Rigidity of ancient solutions to the mean curvature flow

In this talk, we discuss recent developments of ancient solutions to the mean curvature flow in higher dimensions. Consider an ancient flow asymptotic to a cylinder with the number of R factors equal to k, we show that the asymptotic behavior of the flow is characterized by a k x k matrix Q whose eigenvalues can only be 0 and 1. Under the noncollapsing assumption, we further discuss rigidity results when Q is fully degenerate or fully nondegenerate. In the fully degenerate case, we obtain a complete classification. In the fully nondegenerate case, we show that the solutions are determined by only k-1 parameters. This is based on joint work with Beomjun Choi and Wenkui Du.