It follows from work of Perelman that any finite-time singularity of the Ricci flow on a compact 3-manifold is modeled on an ancient ϰ-solution. We prove that every non-compact ancient ϰ-solution in dimension 3 is isometric to a family of shrinking cylinders (or a quotient thereof), or to the Bryant soliton. This confirms a conjecture of Perelman.

In this paper, we show that there is a Cantor set of initial conditions in the planar 4‑body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlevé conjecture for the 4‑body case, and thus settles the last open case of the conjecture.

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!