Lorentz Center - Aperiodic Patterns in Crystals, Numbers and Symbols from 19 Jun 2017 through 23 Jun 2017
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    Aperiodic Patterns in Crystals, Numbers and Symbols
    from 19 Jun 2017 through 23 Jun 2017

 

Aperiodic Patterns in Crystals, Numbers and Symbols

                19-23 June 2017 @ Oort

Description and Aim

Quasicrystals show nearly periodic patterns, which can be seen as the projections of regular crystals in higher dimensional spaces. These regular crystals in higher dimensions are rings of integers in a number field, and that is why there is a connection between crystallography and number theory. There are other connections. Numbers can be represented as strings of digits, while crystals are made out of elementary shapes. The patterns of these building blocks can be studied by symbolic dynamics. Crystallography, number theory, and symbolic dynamics are separate fields. The aim of the workshop was to bring together researchers from these separate fields.

Outcomes of the workshop

Several open questions were raised at the workshop and one of these was even solved during the week. Wieb Bosma (Nijmegen), Michel Dekking (Delft) and Wolfgang Steiner (Paris) settled the Biberstine conjecture, which says that three different integer sequences are in fact one and the same.  These three sequences arise from a certain continued fraction expansion, a rotational dynamical system, and a Fibonacci recurrence. Bosma, Dekking, and Steiner proved that the conjecture is correct. Each one of them took care of a different sequence, using computer algebra, substitution dynamics, and hard analysis. This interaction was representative of the interactive atmosphere of this workshop. Many researchers that had not cooperated before joined hands and achieved considerable progress on different problems. We intend to publish their results in a special issue of Indagationes Mathematicae in 2018.

Organization and format of the workshop

This workshop belongs in a series of Lorentz workshops on Numeration (2010) and Streams (2012, 2013) and workshops in Delft on Probability and Numbers (2012, 2015). These earlier workshops had an emphasis on the interaction between mathematics and computer science. The present workshop was more focused on the interaction between modern continued fraction theory (such as translation surfaces) and quasicrystals (cut-and-project schemes).

The talks in the workshop were a mix of long and short lectures. There were survey lectures as well as short talks by PhD students and postdocs. We scheduled some open afternoons, to allow participants to enter talks until the last moment, so that fresh results could be reported in short talks. This worked very well and led to many discussions that were carried on in front of a blackboard.

Acknowledgements

This workshop was made possible by the generous support of the clusters Diamant and Star, TU Delft, Foundation Compositio, and the Lorentz Center. We would like to thank Nienke Tander of the Lorentz Center for taking excellent care.

Carlo Carminati Pisa, Italy   

Alex Clark Leicester, UK

Robbert Fokkink Delft, The Netherlands   

Cor Kraaikamp Delft, The Netherlands  

Tom Schmidt Corvallis, OR, USA  

 



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