Non-linear elliptic equations and non-associative algebras

The first serious research in the theory of non-linear elliptic partial differential equations was the thesis of Serge Bernstein (1903). He proved that solutions of an elliptic Lagrangian (variational) equation with analytic coefficients are analytic, which is the Hilbert 19th Problem. More generally, L. Nirenberg in 1953 proved that in 2 dimensions solutions of uniformly elliptic equations are classical, i.e. smooth.
That raised the problem whether in higher dimensions there exist non-classical solutions to uniformly elliptic equations. This problem was open until 2007, when the first fully non-linear uniformly elliptic equation without classical solution was constructed using the quaternions. Afterwards, applications of non-associative algebras: the Cayley algebra and Jordan algebras gave a substantial progress towards a classification of non-classical solutions of fully non-linear uniformly elliptic equations.
In the talk which is based on joint work with N. Nadirashvili and V. Tkachev, I will give an exposition of these results and methods of their proofs.
The talk will be held in English