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15 - Motives from Diffraction

Published online by Cambridge University Press:  05 April 2013

J. Stienstra
Affiliation:
Mathematisch Instituut, Universiteit Utrecht, the Netherlands
Jan Nagel
Affiliation:
Université de Lille
Chris Peters
Affiliation:
Université Joseph Fourier, Grenoble
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Summary

Abstract

We look at geometrical and arithmetical patterns created from a finite subset of ℤn by diffracting waves and bipartite graphs. We hope that this can make a link between Motives and the Melting Crystals/Dimer models in String Theory.

Introduction

Why is it that, occasionally, mathematicians studying Motives and physicists searching for a Theory of Everything seem to be looking at the same examples, just from different angles? Should the Theory of Everything include properties of Numbers? Does Physics yield realizations of Motives which have not been considered before in the cohomological set-up of motivic theory? Calabi-Yau varieties of dimensions 1 and 2, being elliptic curves and K3-surfaces, have a long and rich history in number theory and geometry. Calabi-Yau varieties of dimension 3 have played an important role in many developments in String Theory. The discovery of Mirror Symmetry attracted the attention of physicists and mathematicians to Calabi-Yau's near the large complex structure limit [13, 20]. Some analogies between String Theory and Arithmetic Algebraic Geometry near this limit were discussed in [16, 17, 18]. Recently new models appeared, called Melting Crystals and Dimers [14, 11], which led to interesting new insights in String Theory, without going near the large complex structure limit. The present paper is an attempt to find motivic aspects of these new models.

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Publisher: Cambridge University Press
Print publication year: 2007

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  • Motives from Diffraction
    • By J. Stienstra, Mathematisch Instituut, Universiteit Utrecht, the Netherlands
  • Edited by Jan Nagel, Université de Lille, Chris Peters, Université Joseph Fourier, Grenoble
  • Book: Algebraic Cycles and Motives
  • Online publication: 05 April 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107325968.017
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  • Motives from Diffraction
    • By J. Stienstra, Mathematisch Instituut, Universiteit Utrecht, the Netherlands
  • Edited by Jan Nagel, Université de Lille, Chris Peters, Université Joseph Fourier, Grenoble
  • Book: Algebraic Cycles and Motives
  • Online publication: 05 April 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107325968.017
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Motives from Diffraction
    • By J. Stienstra, Mathematisch Instituut, Universiteit Utrecht, the Netherlands
  • Edited by Jan Nagel, Université de Lille, Chris Peters, Université Joseph Fourier, Grenoble
  • Book: Algebraic Cycles and Motives
  • Online publication: 05 April 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107325968.017
Available formats
×