Emergent phenomena are properties of a system of many components which are only evident or even meaningful for the collection as a whole. A typical example is a system of many molecules, whose bulk properties may change from those of a fluid to those of a solid in response to changes in temperature or pressure. The basic mathematical tool for understanding emergent phenomena is the variational principle, most often employed via entropy maximization. The difficulty of analyzing emergent phenomena, however, makes empirical work essential; computations generate conjectures and their results are often our best judge of the truth.

The semester will include three workshops that will concentrate on different aspects of current interest, including unusual settings such as complex networks and quasicrystals, the onset of emergence as small systems grow, and the emergence of structure and shape as limits in probabilistic models. The workshops will (necessarily) bring in researchers in... (more)

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... (more)

Since the days of Boltzmann, it has been well accepted that natural phenomena, when described using tools of statistical mechanics, are governed by various "laws of large numbers." For practitioners of the field this usually means that certain empirical means converge to constants when the limit of a large system is taken. However, evidence has been amassed that such laws apply also to geometric features of these systems and, in particular, to many naturally-defined shapes. Earlier examples where such convergence could be proved include certain interacting particle systems, invasion percolation models and spin systems in equilibrium statistical mechanics.

The last decade has seen a true explosion of "limit-shape" results. New tools of combinatorics, random matrices and representation theory have given us new models for which limit shapes can be determined and further studied: dimer models, polymer models, sorting networks, ASEP (asymmetric exclusion processes), sandpile models,... (more)