This is a personal update on some recent advances on the geometry of moduli spaces of Calabi–Yau manifolds, especially along the finite distance boundary with respect to the Weil–Petersson metric. Two main themes are metric completion and extremal transitions. Besides reviewing the known results, I will also raise some related questions.

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Let V be a hypersurface with an isolated singularity at the origin de- fined by the holomorphic function f : (Cn, 0) → (C, 0). L(V ) is defined to be the Lie algebra of derivations of the moduli algebra A(V ) := On/(f, ∂f , · · · , ∂f ), i.e. ∂x1 ∂xn L(V ) = Der(A(V ), A(V )). The Lie algebra L(V ) is finite dimensional solvable alge- bra and plays an important role in singularity theory. L(V ) is called Yau algebra and λ(V ), the dimension of L(V ), is called Yau number. Fewnomial singularities are those which can be defined by n-nomial in n indeterminates. These singularities were inten- sively studied from a mirror symmetry point of view by many other mathematicians. Here we investigate these singularities from a Yau algebra point of view. In our previous work, we formulated a sharp upper estimate conjecture for the Yau numbers of weighted homogeneous isolated hypersurface singularities and proved that this conjecture holds for binomial isolated hypersurface singularities. In this paper we verify this conjecture for weighted homogeneous fewnomial surface singularities.

Using elementary arguments, we shall show that every bounded disjointness preserving linear functional of the Wiener ring L1 (Z) assumes the form (f)=

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