1. Packings

Inversive Distance Packings
Let $(x_1, y_1, r_1)$ and $(x_2, y_2, r_2)$ be two disks with centers $x_i, y_i$ and radii $r_i$. Then the inversive distance between them is \begin{equation} Inv(i, j) = \frac{(x_j  x_i)^2 + (y_j  y_i)^2  r_i^2  r_j^2}{2r_ir_j}. \end{equation} The inversive distance is invariant under Möbius transformations and preserved by stereographic projection. If two circles $i$ and $j$ are externally tangent, then $Inv(i, j) = 1$ and if they are internally tangent then $Inv(i, j) = 1$.Problem 1.1.
[John Bowers] Let $T$ be a triangulation of the unit sphere with real weights $w_{ij}$ on the edges. Decide if it is possible to produce a a mapping (realization) of vertices to circles such that $Inv(i, j) = w_{ij}$ for all edges $(i, j)$ in the triangulation? If $w_{ij} \in (0, 1]$, for each contact $ij$, then this mapping exists on the sphere and is unique up to Möbius transformation.
 If each $w_{ij} \in [1, 1]$ then there is a nice characterization of existence. If it exists, then it is always unique.
 If some $w_{ij} > 1$, then it does not always exist and when it does it is not necessarily unique.
 Best known approach to (3) is to use the socalled notion of “convexity” for inversive distance circle packing. It is known that nonunitary convex inversive distance circle packings are globally rigid. Nonunitary means no $w_{ij} = 1$.

Disk Packing for Laman Graphs
Problem 1.2.
[Louis Theran] Let $G$ be a planar Laman graph. Does there exist a packing of disks with contact graph $G$ that has generic radii?
Remark. [Louis Theran] Something we do know is that if you can find an infinitesimally rigid packing of a Laman graph, then you can perturb it to a generic one.


Disk Packing on the Unit Sphere
Problem 1.3.
[Robert Connelly] Given a packing of disks on the unit sphere, each with radius $\leq \epsilon$, such that the packing has a nontrivial infinitesimal flex, is the packing unjammed for some $\epsilon$? 
Maximum Contacts in Disk Packing
Problem 1.4.
[Miranda HolmesCerfon] Find a collection of $n$ objects (e.g. disks of given radii) such that all packings have $k$ contacts where $k$ is $\leq$ the number required for isostaticity for that object but there exists a rigid packing that has a nontrivial infinitesimal flex and an equilibrium stress. 
Sticky Sphere Packing in 3D
Definition: Let $S_G$ be the semialgebraic set describing the sticky sphere packing conditions for graph $G$ in $\mathbb{R}^3$
Definition: An edgelength equilibrium stress $\omega$ for a graph $G = (V, E)$ with $V = n$ and $E = m$ is a nonzero vector in $\mathbb{R}^m$ that satisfies $$ \sum_j \omega_{i, j} (p_i  p_j)\ = 0 $$ and $$ \sum_j \omega_{i, j}  p_i  p_j = 0 $$ for all $1\leq i \leq n$.Problem 1.5.
[Meera Sitharam] Can there be an edgelength equilibrium stress of a sticky sphere packing in $\mathbb{R}^3$?
 Under what conditions does the neighborhood of a point in $S_G$ have dimension $4n  m$ where $n$ is the number of vertices and $m$ is the number of edges?
 Is the double banana, or indeed any nucleationfree graph, the contact graph of a sphere packing with generic radii?
For the analogous statement to 3 above, in $\mathbb{R}^2$ for sticky disk packing, there is no edge length equilibrium stress.
Cite this as: AimPL: Rigidity and flexibility of microstructures, available at http://aimpl.org/flexmicro.