ECM26 XXVI European Crystallographic Meeting

MaThCryst Satellite Conference

Darmstadt, Germany, 27 - 29 August 2010

The XXVI European Crystallographic Meeting was held from 29 August to 2 September 2010 in Darmstadt, Germany.

The IUCr Commission on Mathematical and Theoretical Crystallography (MaThCryst) organised a Satellite Conference devoted to the analysis of crystal structure topology and mathematical interpretation of crystal structures.

Program

Lectures were completed with exercises distributed to the participants.

Foundations of aperiodic structures made comprehensible (in cooperation with the IUCr Commission on Aperiodic Crystals).
1. Aperiodic crystals in the higher-dimensional description. Incommensurately modulated structures (IMS), composite structures (CS) and quasiperiodic structures (QS) - similarities and dissimilarities
2. Crystallography of Quasicrystals. Fibonacci sequence, Penrose tiling, octagonal tiling, random tilings - matching rules, symmetry, scaling. nD -embedding, symmetry, structure factor. Description of real quasicrystal structures
3. Some periodic crystal structures get simpler in higher-dimensions.
Introduction to Quaternions and Geometric Algebra and their applications in crystallography.
1. Three dimensional Euclidean space
2. Clifford's geometric algebra of R³
3. Subalgebra of quaternions
4. Reflection in terms of plane normal vector
5. Combination of reflections as geometric products
6. Representations of point groups
7. 3+1 dimensional space time
8. Time reversal as reflection at space hyperplane
9. Magnetic point groups
10. Explicit computations of symmetry transformations
11. Homomorphism between unit quaternions and rotations
12. Neighbouring grains of cubic polycrystals in coincidence misorientation
13. Preferred grain boundaries and twinning of cubic crystals
14. Triple junctions and quadruple nodes in cubic polycrystals
15. Textures of arbitrary polycrystals described as distributions of unit quaternions
Mathematics of minimal surfaces
1. mean curvature
2. variational definition of mean curvature
3. some famous examples: helicoid, catenoid, Scherk, ...
4. triply periodic examples: P, D, G, ...
5. mathematical properties: maximum principle, stability
6. mathematical tools to construct minimal surfaces: Plateau problem, Weierstrass data, perturbation methods
7. significance of minimal versus constant mean curvature, Willmore/Helfrich etc.
8. assumed periodicity versus self organizing structures
9. minimal and cmc surfaces for given space groups: 1-parameter families, bifurcations, distinct families
10. classification problem for minimal surfaces w.r.t. a given space group
11. genus of nets via quotient graphs
12. minimal genus nets with good embeddings
13. catenated net-dual net pairs
14. minimal surfaces as bicontinuous cellular patterns whose labyrinths are defined by these net pairs
15. tricontinuous partitions and branched minima surfaces
16. off-surface properties of minimal surfaces, including domain sizes and the problem of determining a skeletal (medial) representation from a given minimal surface
17. non-cubic minimal surfaces (in particular rhombohedral and tetragonal geometry) as transition surfaces between the cubic TPMS

Speakers

Prof. Hans Grimmer, PSI Villigen (Switzerland)
Prof. Eckhard Hitzer, Fukui, (Japan)
Prof. Walter Steurer, ETH Zürich (Switzerland)
Prof. Karsten Grosse-Brauckmann, Darmstadt (Germany)
Prof. Gerd Schroeder-Turk, Erlangen (Germany)
Prof. Stephen Hyde, Canberra (Australia)

Exposition

A series of models illustrating the minimal surfaces were on display during the whole satellite conference.

Poster presentations

Participants presented posters, which remained on display during the three days of the satellite.

Schedule

9:00-10:30 Morning session I
10:30-11:00 Coffee break
11:00-12:30 Morning session II
12:30-14:00 Lunch break
14:00-15:30 Afternoon session I
15:30-16:00 Coffee break
16:00-18:00 Afternoon session II

Abstracts and didactic material

Abstracts for the posters and for the contributed oral talks as well as didactic material related to the lectures will be available for download shortly.

Venue

The Satellite was held in the same venue as the ECM26 congress: the Darmstadtium congress centre.

Contact

Inquiries .

The Organizers of the ECM26 MaThCryst Satellite Conference observed the basic policy of non-discrimination and affirms the right and freedom of scientists to associate in international scientific activity without regard to such factors as citizenship, religion, creed, political stance, ethnic origin, race, colour, language, age or sex, in accordance with the Statutes of the International Council for Science. At this conference no barriers existed which would have prevented the participation of bona fide scientists.