NCTS(South)/ NCKU Math Colloquium | |
| DATE | 2012-12-20¡@16:10-17:00 |
| PLACE | R204, 2F, NCTS, NCKU |
| SPEAKER | ¶ÀµqÞ³ (Hao-Wei Huang) ¥ý¥Í¡]Indiana University¡^ |
| TITLE | Supports and Regularity for Measures in a Free Additive Convolution Semigroup and Related Problems |
| ABSTRACT | Free probability theory which was initiated by D. Voiculescu around 1986 to investigate the free group factors isomorphism problem is a mathematical theory that studies noncommutative probability spaces. The free independence, connected with free products, is the analogue of the classical independence and the free (additive) convolution $\boxplus$ is the analogue of classical convolution in probability theory. Free probability theory later has deep connections to other mathematical fields, such as random matrix theory, combinatorics, and quantum information theory and is undergoing active research. In this talk, I will briefly introduce some important theorems, such as free central limit theorem, in free probability theory and then explain how the supports of the measures in the semigroup $\{\mu^{\boxplus p}:p>1\}$ vary when $p$ increases, where $\mu$ is a Borel probability measure on $\mathbb{R}$. Moreover, motivated by free central limit theorem I will give equivalent conditions so that the measure $\mu^{\boxplus p}$ has only one component in the support for sufficiently large $p$. I will also talk about other related convolutions, some recent achievements in free probability theory, and some open problems. |