Burnside rings

The Burnside ring of a field $k$ of characteristic zero (introduced in the work of M. Kontsevich and Yu. Chinikel, arXiv:1708.05699) is a free abelian group generated by isomorphism classes of finitely generated extensions $K$ of $k$ with transcendence degree $n \geq 0$, equipped with a multiplication defined by the product of the corresponding $k$-varieties. One can also consider a variant of the Burnside ring where the isomorphism class is equipped with a logarithmic volume form $ \omega \in \Omega^n_{K/k} $ (from the work of A. Chambert-Loir, M. Kontsevich, and Yu. Tschinkel, arXiv:2301.02899). We will discuss the properties of Burnside rings and explore their applications in birational geometry, such as in the specialization of rationality, following the aforementioned works.