Smoothness of critical, fractional Keller-Segel and Burgers equations in 1d

I will present a disproof of a blowup conjecture for the critical, fractional one-dimensional Keller-Segel system. The main tool is a smoothness result for a modified Burgers system; it follows from a modification of the famous method of moduli of continuity by Kiselev, Nazarov, Shterenberg and Volberg. If time permits, I will also present certain regularity results for Keller-Segel system with supercritical fractional diffusion. The talk is based on a joint work with Rafael Granero (Davies).