  We construct a new class of multi-solitary wave solutions for the mass
critical two dimensional nonlinear Schrodinger equation (NLS). Given any
integer K>1, there exists a global (for positive time) solution of (NLS) that
decomposes asymptotically into a sum of solitary waves centered at the vertices
of a K-sided regular polygon and concentrating at a logarithmic rate in large
time. This solution blows up in infinite time with logarithmic rate. Using the
pseudo-conformal transform, this yields the first example of solution blowing
up in finite time with a rate strictly above the pseudo-conformal one. Such
solution concentrates K bubbles at a point. These special behaviors are due to
strong interactions between the waves, in contrast with previous works on
multi-solitary waves of (NLS) where interactions do not affect the blow up
rate.
