This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above-mentioned paper.

We describe the pushforward of a matrix factorization along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and we use this construction to study the convolution of kernels defining integral functors between categories of matrix factorizations. We give an elementary proof of a formula for the Chern character of the convolution generalizing the Hirzebruch–Riemann–Roch formula of Polishchuk and Vaintrob.

We provide a transformation formula of the (Euler characteristic version of the) non-commutative Donaldson–Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an application, we provide alternative proofs of Fomin–Zelevinsky conjectures on cluster algebras.

Let $(X,\omega )$ be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi, the set ${\mathcal{H}}_0$ of Kähler forms cohomologous to $\omega$ has the natural structure of an infinite-dimensional Riemannian manifold. We address the question of whether any two points in ${\mathcal{H}}_0$ can be connected by a smooth geodesic and show that the answer, in general, is “no.”