## Free Probability & Random Matrices Seminar

### Tuesday, March 3rd, 2020

**Time:** 1:30-2:30 p.m. **Place:** Jeffery Hall 422

**Speaker:** Daniel Perales Anaya (Waterloo)

**Title:** Relations between infinitesimal non-commutative cumulants.

**Abstract: **Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory (e.g, using Boolean cumulants to study free infinite divisibility via the Boolean Bercovici-Pata bijection). On the other hand, we have the concept of infinitesimally free cumulants (and the analogue concept for Boolean and monotone under the name of cumulants for differential independence).

In this talk we first are going to discuss the basic properties of non-commutative infinitesimal cumulants. Then we will use the Grassmann algebra to show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Later we will review how the relations between the various types of cumulants are captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. Finally, we will observe how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space, and we will present some interesting formulas in this setting. This is a joint work with Adrian Celestino and Kurusch Ebrahimi-Fard.