Uncategorized – Theorems of the 21st century

You need to know: Euclidean plane ${\mathbb R}^2$ , (second) derivative of a function $x:{\mathbb R}\to {\mathbb R}^2$ .

Background: Let n point particles with masses $m_i > 0$ and positions $x_i \in {\mathbb R}^2$ are moving according to Newton’s laws of motion: $m_j \frac{d^2 x_j}{dt^2} = \sum\limits_{i \neq j} \frac{m_i m_j(x_i - x j)}{r_{ij}^3}, \, 1 \leq j \leq n$ , where $r_{ij}$ is the distance between $x_i$ and $x_j$ . The motion of particles is determined by the initial conditions: their masses and their positions and velocities at time $t=0$ .

The Theorem: On 29th August 2014, Jinxin Xue submitted to Acta Mathematica a paper in which he proved that, for $n=4$ , there is a non-empty set of initial conditions, such that all four points escape to infinity in a finite time, avoiding collisions.

Short context: The problem of describing motion of n bodies under gravitation (n-body problem) in space or plane is a fundamental problem in physics and mathematics, studied by many authors, see, for example, here. In general, the motion can be very complicated even for $n=3$ , but can we at least understand under what initial conditions the solution to the system of Newton equations (presented above) is well-defined for all $t\geq 0$ ? This may be not the case for two reasons: (a) collision happened, and (b) there are no collisions but a point escape to infinity in a finite time. Case (b) is known as non-collision singularity. In 1897, Painlevé proved that there are no such singularities for $n=3$ , but conjectured their existence for all $n>3$ . In 1992, Xia proved this conjecture (for motions in ${\mathbb R}^3$ ) for $n\geq 5$ . The Theorem establishes the remaining (and the hardest) case $n=4$ .