One of the first ideas of topology is to study a complicated topological space by decomposing it into simple blocks. The most important example of a decomposition of this kind is triangulation, which is a decomposition into simplices. If one has a triangulation of a topological space, this makes him much easier to work with this space. For example, to calculate the homology of this space, instead of cumbersome groups of singular chains, one can take much more convenient groups of simplicial chains. In addition, triangulation can be seen as a way of encoding space for computer calculations. It turns out, however, that there is the following surprising fact at first glance. Although it was proved almost 90 years ago that any smooth manifold admits a triangulation (Cairns, 1934), in dimensions 4 and higher, the list of manifolds for which at least one explicit triangulation is known is extremely small. Therefore, known explicit triangulations of relatively simple manifolds are very interesting to study; as a rule, they are beautiful combinatorial objects with interesting symmetry groups. Of particular interest is Kühnel's remarkable 9-vertex triangulation of the complex projective plane $\mathbb{CP}^2$ (1980), which essentially started the science of explicit triangulations. Within the framework of the lecture course, an introduction to the theory of triangulations of manifolds will be given with an emphasis on the constructions of explicit triangulations.

Lecturer
Gaifullin Alexander Aleksandrovich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center