Complex cobordisms and $n$-valued groups

The cobordism ring of stably complex manifolds is isomorphic to a graded ring $\mathbb Z[a_n]$, where $\mathrm{deg}\,a_n = 2n$ for $n = 1, 2,\ldots$. This fundamental result, due to J.Milnor and S.P.Novikov (1960), underlies the realization of the universal one-dimensional formal group (M.Lazard, 1954) as the formal group of geometric cobordisms, defined via the first Chern class of complex vector bundles with values in the complex cobordism theory (A.S.Mishchenko, S.P.Novikov (1967), D.Quillen (1969)). The 2-valued formal group in cobordism was introduced by V.M.Buchstaber and S.P.Novikov (1971) via the first Pontryagin–Borel class of quaternionic bundles with values in the complex cobordism theory. In the same work, a key example of an $n$-valued group $G_n$ was given for each $n$. In 1975, V.M.Buchstaber established the algebraic foundations of $n$-valued group theory. One of the first results of this theory was the proof of the universality of the 2-valued formal group in cobordisms. Numerous researchers are currently developing the theory of n-valued (formal, finite, discrete, topological, and algebro-geometric) groups and their applications in various areas of mathematics and mathematical physics. This talk is devoted to results obtained jointly with A.P.Veselov, A.A.Gaifullin, A.Yu.Vesnin, and M.I.Kornev. We particularly highlight our recent paper with M.I.Kornev, where, for each $n$, the notion of symmetric $n$-algebraic $n$-valued groups was introduced. For $n = 2, 3$, we describe the universal objects in these classes. The group $G_n$ belongs to one of these classes.