Seminar leaders: Professor, Chair of Information Systems J. A. Tussupov,
Associate Professor, Chair of Cryptology, Ye. R. Baisalov
Associate Professor, Chair of Algebra and Geometry, M.I. Bekenov
Associate Professor, Chair of Information Systems, K. Jetpisov
Secretary: Ph.D., Chair of Cryptology, N.D. Markhabatov
L.N. Gumilyov Eurasian National University, st. Kazhymukan 13
16.02.2026, 16.00-18.00 MMF (ЦИСИ) room 305
Serge A. LAWRENCE: What does tell us the action of the automorphism group of a graph on the set of triangulations of a given surface with this graph?
By judiciously using the Pólya's Enumeration Theorem, also known as the Pólya-Redfield Theorem, we can extract not only the number of such triangulations from the action of the automorphism group of a given graph on the set of triangulations of a given surface with this graph, but also write out all these triangulations (with labeled vertices!) explicitly. In particular, we will show how to explicitly write out all twelve triangulations of the torus with the complete 4-partite graph K_2,2,2,2. (This graph is remarkable in that it is the graph of a 4-dimensional hyperoctahedron.) The beauty of the construction becomes simply intriguing when this graph is thought of algebraically, namely, as the Cayley graph of the quaternion group Q_8. Although the emphasis will be on algebra and combinatorics, the geometric component will be illustrated.