Maximal inequalities are of paramount importance in analysis. Here "analysis" is understood in a wide sense and includes harmonic analysis, probability theory and ergodic theory.

We will consider in this survey talk the analogues of some classical inequalities in the noncommutative analysis. Then the usual L_p-spaces are replaced by noncommutative L_p-spaces associated to von Neumann algebras. The theory of noncommutative martingale/ergodic inequalities was remarkable developed in the last 20 years. Many classical results were successfully transferred to the noncommutative setting. This theory has fruitful interactions with operator spaces, quantum stochastic analysis and noncommutative harmonic analysis. We will discuss some of these noncommutative results and explain certain substantial diifficulties in proving them.

After the talk, Prof. Xu will give an introduction to the recruitment of Institute for Advanced Study in Mathematics of Harbin Institute of Technology.