The Ricci flow on a Riemannian manifold $(M,g)$ is determined by the geometric evolution equation $\partial_t g_{ij} = -2R_{ij}$ where $R_{ij}$ is the Ricci curvature. The Ricci flow is the main ingredient in Perelman's proof of the Poincaré conjecture.
The Ricci flow is a type of geometric flow (gradient flow associated to a functional on a manifold) that deform the metric of a Riemannian manifold.
For a metric tensor $g_{ij}$ and Ricci tensor $R_{ij}$, the Ricci flow is defined by the geometric evolution equation \begin{equation*} \partial_t g_{ij}=-2R_{ij}. \end{equation*}
On a $3$-dimensional manifold, Perelman demonstrated how you can get past singularities that Ricci flow produces using surgery on the manifold
