Beatrice de Tiliere
(Université Pierre et Marie Curie, Paris)
Richard Kenyon (Brown University)
(University of Texas)
Peter Winkler (Dartmouth College)
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 combinatorics and probability as well as statistical physics
and related areas. We aim to have experimental contributors for workshops 1 and 2 where we will
highlight the comparison between computational and theoretical modeling and the real world.
This will be combined with computational modules for the student participants.
Crystals, Quasicrystals and Random Networks (Feb 9-13, 2015)
Mark Bowick (Syracuse University)
Persi Diaconis (Stanford University)
Charles Radin (University of Texas, Austin)
Peter Winkler (Dartmouth College)
The densest packing of unit disks in the plane is easily seen to be highly symmetric.
This is exploited in statistical mechanics in arguing that as the density parameter is decreased from its
optimum most packings at fixed density remain quite orderly (`solid'), changing only gradually until at
a specific density they suddenly begin to `melt' into the disordered (`fluid') packings of low density.
This workshop will explore two variants of this fundamental phenomenon. One variant concerns packings of
special shapes, such as the Penrose kites and darts of the accompanying figure, whose densest packings are
aperiodic tilings. The other concerns complex networks for which the optima are certain extremal graphs.
These optimization problems, and especially their associated solid phases and solid/fluid phase transitions,
are the subject of the workshop.
In summary, our workshop will explore two optimization problems on which there is active mathematical research.
It will then focus on their associated solid phases and solid/fluid phase transitions which, on the contrary,
are in dire need of mathematical clarification/understanding. It is hoped that progress can be made by pooling
the expertise of researchers interested in various versions of this phenomenon. To promote cross-disciplinary
information flow between the participants, the workshop format will have long tutorial/discussion sessions in the
mornings, and short, more specialized talks in the afternoons.
The following tutorial/discussion sessions have been arranged so far:
Densest packings by Noam Elkies (Harvard)
Aperiodic tilings by Boris Solomyak (Washington)
Phases from hard spheres by Veit Elser (Cornell)
Extremal graphs by Sasha Razborov (Chicago)
Multipodal phases in graphs by Lorenzo Sadun (Austin)
Nonequilibrium solids by Giulio Biroli (CEA-Saclay)
Other transitions by Remi Monasson (ENS-Paris)
David Aristoff (University of Minnesota)
Jean Bellissard (Georgia Institute of Technology)
Giulio Biroli* (Commissariat à l'Énergie Atomique (CEA))
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.
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, bootstrap percolation models, polynuclear growth models, etc. The goal of the workshop
is to attempt to confront this "ZOO" of combinatorial examples with older foundational work and develop a better understanding of the general
limit shape phenomenon.