The most important elliptic equation in the theory of "geometric PDE" is the "harmonic map equation", which includes Laplace's equation, the equation for geodesics, and the Plateau problem as special cases. We study extending this equation to the case of noncommutative manifolds, where the most basic nontrivial example is the noncommutative 2-torus or irrational rotation algebra. Certain interesting questions about classification of maps between noncommutative 2-tori come up along the way. This is joint work with Mathai Varghese. |