AWM Emmy Noether Lecture
Abstract. Elliptic equations have been and continue to be an important tool in the study of problems in geometry. In recent decades, non-linear second order elliptic equations with critical exponents have played a special role in the solutions of several important problems in conformal geometry; e.g. the problem of prescribing Gaussian curvature and the Yamabe problem. In this talk, I will describe some recent efforts to extend the role played by second order semilinear equations to higher order semilinear equations as well as second order fully non-linear equations. A common feature essential to understanding such equations is the analysis of blow up in these equations. This analysis involves classifying entire solutions to such equations in Euclidean space. I plan to discuss examples of blowup phenomena in several such situations.