Analytic aspects of free convolution

过去十年见证了无线通信理论及其在无线系统中的实现取得了许多进展。由于无线网络的时变特性，具有随机采样条目的矩阵，即所谓的随机矩阵，自然就发挥了作用。研究人员发现，许多多用户无线系统的性能测量与相应的随机矩阵模型的极限行为有关，因为矩阵的大小没有界限。

The last decade has witnessed many advances in wireless communication theory and their implementation in wireless systems. Matrices with randomly sampled entries, the so-called random matrices, come naturally into the play due to the time varying characteristics of the wireless network. Researchers found that the performance measurement for many multiuser wireless systems is relative to the limiting behavior of the corresponding random matrix models as the size of the matrices go without bound. Free probability theory provides a framework for studying such a behavior. This theory was created by Dan Voiculescu in 1985 in order to answer problems in pure mathematics. He later found that it can also be used to study large random matrices. This remarkable discovery changed free probability dramatically and brought new tools and concepts into the random matrix theory. Moreover, in recent years engineers started to apply free probability tools to solve problems in wireless communications, and this has made free probability an interesting and important field for applied scientists. This proposal is mainly concerned with research projects in a special branch of free probability theory, the freeness with amagalmation over a finite dimensional algebra. The motivation of this research is some random matrix models used in wireless communications. I plan to investigate the regularity issues for the operator-valued semicircular variables and for free convolution by methods in matrix analysis and free entropy. I hope that further progress in this line of research would benefit both theoretical and applied sciences.