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BEGIN:VEVENT
UID:8ed2f769f813f6d8b54e281aaa102eb7
CATEGORIES:Number Theory Seminar
CREATED:20210125T122432
SUMMARY:Classification and statistics of cut-and-project sets
LOCATION:zoom
DESCRIPTION: \nAbstract: Cut-and-project point sets are constructed by identifying a st
rip of a fixed n-dimensional lattice (the "cut"), and projecting the lattic
e points in that strip to a d-dimensional subspace (the "project”), and are
a well-studied model of aperiodic order. Dynamical results concerning the
translation action on the hull of a cut-and-project set are known to shed l
ight on certain properties of the point set itself, but what happens when i
nstead of restricting to translations we consider all volume preserving lin
ear actions? \n A homogenous space of cut-and-project sets is defined by fi
xing a cut-and-project construction and varying the n-dimensional grid acco
rding to an ASL(d,R) action. In the talk, which is based on joint work with
René Rühr and Barak Weiss (https://arxiv.org/abs/2012.13299), I will discu
ss this construction and introduce the class of Ratner-Marklof-Strömbergsso
n measures, which are probability measures supported on cut-and-project spa
ces that are invariant and ergodic for the group action. A classification o
f these measures is described in terms of data of algebraic groups, and is
used to prove analogues of results about a Siegel summation formula and ide
ntities and bounds involving higher moments. These in turn imply results ab
out asymptotics, with error estimates, of point-counting and patch-counting
statistics for typical cut-and-project sets.\n \n
X-ALT-DESC;FMTTYPE=text/html: <
/p>
Abstract: Cut-and-project
point sets are constructed by identifying a strip of a fixed n-dimensional
lattice (the "cut"), and projecting the lattice points in that strip to a d
-dimensional subspace (the "project”), and are a well-studied model of aper
iodic order. Dynamical results concerning the translation action on the hul
l of a cut-and-project set are known to shed light on certain properties of
the point set itself, but what happens when instead of restricting to tran
slations we consider all volume preserving linear actions?
<
span style="font-family: Times New Roman; font-size: medium;">A homogenous space of c
ut-and-project sets is defined by fixing a cut-and-project construction and
varying the n-dimensional grid according to an ASL(d,R) action. In the tal
k, which is based on joint work with René Rühr and Barak Weiss (https://arxiv.org/abs/2012.13299), I will discuss this construction and introduce the class o
f Ratner-Marklof-Strömbergsson measures, which are probability measures sup
ported on cut-and-project spaces that are invariant and ergodic for the gro
up action. A classification of these measures is described in terms of data
of algebraic groups, and is used to prove analogues of results about a Sie
gel summation formula and identities and bounds involving higher moments. T
hese in turn imply results about asymptotics, with error estimates, of poin
t-counting and patch-counting statistics for typical cut-and-project sets.<
/span>
CONTACT:Yotam Smilansky (Rutgers University)
X-EXTRAINFO:Zoom Link: https://rutgers.zoom.us/j/95892389306?pwd=WGZNamxBOXlGY1Y4RVpTO
WthWlZVdz09 \nZoom Password: rutgersnts\n
DTSTAMP:20260828T173717
DTSTART;TZID=America/New_York:20210126T140000
DTEND;TZID=America/New_York:20210126T150000
SEQUENCE:0
TRANSP:OPAQUE
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