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BEGIN:VEVENT
SUMMARY:Ujué Etayo (TU Graz)
DTSTART;VALUE=DATE-TIME:20200603T150000Z
DTEND;VALUE=DATE-TIME:20200603T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/1
DESCRIPTION:Title: Astounding connections of the logarithmic energy o
n the sphere\nby Ujué Etayo (TU Graz) as part of Point Distributions
Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract\nDuring t
his talk we will present different problems that are somehow related to th
e following one: find the minimum value of the logarithmic energy of a set
of N points on the sphere of dimension 2. This late problem has been stud
ied for years\, a computational version of it can be found as Problem Numb
er 7 of Steve Smale list "Mathematical Problems for the Next Century". Thi
s computational version of the problem was proposed after Smale and Shub f
ound out a beautiful relation between minimizers of the logarithmic energy
and well conditioned polynomials. Working on this relation\, we are able
to relate these two concepts to yet a new one: a sharp Bombieri type inequ
ality for univariate polynomials. The problem can also be rewritten as a f
acility location problem\, as proved by Beltrán\, since the logarithmic e
nergy is just a normalization of the Green function for the Laplacian on t
he sphere.\n\nThis talk will be recorded and posted on the webinar homepag
e. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josiah Park (Georgia Institute of Technology)
DTSTART;VALUE=DATE-TIME:20200610T150000Z
DTEND;VALUE=DATE-TIME:20200610T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/2
DESCRIPTION:Title: Optimal measures for three-point energies and semi
definite programming\nby Josiah Park (Georgia Institute of Technology)
as part of Point Distributions Webinar\n\nLecture held in Zoom\, password
: 600Cell.\n\nAbstract\nGiven a potential function of three vector argumen
ts\, \, which is -invariant\, for all orthogonal\, we find that surface me
asure minimizes those interaction energies of the form over the sphere whe
never the potential function satisfies a positive definiteness criteria. W
e use semidefinite programming bounds to determine optimizing probability
measures for other energies. This latter approach builds on previous use o
f such bounds in the discrete setting by Bachoc-Vallentin\, Cohn-Woo\, and
Musin\, and is successful for kernels which can be shown to have expansio
ns in a particular basis\, for instance certain symmetric polynomials in i
nner products \, \, and . For other symmetric kernels we pose conjectures
on the behavior of optimizers\, partially inferred through numerical studi
es. This talk is based on joint work with Dmitriy Bilyk\, Damir Ferizovic\
, Alexey Glazyrin\, Ryan Matzke\, and Oleksandr Vlasiuk.\n\nThis talk will
be recorded and posted on the webinar homepage. Slides will be available
too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Dostert (EPFL)
DTSTART;VALUE=DATE-TIME:20200617T150000Z
DTEND;VALUE=DATE-TIME:20200617T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/3
DESCRIPTION:Title: Semidefinite programming bounds for the average ki
ssing number\nby Maria Dostert (EPFL) as part of Point Distributions W
ebinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract\nThe avera
ge kissing number of $R^n$ is the supremum of the average degrees of conta
ct graphs of packings of finitely many balls (of any radii) in $R^n$.\nIn
this talk I will provide an upper bound for the average kissing number bas
ed on semidefinite programming that improves previous bounds in dimensions
3\, . . . \, 9.\nA very simple upper bound for the average kissing number
is twice the kissing number\; in dimensions 6\, . . . \, 9 our new bound
is the first to improve on this\nsimple upper bound. This is a joined work
with Alexander Kolpakov and Fernando Mário de Oliveira Filho.\n\nThis ta
lk will be recorded and posted on the webinar homepage. Slides will be ava
ilable too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Phillipe Moustrou (UiT - The Arctic University of Norway)
DTSTART;VALUE=DATE-TIME:20200624T150000Z
DTEND;VALUE=DATE-TIME:20200624T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/4
DESCRIPTION:Title: Exact semidefinite programming bounds for packing
problems\nby Phillipe Moustrou (UiT - The Arctic University of Norway)
as part of Point Distributions Webinar\n\nLecture held in Zoom\, password
: 600Cell.\n\nAbstract\nIn the first part of the talk\, we present how sem
idefinite programming methods can provide upper bounds for various geometr
ic packing problems\, such as kissing numbers\, spherical codes\, or packi
ngs of spheres into a larger sphere. When these bounds are sharp\, they gi
ve additional information on optimal configurations\, that may lead to pro
ve the uniqueness of such packings. For example\, we show that the lattice
E8 is the unique solution for the kissing number problem on the hemispher
e in dimension 8.\n\nHowever\, semidefinite programming solvers provide ap
proximate solutions\, and some additional work is required to turn them in
to an exact solution\, giving a certificate that the bound is sharp. In th
e second part of the talk\, we explain how\, via our rounding procedure\,
we can obtain an exact rational solution of semidefinite program from an a
