This talk will be divided into two parts. The first part will focus on infinitesimal operators, distinguished by a spectral measure concentrated at 0 but with a non-trivial infinitesimal law. We will delve into their properties; in particular, we will show how they can be applied to construct Boolean and monotone independent elements. In the second part of the presentation, we will introduce a methodology for computing polynomials in monotone independent elements. Specifically, we will derive the explicit distribution of $\alpha ab+\beta ba$ whenever $a$ and $b$ are monotone independent.

The talk is based on my recent works with Jamie Mingo (arXiv: 2308.02064) and Marwa Banna (to appear soon).

Bio: Pei-Lun Tseng currently serves as a postdoctoral researcher at NYU Abu Dhabi, working under the guidance of Marwa Banna. He successfully completed his Ph.D. in 2016 at Queen's University under the supervision of Jamie Mingo. His research is centered around free probability theory, with a particular emphasis on infinitesimal free probability. Additionally, Pei-Lun holds an interest in the broader scope of non-commutative probability theory.