Organizing Committee
Abstract

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 Alexander Razborov (Chicago)
  • Multipodal phases in graphs by Lorenzo Sadun (Austin)
  • Nonequilibrium solids by Giulio Biroli (CEA-Saclay)
  • Other transitions by Remi Monasson (ENS-Paris)

Confirmed Speakers & Participants

Workshop Schedule

Monday, February 9, 2015
TimeEventLocationMaterials
8:30 - 8:55Registration11th Floor Collaborative Space 
8:55 - 9:00Welcome - ICERM Director11th Floor Lecture Hall 
9:00 - 9:10Theme of workshop - Charles Radin, University of Texas11th Floor Lecture Hall
9:10 - 10:10Densest packing of, and on, Euclidean spheres - Noam Elkies, Harvard University 11th Floor Lecture Hall
10:10 - 10:40Discussion Period11th Floor Lecture Hall 
10:40 - 11:10Coffee/Tea Break11th Floor Collaborative Space 
11:10 - 12:10Impromptu Discussion of Statistical Formalism - Charles Radin, University of Texas11th Floor Lecture Hall
12:10 - 12:40Discussion Period11th Floor Lecture Hall 
12:40 - 2:20Break for Lunch  
2:20 - 3:00The role of symmetry in sphere packing in high dimensions - Henry Cohn, Microsoft (New England)11th Floor Lecture Hall
3:00 - 3:30Short Break  
3:30 - 4:10The fluid-solid phase transition on a curved surface - Vinothan Manoharan, Harvard University 11th Floor Lecture Hall
4:30 - 6:00Welcome Reception11th Floor Collaborative Space 
Tuesday, February 10, 2015
TimeEventLocationMaterials
9:00 - 10:00Syntactic methods in continuous combinatorics - Alexander Razborov, University of Chicago11th Floor Lecture Hall
10:00 - 10:30Discussion Period11th Floor Lecture Hall 
10:30 - 11:00Coffee/Tea Break11th Floor Collabortive Space 
10:30 - 11:50Statistical Mechanics and Combinatorics (ongoing semester course) - Richard Kenyon, Brown University10th Floor Classroom 
11:00 - 12:00Multipodal phases in graphs - Lorenzo Sadun, University of Texas11th Floor Lecture Hall
12:00 - 12:30Discussion Period11th Floor Lecture Hall 
12:30 - 2:10Break for Lunch  
2:10 - 2:40Fixed-energy harmonic functions - Richard Kenyon, Brown University 11th Floor Lecture Hall
2:40 - 2:50Short Break  
2:50 - 3:20Harmonic pinnacles in the discrete Gaussian model - Eyal Lubetzky, New York University 11th Floor Lecture Hall
3:20 - 3:50Coffee/Tea Break11th Floor Collaborative Space 
3:50 - 4:20Random walk on the random graph - Yuval Peres, Microsoft (Redmond)11th Floor Lecture Hall
4:20 - 4:30Short Break  
4:30 - 5:00Various types of phase transitions in extremal graph theory - Miklos Simonovits, Hungarian Academy of Sciences11th Floor Lecture Hall
Wednesday, February 11, 2015
TimeEventLocationMaterials
9:00 - 10:00Non-equilibrium solids - Giulio Biroli, CEA Saclay11th Floor Lecture Hall
10:00 - 10:30Discussion Period11th Floor Lecture Hall 
10:30 - 10:50Coffee/Tea Break11th Floor Collaborative Space 
10:50 - 11:50Aperiodic tilings - Boris Solomyak, University of Washington11th Floor Lecture Hall
11:50 - 12:20Discussion Period11th Floor Lecture Hall 
12:20 - 1:40Break for Lunch  
1:40 - 2:10Poster Session11th Floor Lecture Hall and Collabortive Space 
2:10 - 2:40How the Ising crystal grows - Senya Shlosman, Aix-Marseille University11th Floor Lecture Hall
2:40 - 2:50Short Break  
2:50 - 3:20Characterizing extremal limits - Oleg Pikhurko, University of Warwick11th Floor Lecture Hall
