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SUMMARY:Approximate lattices in nilpotent Lie groups
DTSTART;VALUE=DATE-TIME:20181008T123000Z
DTEND;VALUE=DATE-TIME:20181008T134500Z
DTSTAMP;VALUE=DATE-TIME:20210623T125207Z
UID:indico-event-3912@indico.math.cnrs.fr
DESCRIPTION:In order to analyze mathematical and physical systems it is of
ten necessary to assume some form of order\, e.g. perfect symmetry or comp
lete randomness. Fortunately\, nature seems to be biased towards such form
s of order as well. During the second half of the 20th century\, a new par
adigm of "aperiodic order" was suggested. Instances of aperiodic order wer
e discovered in different areas of mathematics\, such as harmonic analysis
and diophantine approximation (Meyer)\, tiling theory (Wang\, Penrose) an
d additive combinatorics (Freiman\, Erdös-Szemeredi)\; after some initial
resistance is has now been accepted that aperiodic order also exists in n
ature in the form of quasicrystals. \n\nTogether with Michael Björklund w
e have proposed a general mathematical framework for the study of aperiodi
c structures in metric spaces\, based on the notion of an "approximate lat
tices". Roughly speaking\, approximate lattices generalize lattices in the
same way that approximate subgroups (in the sense of Tao) generalize subg
roups. Approximate lattices in Euclidean space are essentially the "harmon
ious sets" of Meyer (a.k.a. mathematical quasi-crystals)\, but there are i
nteresting examples in other geometries\, such as symmetric spaces\, Bruha
t-Tits buildings or nilpotent Lie groups. It turns out that with every app
roximate lattice one can associate a dynamical system\, which replaces the
homogeneous space associated with a lattice - thus the study of approxima
te lattices can be considered as "geometric group theory enriched over dyn
amical systems".\n\nIn this talk I will (1) give an overview over the basi
c framework of approximate lattices and geometric approximate group theory
\; (2) illustrate the framework by formulating Meyer's theory of harmoniou
s sets in this language\; (3) time permitting\, discuss some recent struct
ure theory of approximate lattices in nilpotent Lie groups and application
s to Bragg peaks in the Schrödinger spectrum of magnetic quasicrystals.\n
\nBased on joint works with Michael Björklund (Chalmers)\, Matthew Cordes
(ETH)\, Felix Pogorzelski (Leipzig) and Vera Tonić (Rijeka).\n\nhttps://
indico.math.cnrs.fr/event/3912/
LOCATION:IHES Amphithéâtre Léon Motchane
URL:https://indico.math.cnrs.fr/event/3912/
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