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\title{19th Workshop: Noncommutative Probability, Noncommutative Harmonic 
Analysis \\ and Related Topics with Applications, 
{31.07-6.08.2022, Będlewo}}

\author{\titsbf{ABSTRACT}}
\date{}
%\date{\nodate}#{\sc 31.07-6.08.2022, Będlewo}}

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\namesf{Franz Lehner (Technische Universitat Graz),}

	\namesf{Kamil Szpojankowski (Warsaw University of Technology)}

\titsbf{Subordination of (some) polynomials via Boolean cumulants}

\titsbf{Abstract}: 
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In this joint talk our goal is to show that Boolean cumulants are an effective tool, which allow to calculate conditional expectations of resolvents of polynomials if free random variables. In particular using Boolean cumulants one can get a ''subordination'' for any polynomial in free random variables, this generalizes the observation made by Biane about conditional expectations of $\mathbb{E}_a(z-a-b)^{-1}$ and $\mathbb{E}_a(z-a^{1/2}ba^{1/2})^{-1}$.

The talk is divided into two parts: 
\begin{enumerate}
	\item In the first part we present a new (more algebraic) approach to our previous results from \cite{Paper1}, which relate conditional expectations and Boolean cumulants.
	\item In the second part we will explain how one can use the algebraic machinery from the first part in order to calculate conditional expectations for any polynomial in free random variables.
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