3:20 - 3:50Coffee/Tea Break11th Floor Collaborative Space 
3:50 - 4:20Universal behavior near Erdoes-Renyi - Lorenzo Sadun, University of Texas11th Floor Lecture Hall
4:20 - 4:30Short Break  
4:30 - 5:00Problems on convergent sequences of dense graphs - Vera Sos, Hungarian Academy of Sciences11th Floor Lecture Hall
Thursday, February 12, 2015
TimeEventLocationMaterials
9:00 - 10:00Other transitions in statistical physics and beyond- from the liquid-gas transition to the representation of space in the brain - Remi Monasson, Ecole Normale Superieure (Paris)11th Floor Lecture Hall 
10:00 - 10:30Discussion Period11th Floor Lecture Hall 
10:30 - 11:50Statistical Mechanics and Combinatorics (ongoing semester course) - Richard Kenyon, Brown University10th Floor Classroom 
11:00 - 12:00The Phases of Hard Sphere Systems - Veit Elser, Cornell University 
12:00 - 12:30Discussion Period11th Floor Lecture Hall 
12:30 - 2:10Break for Lunch  
2:10 - 2:40Phase transitions in models of quasicrystals - David Aristoff, Colorado State University11th Floor Lecture Hall
2:40 - 2:50Short Break  
2:50 - 3:20Asymptotics for sparse exponential graph models - Mei Yin, University of Denver11th Floor Lecture Hall
3:00 - 3:10Post Doc & Grad Student Group Photos11th Floor Lecture Hall 
3:20 - 3:25Workshop Group Photo11th Floor Lecture Hall 
3:20 - 3:25Workshop Group Photo11th Floor Lecture Hall 
3:25 - 3:50Coffee/Tea Break11th Floor Collaborative Space 
3:50 - 4:20Lattices, packings and energy minimization - Abhinav Kumar, SUNY Stony Brook11th Floor Lecture Hall 
4:20 - 4:30Short Break  
4:30 - 5:00Large deviations in random graphs - Yufei Zhao, Massachusetts Institute of Technology11th Floor Lecture Hall
Friday, February 13, 2015
TimeEventLocationMaterials
9:00 - 10:30Discussion Period - Topics determined from previous talks11th Floor Lecture Hall 
10:30 - 11:00Coffee/Tea Break11th Floor Lecture Hall 
11:00 - 12:30Discussion Period - Topics determined from previous talks11th Floor Lecture Hall 
12:30 - 2:10Break for Lunch  
2:10 - 2:40L^p theory of sparse graph limits - Jennifer Chayes, Microsoft (New England)11th Floor Lecture Hall
2:40 - 2:50Short Break  
2:50 - 3:20Applications of L^p theory of sparse graph limits - Christian Borgs, Microsoft (New England)11th Floor Lecture Hall
3:20 - 3:50Coffee/Tea Break11th Floor Collaborative Space 
3:50 - 4:20Phase transitions in tilings constructed from stationary fusion rules - Natalie Frank, Vassar College11th Floor Lecture Hall
4:30 - 5:00Crystallization of hard shapes- from disks to tetrahedra - Michael Engel, University of Michigan11th Floor Lecture Hall 

Associated Semester Workshops

Phase Transitions and Emergent Properties
Limit Shapes

Lecture Videos

Universal behavior near Erdoes-Renyi

Lorenzo Sadun
University of Texas
February 11, 2015

Characterizing extremal limits

Oleg Pikhurko
University of Warwick
February 11, 2015

How the Ising crystal grows

Senya Shlosman
Aix-Marseille University
February 11, 2015

Aperiodic tilings

Boris Solomyak
University of Washington
February 11, 2015

Fixed-energy harmonic functions

Richard Kenyon
Brown University
February 10, 2015

Multipodal phases in graphs

Lorenzo Sadun
University of Texas
February 10, 2015

Syntactic methods in continuous combinatorics

Alexander Razborov
University of Chicago
February 10, 2015