I will discuss the classification of local rings of finite representation type. These are the commutative local rings having only finitely many indecomposable representations (or modules) satisfying a certain reasonable condition. (The modules in question are called Cohen-Macaulay modules. They are the natural generalization to higher dimensions of the lattices one encounters in algebraic number theory.)
Until recently the classification applied primarily to rings of the form S/(f), where f is a non-zero element of the formal power series ring S in several variables over an algebraically closed field. This classification has mysterious connections with Dynkin diagrams, and the proof uses ideas from algebraic geometry, topology and commutative algebra.
In the last few months we have obtained a much more general classification, which applies, in particular, to any excellent Gorenstein ring containing the field of rational numbers.