The recent developments in the research area have significantly advanced the understanding and computational techniques in manifold analysis and surface mapping. There is a notable shift towards intrinsic methods in finite element analysis, particularly in handling partial differential equations on manifolds with approximate metrics, such as Regge metrics. These methods aim to avoid the use of preferred coordinates or embeddings, offering both conceptual clarity and potential computational benefits. Additionally, there is a growing interest in extending density-equalizing mapping techniques to surfaces with more complex topologies, such as toroidal surfaces, which opens new avenues for applications in geometry processing and imaging science. The field is also witnessing advancements in the numerical approximation of hyperbolic mean curvature flows, with new finite element and finite difference schemes being proposed for axially symmetric surfaces. Furthermore, significant progress has been made in the computation of symmetries for rational surfaces, with general and specific algorithms developed for sparse parametrizations and ruled surfaces, respectively. These algorithms have been implemented in computer algebra systems, demonstrating their practical efficiency.

Noteworthy papers include one that introduces a novel algorithm for computing density-equalizing maps on toroidal surfaces, and another that proposes innovative finite element and finite difference schemes for hyperbolic mean curvature flows.