Lorentz Center

International center for scientific workshops

International center for scientific workshops

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## Mathematical Facets of Crystallization |

Solid-state physics is
largely about crystals. In spite of the enormous achievements and numerous
applications of solid-state physics, it remains unclear mathematically how
crystals come into existence. Different areas of
mathematics have contributed towards a rigorous picture of crystallization, but
there still is a long way to go. This workshop focused on four topics that are
closely connected with crystallization, and approached these topics from different mathematical angles. These four topics are: • Spatial
symmetry breaking at positive temperature • Crystallization
and surface effects at zero temperature • Metastability of continuum particle systems • Elasticity
and quasi-crystals. Any mathematical theory of
crystallization must be rooted in analysis, probability theory and statistical
physics. Analytic approaches typically consider only ground states of crystals
(i.e., crystals at zero temperature). At positive temperature, defects occur
and probabilistic approaches are needed. The workshop focused on the modeling
of thermal effects through Gibbs measures, which requires a deep understanding
of statistical mechanical techniques as well. The workshop brought
together researchers from different backgrounds: Analysis, Probability, Stochastic Geometry
and Statistical Mechanics. In order to facilitate the different backgrounds, we
asked four prominent researchers from different communities to give an extended
overview talk and identify open problems. These keynote speakers were Michael Baake (Bielefeld), Aernout van Enter (Groningen), Marek
Biskup (Los Angeles), and Florian Theil (Warwick). Apart from that, we kept a light schedule of
talks, with ample room for individual discussion and three plenary discussion
rounds. In particular the overview talks and the plenary discussions were very
fruitful and were well received by the participants. Our goal was to make
participants aware of relevant developments and to stimulate collaboration
between different mathematical communities. Indeed, several new research
projects were initiated during the week. Moreover, a list of key challenges
towards a rigorous understanding of crystallization has been comprised. There were 25 participants
in total, most of them from the United Kingdom, Germany and The Netherlands,
but also from Czech Republic, France and the US.
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