Mathematics

In the talk I would like to discuss the links between the Allen-Cahn equation in R^N and minimal graphs. A celebrated conjecture due to De Giorgi asserts that any solition u satisfying |u|<1 and monotone in one direction must be one dimensional, in the sense that the level sets are hyperplanes, at least in dimension N ≤ 8. In dimension N=9, a counter example was constructed by Del Pino, Kowalczyk and Wei. I would like to give the outlines of the situation about the problem and to discuss what may happen in higher dimension.