Vladislav Goldberg (joint with V. Lychagin) - "Algebraic aspects of compatibility of PDEs".

We find an invariant characterization of planar webs of maximum rank.
For 4-webs, we prove that a planar 4-web is of maximum rank if and only if it is linearizable and its curvature vanishes. 
This result leads to the direct web-theoretical proof of the Poincare's theorem: 
A planar 4-web of maximum rank is linearizable. 

We also find an invariant intrinsic characterization of planar 4-webs of rank two and one and 
prove that in general such webs are not linearizable. This solves the Blaschke problem 
``to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3".
Finally, we find invariant characterization of planar 5-webs of maximum rank and prove that
 in general such webs are not linearizable.