Research Summary

This project started with the investigation of the space of all Alexandrov surfaces. The importance of Alexandrov spaces stems mainly from the generality of the concept, which allows both differentiable and non-differentiable manifolds to be included in the investigation. An important class of Alexandrov surfaces is that of convex surfaces. Part of our research is devoted to them. In many applications, precisely the non-differentiable case, particularly the theory of polyhedra, including the graphs which are their 1-skeleta, is the prevailing one. Thus, the study of graphs essentially appearing in computer networks is also included in our project. Generalized Halin graphs and Toeplitz graphs are considered as well. Moreover, generalized convexity is also investigated. Concerning the space of Alexandrov surfaces, we tackle approximation issues and undertake a generic study (via Baire categories).

Expected results

We want to obtain new results mainly in, but not limited to, the topics mentioned in the above brief summary.