The Keller-Segel equations model chemotaxis of bio-organisms. In a reduced form, considered in this talk, they are related to Vlasov equation for self-gravitating systems and are used in social sciences in description of crime patterns.  

It is relatively easy to show that in the critical dimension 2 and for mass of the initial condition greater than 8 \pi, the solutions 'blowup' (or 'collapse') in finite time. This blowup is supposed to describe the chemotactic aggregation of the organisms and understanding its mechanism, especially its universal features, would allow to compare theoretical results with experimental observations. Understanding this mechanism turned out to be a very subtle problem defying solution for a long time. 

In this talk I discuss recent results on dynamics of solutions of the (reduced) Keller-Segel equations in the critical dimension 2 which include a formal derivation and partial rigorous results on the blowup dynamics of solutions. The talk is based on the joint work with S. I. Dejak, D. Egli and  P.M. Lushnikov.