2026.06.10 Colloquia Seminars
| Speaker |
Prof. Alexander A. Ivanov, 河北师范大学 |
| Title |
Local Graph Theory for the Higman–Sims Group |
| Time |
2026.06.15 14:00-15:00 |
| Venue |
N219 |
| Abstract |
During the last quarter of the twentieth century, an approach to proving the existence and uniqueness of sporadic and other simple groups was developed. This approach is based on an ingenious combination of group-theoretical techniques with methods from graph theory and topology. It is known as the Group Amalgam Method. The graph-theoretical aspect of the Group Amalgam Method begins with a graph Γ on which the group G in question acts as a group of automorphisms, with the action being transitive on both vertices and edges. Consequently, for every vertex x of Γ, the isomorphism type of the subgraph ∆ induced by the vertices adjacent to x is independent of the choice of x. It is assumed that ∆ is connected. This local data, although possibly incomplete or even hypothetical, serves as a starting point. The group G can be constructed and proved unique provided that Γ is shown to be the only connected graph which is locally ∆ in the above sense. This leads to the following general problem of Local Graph Theory.
Problem A. For a given graph ∆:
(a) prove or disprove the existence of a graph Γ which is locally ∆;
(b) construct the universal cover of Γ with respect to the local isomorphism;
(c) classify, up to isomorphism, all connected graphs which are locally ∆.
In general, even Problem A (a) is unsolvable: it has been proved that no algorithm exists which determines whether a graph locally isomorphic to a given graph ∆ exists (that is, existence is not a recursive function of ∆). Nevertheless, this does not prevent the solution of Problem A for specific graphs including the one related to the Higman–Sims sporadic simple group which is the subject of my lecture. |
| Biography |
Alexander A. Ivanov is currently a professor of Hebei Normal University. He earned his doctoral degree from the Moscow Institute of Physics and Technology and previously served as a professor at Imperial College London. His main research interests include geometric structures of sporadic groups, distance-regular graphs, and arc-transitive graphs. Professor Ivanov is the author of the monographs 《Geometry of Sporadic Groups I》 (Cambridge University Press, 1999) and 《Geometry of Sporadic Groups II》 (Cambridge University Press, 2001). He was a 45-minute invited speaker of the 1990 International Congress of Mathematicians in Tokyo. He has published more than 80 research papers in renowned international academic journals such as Inventiones mathematicae, Trans. of AMS, and J. of Algebra etc. | |
