Recently, I read the
Angenent, Sigurd B., Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, Proc. 3rd Conf., Gregynog/UK 1989, Prog. Nonlinear Differ. Equ. Appl. 7, 21-38 (1992). ZBL0762.53028.
and
Daskalopoulos, Panagiota; Hamilton, Richard; Sesum, Natasa, Classification of compact ancient solutions to the curve shortening flow, J. Differ. Geom. 84, No. 3, 455-464 (2010). ZBL1205.53070.
I don't know why they study the ancient solution ? What is the mean of the class of ancient solution ?
Long story short. The study of ancient solution to MCF is vital in understanding the formation of singularity in MCF.
Let $\{ M_t : [0,T) \}$ be a family of smooth MCF in $\mathbb R^n$ which becomes singular at time $T$ at $X\in \mathbb R^n$ (that is, the curvature is unbounded around $X$). Then for any $T_i \to T$, $X_i \to X$ and $\lambda_i \to +\infty$, the sequence of MCF
$$\{ \lambda_i (M_{ \lambda_i^{-2}t + T_i} -X_i) : t \in (-\lambda_i^2 T_i, T_i -T)\}$$
converges to an ancient solution to MCF $\{ M^\infty_ t: t\in (-\infty, 0)\}$. This is called a blow up of the original MCF at $X$.
The idea (indeed, the central idea) is that if you know enough about those ancient solutions that arise from this blowup procedure, then you know a lot about the singularity.
There are lots of such example, but I just point you to the recent advance, where they proved the mean convex neighborhood conjecture in $\mathbb R^3$ by classifying all ancient solutions with low entropy.