Ryan Budney recently constructed an operad that encodes splicing of knots and extends his little 2-cubes action on the space of (long) knots. He further showed that the space of knots is freely generated over the splicing operad by the subspace of torus and hyperbolic knots. Infection of knots (or links) by string links is a generalization of splicing from knots to links and is useful for studying concordance of knots. In joint work with John Burke, we construct a colored operad that encodes this infection operation.

I shall report about my 2011 results, which is the resolution of one long standing problem in the theory of Darboux transformations. It is known that many Darboux transformations can be constructed using Darboux Wronskian formulas. The only known exceptions have been two transformations of order one - Laplace transformations, which are often used in applications. I shall show that for order one there is no other exceptions and that for order two Wronskian formulas are complete. History of the question as well as an introduction into the area will be provided.

A linear algebraic group is called a Cayley group if it is equivariantly birationally isomorphic to its Lie algebra. It is stably Cayley if the product of the group and some torus is Cayley. Cayley gave the first examples of Cayley groups with his Cayley map back in 1846. In joint work with Blunk, Borovoi, Kunyavskii and Reichstein, we classify the simple stably Cayley groups over an arbitrary field of characteristic $0$.