This workshop will explore emergent phenomena in the context of small clusters, supramolecular
self-assembly and the shape of self-assembled structures such as polymer vesicles. The
emphasis will be on surprises which arise when common conditions are not satisfied, for instance
when the number of components is small, or they are highly non-spherical, or there are several
types of components. Interactions vary from hard sphere repulsion to competition between
coarse-grained liquid-crystalline ordering competing with shape deformation. Examples of this
behavior are common in materials such as bulk homopolymers (rubber), copolymers, liquid crystals
and colloidal aggregates. A basic mathematical setting would be to consider small clusters
of hard spheres with isotropic short-range attractions and study the shape of the clusters as a
function of the number of components. One known surprise is that highly symmetric structures
are suppressed by rotational entropy. This emphasizes the need to accurately count the number
of particle configurations that lead to the same final state. Small clusters can also generate
anisotropic building blocks which can in turn serve as nano- or meso-scale building blocks for
supermolecules and bulk materials (supramolecular chemistry) freed from the limited scope of
atoms and quantum-mechanical bonding. These structures frequently possess topological defects
in their ground states because they lower the energy. The challenge is to determine the shape and
equilibrium defect structure of such superatoms and the number and geometry of their arrangement.
The number of defects determines the effective valence of the super atoms and the global
geometry of their arrangement determines the types of directional bonding possible when defects
are linked together. The phenomenon of the appearance of singularities/defects because they
are minimizers not necessarily required by topology or boundary conditions is also encountered
in the study of harmonic maps. Moving up to self-assembly of large numbers of units, block
copolymers self-assemble into a wide variety of structures including vesicles, nano-fibers and tori.
Many of the structures formed are essentially two-dimensional surfaces embedded in R3. The
mathematical challenge is to find both the shape and the order of the assembled object. This
requires minimizing of a functional that depends on both the local and global order of the relevant
matter fields and the shape of the surface.