Speaker: Victor Turchin

Abstract: The problem of classification of knots in R^3 is notoriously difficult. The first interest to knots goes back to prehistoric times when knots started to appear in the arts. They were also used as the only way of "writing" in the Inca Empire (1438-1533) in South America. Lots of easily computable knot invariants are known nowadays, and many (in fact, infinite series of) knots have been distinguished one from another, though the complete classification is still out of reach. It thus appeared as a big surprise in the 1960s that all higher-dimensional knots i.e., embeddings S^m --> S^n or D^m --> D^n, are always trivial in the piecewise linear and topological locally flat settings, provided the codimension n-m>2 (due to Zeeman and Stallings). In the smooth setting, knots S^m --> S^n, n-m>2, are always trivial only provided n>(3m+2)/2. In particular there are non-trivial knots of codimension n-m > 2. This case was studied by André Haefliger in about the same time in the 1960s. He showed that the isotopy classes of such knots always form a commutative group of rank at most one. His result reduces the geometrical problem of classification of knots to a homotopy theoretical problem of computation of homotopy groups of some standard spaces. I will focus on explaining this higher dimensional knot theory rationally ignoring the torsions. At the end of my talk, I will explain relatively recent results of myself and my coauthors that fully describe all the rational homotopy groups of the spaces of smooth relative to the boundary embeddings of discs D^m --> D^n, n-m>2.