pproximate solution in floating point given by the solver.\n\nJoint work w
ith Maria Dostert and David de Laat.\n\nThis talk will be recorded and pos
ted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David de Laat (TU Delft)
DTSTART;VALUE=DATE-TIME:20200701T150000Z
DTEND;VALUE=DATE-TIME:20200701T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/5
DESCRIPTION:Title: High-dimensional sphere packing and the modular bo
otstrap\nby David de Laat (TU Delft) as part of Point Distributions We
binar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract\nRecently\,
Hartman\, Mazáč\, and Rastelli discovered a connection between the Cohn
-Elkies bound for sphere packing and problems in the modular bootstrap. In
this talk I will explain this connection and discuss our numerical study
into high dimensional sphere packing and the corresponding problems in the
modular bootstrap. The numerical results indicate an exponential improvem
ent over the Kabatianskii-Levenshtein bound. I will also discuss implied k
issing numbers and how these relate to improvements over the Cohn-Elkies b
ound.\n\nJoint work with Nima Afkhami-Jeddi\, Henry Cohn\, Thomas Hartman\
, and Amirhossein Tajdini.\n\nThis talk will be recorded and posted on the
webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matthew de Courcy-Ireland (EPFL)
DTSTART;VALUE=DATE-TIME:20200708T150000Z
DTEND;VALUE=DATE-TIME:20200708T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/6
DESCRIPTION:Title: Lubotzky-Phillips-Sarnak points on a sphere\nb
y Matthew de Courcy-Ireland (EPFL) as part of Point Distributions Webinar\
n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract\nWe will discuss
work of Lubotzky-Phillips-Sarnak on special configurations of points on th
e two-dimensional sphere: what these points achieve\, the sense in which i
t is optimal\, and aspects of the construction that are specific to the sp
here.\n\nThis talk will be recorded and possibly posted on the webinar hom
epage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mateus Sousa (Ludwig Maximilian University of Munich)
DTSTART;VALUE=DATE-TIME:20200717T150000Z
DTEND;VALUE=DATE-TIME:20200717T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/7
DESCRIPTION:Title: Uncertainty principles\, interpolation formulas an
d packing problems\nby Mateus Sousa (Ludwig Maximilian University of M
unich) as part of Point Distributions Webinar\n\nLecture held in Zoom\, pa
ssword: 600Cell.\n\nAbstract\nIn this talk we will discuss how certain unc
ertainty principles and interpolation formulas are connected to packing pr
oblems and talk about some recent developments on these fronts.\n\nThis ta
lk will be recorded and posted on the webinar homepage. Slides will be ava
ilable too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giuseppe Negro (U of Birmingham)
DTSTART;VALUE=DATE-TIME:20200722T150000Z
DTEND;VALUE=DATE-TIME:20200722T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/8
DESCRIPTION:Title: Sharp estimates for the wave equation via the Penr
ose transform\nby Giuseppe Negro (U of Birmingham) as part of Point Di
stributions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstra
ct\nIn 2004\, Foschi found the best constant\, and the extremizing functio
ns\, for the Strichartz inequality for the wave equation with data in the
Sobolev space \n$\\dot{H}^{1/2}\\times\\dot{H}^{-1/2}(\\mathbb{R}^3)$. He
also formulated a conjecture\, concerning the extremizers to this Strichar
tz inequality in all spatial dimensions $d\\geq 2$. We disprove such conje
cture for even $d$\, but we provide evidence to support it for odd $d$. Th
e proofs use the conformal compactification of the Minkowski space-time gi
ven by the Penrose transform. \n\nPart of this talk is based on joint work
with Felipe Gonçalves (Univ. Bonn).\n\nThis talk will be recorded and po
sted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tania Stepaniuk (U of Lübeck)
DTSTART;VALUE=DATE-TIME:20200729T150000Z
DTEND;VALUE=DATE-TIME:20200729T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/9
DESCRIPTION:Title: Estimates for the discrete energies on the sphere<
/a>\nby Tania Stepaniuk (U of Lübeck) as part of Point Distributions Webi
nar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract\nWe find uppe
r and lower estimate for the discrete energies whose Legendre-Fourier coef
ficients decrease to zero approximately as power functions.\n\nThis talk w
ill be recorded and posted on the webinar homepage. Slides will be availab
le too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mathias Sonnleitner (JKU Linz)
DTSTART;VALUE=DATE-TIME:20200731T150000Z
DTEND;VALUE=DATE-TIME:20200731T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/10
DESCRIPTION:Title: Uniform distribution on the sphere and the isotro
pic discrepancy of lattice point sets\nby Mathias Sonnleitner (JKU Lin
z) as part of Point Distributions Webinar\n\nLecture held in Zoom\, passwo
rd: 600Cell.\n\nAbstract\nAistleitner\, Brauchart and Dick showed in 2012
how the spherical cap discrepancy of mapped point sets may be estimated in
terms of their isotropic discrepancy. We provide a characterization of th
e isotropic discrepancy of lattice point sets in terms of the spectral tes
t\, the inverse length of the shortest vector in the corresponding dual la
ttice. This is used to give a lower bound on the discrepancy in question.
