It was observed in the 1980's that the entropy of an action by two or more commuting automorphisms on a compact abelian group is sometimes given by a zeta- or L-value. For example zeta(3) arises in this way. In the talk I will report on joint work with Thu Hà Trieu, which at least for expansive actions explains the deeper reason for these observations by a combination of operator algebra theory, in particular the theory of their determinants, K-theory and cyclic and Deligne cohomology. The talk is addressed to a general audience and will not go into technical details. Instead we will sketch the relevant theories and explain how to fit them together for our purpose.

The goal of this talk is to present the construction of new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the ordinary double point singularity in dimension 3. These metrics have tangent cone $\mathbb{C} ×(\mathbb{C}^2/\mathbb{Z}_2)$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a singular Calabi-Yau metric with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point, thereby providing a new metric realisation of the Atiyah flop. As we will explain, this construction is meant to provide a local adiabatic model for certain degenerations of compact Calabi-Yau manifolds.