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ICERM Semester Program on "Phase Transitions and Emergent Properties"
(February 2 - May 8, 2015)

CLICK HERE TO PARTICIPATE
Organizing Committee
  • Mark Bowick
    (Syracuse University)
  • Beatrice de Tiliere
    (Université Pierre et Marie Curie, Paris)
  • Richard Kenyon
    (Brown University)
  • Charles Radin
    (University of Texas)
  • Peter Winkler
    (Dartmouth College)

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Introduction

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 di fferent 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.



**Long-Term Participants
View in-residence dates for long-term visitors
  • David Aristoff **
    (University of Minnesota)
  • Jean Bellissard
    (Georgia Institute of Technology)
  • Giulio Biroli
    (Commissariat à l'Énergie Atomique (CEA))
  • Marek Biskup **
    (University of California, Los Angeles)
  • Christian Borgs
    (Microsoft Research)
  • Cedric Boutillier
    (Université de Paris VI (Pierre et Marie Curie))
  • Mark Bowick **
    (Syracuse University)
  • Michael Brenner **
    (Harvard University)
  • Graham Brightwell **
    (London School of Economics and Political Science)
  • Angelo Cacciuto **
    (Columbia University)
  • Maria Cameron
    (University of Maryland)
  • Sourav Chatterjee **
    (Stanford University)
  • Jennifer Chayes
    (Microsoft Research)
  • Mihai Ciucu **
    (Indiana University)
  • Henry Cohn
    (Microsoft Research)
  • Ivan Corwin **
    (Massachusetts Institute of Technology)
  • Amir Dembo **
    (Stanford University)
  • Béatrice de Tilière **
    (Université de Paris VI (Pierre et Marie Curie))
  • Persi Diaconis
    (Stanford University)
  • Noam Elkies **
    (Harvard University)
  • Veit Elser **
    (Cornell University)
  • Patrik Ferrari
    (Rheinische Friedrich-Wilhelms-Universität Bonn)
  • Vadim Gorin **
    (Massachusetts Institute of Technology)
  • Olle Haggstrom **
    (Chalmers University of Technology)
  • Tyler Helmuth **
    (University of British Columbia)
  • Miranda Holmes-Cerfon **
    (Courant Institute of Mathematical Sciences)
  • Ander Holroyd **
    (Microsoft Research)
  • Sabine Jansen **
    (Ruhr-Universität Bochum)
  • Randy Kamien **
    (University of Pennsylvania)
  • Richard Kenyon **
    (Brown University)
  • Roman Kotecky **
    (University of Warwick)
  • Abhinav Kumar
    (Massachusetts Institute of Technology)
  • Robert Kusner **
    (University of Massachusetts)
  • Jeffrey Lagarias **
    (University of Michigan)
  • Lionel Levine
    (Cornell University)
  • Marcin Lis **
    (Vrije Universiteit Amsterdam)
  • Andrea Liu **
    (University of Pennsylvania)
  • Oren Louidor **
    (Technion-Israel Institute of Technology)
  • Eyal Lubetzky
    (Microsoft Research)
  • Malwina Luczak **
    (Queen Mary and Westfield College)
  • Russell Lyons
    (Indiana University)
  • Vinothan Manoharan
    (Harvard University)
  • Fabio Martinelli **
    (Consiglio Nazionale delle Ricerche (CNR))
  • Elisabetta Matsumoto
    (Princeton University)
  • Sevak Mkrtchyan **
    (Carnegie Mellon University)
  • Remi Monasson **
    (École Normale Supérieure)
  • Greta Panova
    (University of California, Los Angeles)
  • Robin Pemantle
    (University of Pennsylvania)
  • Yuval Peres
    (Microsoft Research)
  • Oleg Pikhurko
    (University of Warwick)
  • Charles Radin **
    (University of Texas at Austin)
  • Dana Randall **
    (Georgia Institute of Technology)
  • Alexander Razborov **
    (University of Chicago)
  • Dan Romik
    (University of California, Davis)
  • Lorenzo Sadun **
    (University of Texas at Austin)
  • Itai Shafrir
    (Technion-Israel Institute of Technology)
  • Senya Shlosman **
    (Aix-Marseille University)
  • Vladas Sidoravicius **
    (Institute of Pure and Applied Mathematics (IMPA))
  • Miklós Simonovits **
    (Hungarian Academy of Sciences (MTA))
  • Boris Solomyak
    (University of Washington)
  • Vera Sos **
    (Hungarian Academy of Sciences (MTA))
  • Herbert Spohn **
    (TU München)
  • Andrea Sportiello
    (Università di Milano)
  • Jeff Steif
    (Chalmers University of Technology)
  • Daniel Stein **
    (New York University)
  • Jessica Striker **
    (University of Minnesota)
  • Cristina Toninelli **
    (Université de Paris VII (Denis Diderot) et Université de Paris VI (Pierre et Marie Curie))
  • Mirjana Vuletic **
    (University of Massachusetts)
  • David Wales
    (Downing College)
  • Xuan Wang **
    (University of North Carolina)
  • Samuel Watson **
    (Massachusetts Institute of Technology)
  • Peter Winkler **
    (Dartmouth College)
  • Matthieu Wyart
    (New York University)
  • Mei Yin **
    (Brown University)
  • Thomas Yu
    (Drexel University)
  • Yufei Zhao **
    (Massachusetts Institute of Technology)

Workshops and Associated Events:

Crystals, Quasicrystals and Random Networks (Feb 9-13, 2015)


Organizing Committee
  • Mark Bowick
    (Syracuse University)
  • Persi Diaconis
    (Stanford University)
  • Charles Radin
    (University of Texas, Austin)
  • Peter Winkler
    (Dartmouth College)
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Description

