A sharp Schrodinger maximal estimate in $\mathbb{R}^2$
  We show that $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>1/3$. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling. 
Xiumin Du, Larry Guth, and Xiaochun Li
math.CA