New definability lattices

This talk will present recent results on definability lattices of nonhomogeneous structures. The definability lattice of an infinite undirected tree with a given finite degree of all vertices no less than $3$ was recently discovered by S. M. Dudakov. It contains $3$ elements. This result contrasts with the authors' result: for degree $2$, the lattice is infinite. For integer order, a naturally structured lattice has been found, but it is not known whether it coincides with the entire definability lattice of this structure. The necessary definitions will be given, the history of the problem and P. S. Novikov's formulation will be discussed, the proofs will be commented on, and open questions will be formulated. Joint work with S. F. Soprunov.