讲座内容：

Let $\mathcal{S}$ be a class of subsets of $\mathbb{R}^n$ and $\langle \mathcal{A,+} \rangle$ be an Abelian semigroup.A mapping $Z:\mathcal{S} \to \langle \mathcal{A,+} \rangle$ is called a \emph{valuation} provided\begin{equation*}Z K+Z L=Z(K\cup L)+Z(K \cap L),\end{equation*}for all $K,L\in \mathcal{S}$ with $K\cup L,K\cap L \in \mathcal{S}$.

The role of valuations is pivotal in addressing Hilbert's third problem, as the comprehensive resolution of this problem is equivalent to the classifications of real-valued valuations.

In this talk, I will show some recent studies on Minkowski valuation and function-valued valuations encompassing important geometric operators such as the projection body, moment body, the Legendre ellipsoid, and the Laplace transform.