^3
of higher genus; he also constructed numerous examples of immersed minimal
tori. Motivated by these results, Lawson conjectured that any embedded
minimal surface in S^3 of genus 1 must be congruent to the Clifford
torus.

In this talk, I will describe a proof of Lawson's conjecture. The proof
involves an application of the maximum principle to a function that depends
on a pair of points on the surface.