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))
  • Marek Biskup
    (University of California, Los Angeles)
  • Christian Borgs *
    (Microsoft Research)
  • Mark Bowick *
    (Syracuse University)
  • Michael Brenner
    (Harvard University)
  • Sourav Chatterjee *
    (Stanford University)
  • Jennifer Chayes *
    (Microsoft Research)
  • Henry Cohn *
    (Microsoft Research)
  • Ivan Corwin
    (Massachusetts Institute of Technology)
  • Amir Dembo
    (Stanford University)
  • Béatrice de Tilière
    (Université de Paris VI (Pierre et Marie Curie))
  • Persi Diaconis *
    (Stanford University)
  • Noam Elkies *
    (Harvard University)
  • Veit Elser*
    (Cornell University)
  • Vadim Gorin
    (Massachusetts Institute of Technology)
  • Tyler Helmuth
    (University of British Columbia)
  • Ander Holroyd
    (Microsoft Research)
  • Sabine Jansen
    (Ruhr-Universität Bochum)
  • Richard Kenyon *
    (Brown University)
  • Roman Kotecky*
    (University of Warwick)
  • Abhinav Kumar *
    (Massachusetts Institute of Technology)
  • Robert Kusner
    (University of Massachusetts)
  • Jeffrey Lagarias
    (University of Michigan)
  • Marcin Lis
    (Vrije Universiteit Amsterdam)
  • Oren Louidor
    (Technion-Israel Institute of Technology)

Small Clusters, Polymer Vesicles and Unusual Minima (March 16-20, 2015)


Organizing Committee

Description

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.


  • Marcin Lis
    (Vrije Universiteit Amsterdam)
  • Oren Louidor
    (Technion-Israel Institute of Technology)
  • Malwina Luczak
    (Queen Mary and Westfield College)
  • Elisabetta Matsumoto*
    (Princeton University)
  • Sevak Mkrtchyan
    (Carnegie Mellon University)
  • Charles Radin
    (University of Texas at Austin)
  • Emily Russell
    (Harvard University)
  • Lorenzo Sadun
    (University of Texas at Austin)
  • Itai Shafrir *
    (Technion-Israel Institute of Technology)
  • Senya Shlosman
    (Aix-Marseille University)
  • Daniel Stein
    (New York University)
  • Mirjana Vuletic
    (University of Massachusetts)
  • David Wales *
    (Downing College)
  • Xuan Wang
    (University of North Carolina)
  • Samuel Watson
    (Massachusetts Institute of Technology)
  • Peter Winkler
    (Dartmouth College)
  • Matthieu Wyart*
    (New York University)
  • Thomas Yu*
    (Drexel University)

Limit shapes (April 13-17, 2015)


Organizing Committee
  • Marek Biskup
    (University of California, Los Angeles)
  • Alexei Borodin
    (MIT)
  • Béatrice de Tilière
    (Université Pierre et Marie Curie, Paris 6)
  • Richard Kenyon
    (Brown University)
  • Senya Shlosman
    (Aix-Marseille University)

Description

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.


  • David Aristoff
    (University of Minnesota)
  • Marek Biskup
    (University of California, Los Angeles)
  • Cedric Boutillier
    (Université de Paris VI (Pierre et Marie Curie))
  • Mark Bowick
    (Syracuse University)
  • Michael Brenner
    (Harvard University)
  • Graham Brightwell
    (London School of Economics and Political Science)
  • Ivan Corwin*
    (Massachusetts Institute of Technology)
  • Amir Dembo
    (Stanford University)
  • Béatrice de Tilière
    (Université de Paris VI (Pierre et Marie Curie))
  • Persi Diaconis
    (Stanford University)
  • Patrik Ferrari*
    (Rheinische Friedrich-Wilhelms-Universität Bonn)
  • Vadim Gorin*
    (Massachusetts Institute of Technology)
  • Tyler Helmuth
    (University of British Columbia)
  • Ander Holroyd
    (Microsoft Research)
  • Randy Kamien
    (University of Pennsylvania)
  • Richard Kenyon
    (Brown University)
  • Abhinav Kumar
    (Massachusetts Institute of Technology)
  • Robert Kusner
    (University of Massachusetts)
  • Jeffrey Lagarias
    (University of Michigan)
  • Lionel Levine*
    (Cornell University)
  • Marcin Lis
    (Vrije Universiteit Amsterdam)
  • Malwina Luczak
    (Queen Mary and Westfield College)
  • Fabio Martinelli
    (Consiglio Nazionale delle Ricerche (CNR))
  • Sevak Mkrtchyan
    (Carnegie Mellon University)
  • Greta Panova*
    (University of California, Los Angeles)
  • Robin Pemantle
    (University of Pennsylvania)
  • Yuval Peres
    (Microsoft Research)
  • Charles Radin
    (University of Texas at Austin)
  • Dan Romik*
    (University of California, Davis)
  • Emily Russell
    (Harvard University)
  • Lorenzo Sadun *
    (University of Texas at Austin)
  • Senya Shlosman
    (Aix-Marseille University)
  • Vladas Sidoravicius
    (Institute of Pure and Applied Mathematics (IMPA))
  • Andrea Sportiello*
    (Università di Milano)
  • Jeff Steif
    (Chalmers University of Technology)
  • Daniel Stein
    (New York University)
  • Jessica Striker
    (University of Minnesota)
  • Cristina Toninelli
    (Université de Paris VII (Denis Diderot) et Université de Paris VI (Pierre et Marie Curie))
  • Mirjana Vuletic
    (University of Massachusetts)
  • Xuan Wang
    (University of North Carolina)
  • Samuel Watson
    (Massachusetts Institute of Technology)
  • Peter Winkler
    (Dartmouth College)