13 problems

Let $F$ be the function defined in the paper's equation for the twisted $(2,n)$-crowns, with $n\geq 4$. Its positive zeros are denoted by $z_1$ and $z_2$. Uniqueness conjecture. Th…

Let four bodies with positive masses be arranged in a specified ordering along the boundary of their convex hull. A planar configuration is convex if no body lies inside or on the…

Let $n$ bodies have equal masses and lie on a common circle, with their center of mass coinciding with the center of the circle. A central configuration is one in which the acceler…

Let $n$ bodies have masses $m_1,\dots,m_n$ and pairwise distances $r_{ij}$ satisfying the Albouy–Chenciner equations and the Cayley–Menger equations for planar configurations. A nor…

Consider planar equilateral 5-body central configurations with all masses positive in the Newtonian case. The regular pentagon and the star are the configurations with equal masses…

Consider the planar equal-mass $N$-body problem with homogeneous potential exponent $A$, and let $C_N(A)$ denote the number of equal-mass central configurations. Monotonicity conje…

Let $f$ be the function for the equal-mass planar $N$-body problem, and let $M(t)$ denote its Morse polynomial. Newtonian Morse-polynomial conjecture. For $A=3$, the Morse polynomi…

Consider the equal-mass planar six-body problem, with exponent $A$, function $f$, and Morse polynomial $M(t)$; let $P(t)$ be the Poincaré polynomial of the quotient configuration s…

Let $f$ be the function for the equal-mass planar five-body problem, let $\mathcal{C}_5$ be the quotient configuration space, and let $P(t)$ be its Poincaré polynomial. Five-body M…

Let $f$ be the function whose critical points represent central configurations in the four-body problem, and let $\mathcal{C}_4$ be the quotient configuration space with Poincaré p…

Let $N>4$, let $A$ be the exponent of the homogeneous potential, and let $f$ be the function whose Morse index is considered on the quotient configuration space $\mathcal{C}_N$. Fo…

Convexity and endpoint maximum conjecture. The sequence $(a_i)_{1\leq i\leq n-1}$ is convex. It is strictly increasing when $\ell=2$, and only when $\ell=2$. Moreover, there exists…

Let a central configuration consist of $L$ homothetic regular polygons, with the masses assigned to the vertices of each polygon. Equal-mass conjecture. Such a central configuratio…