By Kelei Wang, Wuhan University

In this talk I will discuss an elliptic system involving fractional Laplacians. This system is used to model the "phase separation" phenomena with anomalous diffusion. We are mainly concerned with the uniqueness of one dimensional profiles. It turns out that some trivial facts in the classical Laplacian case can become rather serious in this fractional setting. To attack this problem we need various tools: Yau's gradient estimate and hyperbolic geometry, Almgren's monotonicity formula and blowing down analysis, the method of moving plane and so on.