Approximation of mean curvature flow: a conjecture of Ennio De Giorgi.

In this talk I will introduce a regularization method
of the mean curvature flow of a submanifold
(of any dimension and codimension) in the Euclidean space, by means of
approximating flows of higher order.
It is well known that the mean curvature flow develops singularities in
finite time, at the moment we are able to show that these regularized
evolutions converge for every time before the first singularity.
An interesting problem is then to study the convergence at the subsequent
times, a positive answer to this question will lead to a new
definition of generalized mean curvature flow in any codimension.
All this line of research and the associate conjectures were suggested by
Ennio~De~Giorgi in one of his last papers.