A finitely generated projective module is stably free if it becomes free after taking the direct sum with a finitely generated free module. A commutative ring R is called a Hermite ring if all stably free R-modules are free. Examples include rings where all finitely generated projective modules are free (such as polynomial rings over a field), and all Dedekind domains.

 In his book on Serre's problem, Lam asked if the following conjecture is true: if R is a Hermite ring, then the polynomial ring in one variable R[x] is also Hermite. This conjecture is known as the Hermite ring conjecture.

 In my talk I will explain how certain computations in symplectic K-theory can be used to construct a counterexample to the Hermite ring conjecture.