The quantisation of the spectral action for spectral triples remains a largely open problem. Even within a perturbative framework, serious challenges arise when in the presence of non-abelian gauge symmetries. This is precisely where the Batalin–Vilkovisky (BV) formalism comes into play: a powerful tool specifically designed to handle the perturbative quantisation of gauge theories. The central question I will address is whether it is possible to develop a BV formalism entirely within the framework of noncommutative geometry (NCG). After a brief introduction to the key ideas behind BV quantisation, I will report on recent progress toward this goal, showing that the BV formalism can be fully formulated within the language of NCG in the case of finite spectral triples. [-]

The right side of the Baum-Connes conjecture is the $K$-theory of the reduced $C^*$-algebra $C^*_{red} (G)$ of the group $G$. This algebra is the completion of the algebra $L^1(G)$ in the norm of the algebra of operators acting on $L^2(G)$. If we complete the algebra $L^1(G)$ in the norm of the algebra of operators acting on $L^p(G)$ we will get the Banach algebra $C^{*,p}_{red}(G)$. The $K$-theory of this algebra serves as the right side of the $L^p$-version of the Baum-Connes conjecture. The construction of the left side and the assembly map in this case requires a little bit of techniques of asymptotic morphisms for Banach algebras. A useful category of Banach algebras for this purpose includes all algebras of operators acting on $L^p$-spaces (which may be called $L^p$-algebras).
The current joint work in progress with Guoliang Yu aims at proving the following result:
The $L^p$-version of the Baum-Connes conjecture with coefficients in any $L^p$-algebra is true for any discrete group $G$ which admits an affine-isometric, metrically proper action on the space $X = l^p(Z)$, where $Z$ is a countable discrete set, so that the linear part of this action is induced by a measure-preserving action of $G$ on $Z$.
I will discuss the techniques involved in this work.[-]