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UID:7069@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20190408T140000
DTEND;TZID=Europe/Paris:20190408T150000
DTSTAMP:20241120T203411Z
URL:https://www.i2m.univ-amu.fr/evenements/a-homotopical-approach-to-symme
 tries-of-aperiodic-patterns/
SUMMARY:Jamie Walton (University of Glasgow): A homotopical approach to sym
 metries of aperiodic patterns
DESCRIPTION:Jamie Walton: The translational structure of an aperiodically o
 rdered pattern can be studied via the topology of an associated space\, ca
 lled the translational hull. Much can be said about these spaces\; for exa
 mple there are now several techniques for computing their cohomology. Howe
 ver\, information on rotational symmetry is lost by passing to the transla
 tional hull. To study this rotational structure topologically\, we conside
 r instead a more complicated space\, the rotational hull of the pattern\, 
 denoted $\\Omega_r$. This talk shall provide an introduction to the study 
 of these spaces and discuss recent joint work with John Hunton which provi
 des tools for studying the topologies of rotational hulls\, for example in
  calculating their cohomology. In this work a space $\\Omega_G$ was introd
 uced\, initially as an intermediate for studying $\\Omega_r$. We show that
  it is perhaps more natural from a homotopical viewpoint by showing that f
 or periodic patterns the fundemental group of $\\Omega_G$ recovers the cla
 ssical space group of Euclidean symmetries of the pattern. We thus introdu
 ce a new invariant\, the pro- or space-fundamental group of $\\Omega_G$\, 
 which extends the classical space group to aperiodic patterns. For certain
  cut and project patterns an alternative definition for the aperiodic spac
 e group was given by crystallographers. We compare the two notions by show
 ing that our topological space group naturally projects to the aperiodic s
 pace group. The map is always non-injective and so our invariant appears t
 o contain further information.\n\n\n
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 020/01/James_Walton_University_of_Glasgow.jpg
CATEGORIES:Séminaire,Dynamique et Topologie
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DTSTART:20190331T030000
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