\n\nThe talk is based on joint work with F. Pillichshammer.\n\nThis talk w
ill be recorded and posted on the webinar homepage. Slides will be availab
le too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oscar Quesada (IMPA)
DTSTART;VALUE=DATE-TIME:20200805T150000Z
DTEND;VALUE=DATE-TIME:20200805T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/11
DESCRIPTION:Title: Developments on the Fourier sign uncertainty prin
ciple\nby Oscar Quesada (IMPA) as part of Point Distributions Webinar\
n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract\nCan we control t
he signs of a function and its Fourier transform\, simultaneously\, in an
arbitrary way? \n\n\nAn uncertainty principle in Fourier analysis is the a
nswer to this type of question. They lie at the heart of Fourier optimizat
ion problems\, such as the Cohn-Elkies linear program for sphere packings.
We will discuss some answers to this question from a new perspective\, an
d why it might be relevant for problems in diophantine geometry and optima
l configurations. (Joint work with Emanuel Carneiro).\n\nThis talk will be
recorded and posted on the webinar homepage. Slides will be available too
.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Louis Brown (Yale)
DTSTART;VALUE=DATE-TIME:20200812T150000Z
DTEND;VALUE=DATE-TIME:20200812T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/12
DESCRIPTION:Title: Positive-definite Functions\, Exponential Sums an
d the Greedy Algorithm: a Curious Phenomenon\nby Louis Brown (Yale) as
part of Point Distributions Webinar\n\nLecture held in Zoom\, password: 6
00Cell.\n\nAbstract\nWe describe a curious dynamical system that results i
n sequences of real numbers in [0\,1] with seemingly remarkable properties
. Let the even function $f:\\mathbb{T} \\rightarrow \\mathbb{R}$ satisfy $
\\widehat{f}(k) \\geq c|k|^{-2}$ and define a sequence via\n\n$$x_n = \\ar
g\\min_x \\sum_{k=1}^{n-1}{f(x-x_k)}.$$\n\nSuch greedy sequences seem to b
e astonishingly regularly distributed in various ways. We explore this\,
and generalize the algorithm (and results on it) to higher-dimensional man
ifolds\, where the setting is even nicer.\n\nThis talk will be recorded an
d posted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julian Hofstadler (JKU Linz)
DTSTART;VALUE=DATE-TIME:20200814T150000Z
DTEND;VALUE=DATE-TIME:20200814T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/13
DESCRIPTION:Title: On a subsequence of random points\nby Julian
Hofstadler (JKU Linz) as part of Point Distributions Webinar\n\nLecture he
ld in Zoom\, password: 600Cell.\n\nAbstract\nWe want to study the ideas of
R. Dwivedi\, O. N. Feldheim\, O. Guri-Gurevich and A. Ramdas from their p
aper 'Online thinning in reducing discrepancy'\, where they give a criteri
on for choosing points of a random sequence. This technique\, called thinn
ing\, shall improve the distribution of random points\, and we also want t
o discuss their attempt to create thinned samples with small discrepancy.\
n\nThis talk will be recorded and posted on the webinar homepage. Slides w
ill be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felipe Gonçalves (U of Bonn)
DTSTART;VALUE=DATE-TIME:20200819T150000Z
DTEND;VALUE=DATE-TIME:20200819T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/14
DESCRIPTION:Title: Sign Uncertainty\nby Felipe Gonçalves (U of
Bonn) as part of Point Distributions Webinar\n\nLecture held in Zoom\, pas
sword: 600Cell.\n\nAbstract\nWe will talk about recent developments of the
sign uncertainty principle and its relation with sphere packing bounds an
d spherical designs. This is joint work with J. P. Ramos and D. Oliveira e
Silva.\n\nThis talk will be recorded and posted on the webinar homepage.
Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Krieg (JKU Linz)
DTSTART;VALUE=DATE-TIME:20200828T153000Z
DTEND;VALUE=DATE-TIME:20200828T163000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/16
DESCRIPTION:Title: Order-optimal point configurations for function a
pproximation\nby David Krieg (JKU Linz) as part of Point Distributions
Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract\nWe show
that independent and uniformly distributed sampling points are as good as
optimal sampling points for the approximation (and integration) of functi
ons from the Sobolev space $W_p^s(\\Omega)$ on domains $\\Omega\\subset \\
mathbb{R}^d$ in the $L_q(\\Omega)$-norm whenever $q< p$\, where we take $q
=1$ if we only want to compute the integral. In the case $q\\ge p$ there i
s a loss of a logarithmic factor. More generally\, we characterize the qua
lity of arbitrary sampling points $P\\subset \\Omega$ via the $L_\\gamma(\
\Omega)$-norm of the distance function ${\\rm dist}(\\cdot\,P)$\, where $\
\gamma=s(1/q-1/p)_+^{-1}$. This improves upon previous characterizations b
ased on the covering radius of $P$. \n\nThis is joint work with M. Sonnlei
tner.\n\nThis talk will be recorded and posted on the webinar homepage. Sl
ides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitriy Bilyk (U of Minnesota)
DTSTART;VALUE=DATE-TIME:20200916T140000Z
DTEND;VALUE=DATE-TIME:20200916T150000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/17
DESCRIPTION:Title: Stolarsky principle: generalizations\, extensions
\, and applications\nby Dmitriy Bilyk (U of Minnesota) as part of Poin
t Distributions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAb
stract\nIn 1973 Kenneth Stolarsky proved a remarkable identity\, which con
nected two classical quantities\, which measure the quality of point distr
ibutions on the sphere: the $L^2$ spherical cap discrepancy and the pairwi
se sum of Euclidean distances between points. This fact\, which came to be
known as the Stolarsky Invariance Principle\, established a certain dual
ity between problems of discrepancy theory on one hand\, and distance geom
etry or energy optimization on the other\, and allowed one to transfer met
hods and results of one field to the other. Since then numerous versions\
, extensions\, and generalizations of this principle have been found\, lea
ding to connections between various notions of discrepancy and discrete en
ergies in different settings and to a number of applications to various pr
oblems of discrete geometry. In this talk we shall survey known work on
the Stolarsky principle\, as well as some related problems.\n\nThis talk w
ill be recorded and posted on the webinar homepage. Slides will be availab
le too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Steinerberger (U of Washington)
DTSTART;VALUE=DATE-TIME:20200930T170000Z
DTEND;VALUE=DATE-TIME:20200930T180000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/18
DESCRIPTION:Title: Optimal Transport and Point Distributions on the
Torus\nby Stefan Steinerberger (U of Washington) as part of Point Dist
ributions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract
\nThere are lots of ways of measuring the regularity of a set\nof points o
n the Torus. I'll introduce a fundamental notion from Optimal\nTransport\
, the Wasserstein distance\, as another such measure. It \ncorresponds qui
te literally over what distance one has to spread the\npoints to be evenly
distributed\, it has a natural physical intuition\n(the notion itself was
derived in Economics modeling transport) and is\nnaturally related to oth
er notions such as discrepancy or Zinterhof's\ndiaphony. Classical Fourie
r Analysis allows us to bound this transport \ndistance via exponential su
ms which are well studied\; this allows us to revisit\nmany classical cons
tructions and get transport bounds basically for free. \nWe'll finish by r
evisiting a classical problem from numerical integration \nfrom this new a
ngle. There will be many open problems throughout the talk.\n\nThis talk
will be recorded and posted on the webinar homepage. Slides will be availa
ble too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stefan Steinerberger (U of Washington)
DTSTART;VALUE=DATE-TIME:20201002T170000Z
DTEND;VALUE=DATE-TIME:20201002T180000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/19
DESCRIPTION:Title: Optimal Transport and Point Distributions on Mani
folds\nby Stefan Steinerberger (U of Washington) as part of Point Dist
ributions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract
\nWe'll go somewhat deeper into the connection between the\nWasserstein di
stance and notions from potential theory: in particular\,\nhow the classic
al Green function can be used to derive bounds on \nWasserstein transport
on general manifolds. On the sphere\, our results\nsimplify and the Riesz
energy appears in a nice form. We conclude with\na fundamental new idea: t
he Wasserstein Uncertainty Principle which\nsays that if it's terribly eas
y to buy milk wherever you are\, then there\nmust be many supermarkets --
the precise form of this isoperimetric\nprinciple is not known and\, despi
te being purely geometric\, it would \nhave immediate impact on some PDE p
roblems.\n\nThis talk will be recorded and posted on the webinar homepage.
Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michelle Mastrianni (U of Minnesota)
DTSTART;VALUE=DATE-TIME:20201021T140000Z
DTEND;VALUE=DATE-TIME:20201021T150000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/20
DESCRIPTION:Title: Bounds for Star-Discrepancy with Dependence on th
e Dimension\nby Michelle Mastrianni (U of Minnesota) as part of Point
Distributions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbst
ract\nThe question of how the star-discrepancy (with respect to corners) o
f an n-point set in the d-dimensional unit cube depends on the dimension d
was studied in 2001 by Heinrich\, Novak\, Wasilkowski and Wozniakowski. T
hey established an upper bound that depends only polynomially on d/n. The
proof makes use of the fact that the set of corners in the d-dimensional u
nit cube is a VC-class\, and employs a result by Talagrand (1994) that use
s a partitioning scheme to study the tails of the supremum of a Gaussian p
rocess under certain conditions that are always satisfied by VC-classes. I
n 2011\, Aistleitner produced a simpler proof of this upper bound using a
direct dyadic partitioning argument. The best lower bound was achieved by
Hinrichs (2003)\, who built upon the ideas of using VC-inequalities to ach
ieve a lower bound with polynomial behavior in d/n as well. In this talk I
will introduce the notion of VC dimension and discuss how it is employed
in the above proofs\, and outline how the direct partitioning argument for
the upper bound uses the same underlying ideas about where the bulk of th
e contribution to the tails arises.\n\nThis talk will be recorded and post
ed on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Reznikov (Florida State)
DTSTART;VALUE=DATE-TIME:20200923T140000Z
DTEND;VALUE=DATE-TIME:20200923T150000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/21
DESCRIPTION:Title: Minimal discrete energy on fractals\nby Alexa
nder Reznikov (Florida State) as part of Point Distributions Webinar\n\nLe
cture held in Zoom\, password: 600Cell.\n\nAbstract\nWe will survey some o
ld and new results on the existence of\nasymptotic behavior of minimal dis
crete Riesz energy of many particles\nlocated in a fractal set. Unlike in
the case of a rectifiable set\,\nwhen the asymptotic behavior always exist
s\, we will show that on a\nlarge class of somewhat "balanced" fractals th
e energy (and\nbest-packing) does not have any asymptotic behavior.\n\nThi
s talk will be recorded and posted on the webinar homepage.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Grabner (TU Graz)
DTSTART;VALUE=DATE-TIME:20201014T140000Z
DTEND;VALUE=DATE-TIME:20201014T150000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/22
DESCRIPTION:Title: Fourier-Eigenfunctions and Modular Forms\nby
Peter Grabner (TU Graz) as part of Point Distributions Webinar\n\nLecture
held in Zoom\, password: 600Cell.\n\nAbstract\nEigenfunctions of the Fouri
er-transform play a major role in Viazovska's\nproof of the best packing o
f the $E_8$ lattice in dimension 8 and the\nsubsequent determination of th
e Leech lattice as best packing\nconfiguration dimension 24 by Cohn\, Kuma
r\, Miller\, Radchenko\, and\nViazovska. In joint work with A. Feigenbaum
and D. Hardin we have shown\nthat the constructions as used for these res
ults are unique\; we could\nshed more light on the underlying modular and
quasimodular forms and\ndetermine linear recurrence relations and differen
tial equations\ncharacterising these forms.\n\nThis talk will be recorded
and posted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Beltran (U of Cantabria)
DTSTART;VALUE=DATE-TIME:20201028T140000Z
DTEND;VALUE=DATE-TIME:20201028T150000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/23
DESCRIPTION:Title: Smale’s motivation in describing the 7th proble
m of his list\nby Carlos Beltran (U of Cantabria) as part of Point Dis
tributions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstrac
t\nIn 1993\, Mike Shub and Steve Smale posed a question that would be late
r included in Smale’s list as 7th problem. Although this last problem ha
s became so famous\, the exact reason for its form and the consequences th
at its solution would have for the initial goal are not so well known in t
he mathematician community. In this seminar\, I will describe the thrillin
g story of these origins: where the problem came from\, would it still be
useful for that task\, and what is left to do. I will probably talk a lot
and show very few formulas\, and I will also present some open problems.\n
\nThis talk will be recorded and posted on the webinar homepage. Slides wi
ll be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adrian Ebert (RICAM)
DTSTART;VALUE=DATE-TIME:20201104T150000Z
DTEND;VALUE=DATE-TIME:20201104T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/24
DESCRIPTION:Title: Construction of (polynomial) lattice rules by smo
othness-independent component-by-component digit-by-digit constructions\nby Adrian Ebert (RICAM) as part of Point Distributions Webinar\n\nLectu
re held in Zoom.\n\nAbstract\nIn this talk\, we introduce component-by-com
ponent digit-by-digit algorithms (CBC-DBD)\nfor the construction of (polyn
omial) lattice rules in weighted Korobov/Walsh spaces with\nprescribed dec
ay of the involved series coefficients and associated smoothness α > 1. T
he\npresented methods are extensions of a construction algorithm establish
ed by Korobov\nin [1] to the modern quasi-Monte Carlo (QMC) setting. We sh
ow that the introduced\nCBC-DBD algorithms construct QMC rules with N = 2
n points which achieve the almost\noptimal worst-case error convergence ra
tes in the studied function spaces. Due to the used\nquality functions\, t
he algorithms can construct good (polynomial) lattice rules indepen-\ndent
of the smoothness α of the respective function class. Furthermore\, we d
erive suitable\nconditions on the weights under which the mentioned error
bounds are independent of the\ndimension. The presented algorithms can be
implemented in a fast manner such that the\nconstruction only requires O(s
N ln N ) operations\, where N = 2 n is the number of lattice\npoints and s
denotes the dimension. We stress that these fast constructions achieve th
is\ncomplexity without the use of fast Fourier transformations (FFTs)\, as
in\, e.g.\, [2]. We\npresent extensive numerical results which confirm ou
r theoretical findings.\n\n[1] N.M. Korobov. On the computation of optimal
coefficients. Dokl. Akad. Nauk SSSR\,\n26:590–593. 1982.\n\n[2] D. Nuye
ns\, R. Cools. Fast component-by-component construction of rank-1 lattice\
nrules with a non-prime number of points. J. Complexity 22(1)\, 4–28. 20
06.\n\nJoint work with: Peter Kritzer (RICAM Linz)\,\nDirk Nuyens (NUMA KU
Leuven)\, \nOnyekachi Osisiogu (Ricam Linz) and \nTetiana Stepaniuk (U. o
f Lübeck)\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Glazyrin (U of Texas Rio Grande Valley)
DTSTART;VALUE=DATE-TIME:20201111T150000Z
DTEND;VALUE=DATE-TIME:20201111T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/25
DESCRIPTION:Title: Mapping to the space of spherical harmonics\n
by Alexey Glazyrin (U of Texas Rio Grande Valley) as part of Point Distrib
utions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract\nF
or a variety of problems for point configurations in spheres\, the space o
f spherical harmonics plays an important role. In this talk\, we will disc
uss maps from point configurations to the space of spherical harmonics. Su
ch maps can be used for finding bounds on packings\, energy bounds\, and c
onstructing new configurations. We will explain classical results from thi
s perspective and prove several new bounds. Also we will show a new short
proof for the kissing number problem in dimension 3.\n\nThis talk will be
recorded and posted on the webinar homepage. Slides will be available too.
\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Vlasiuk (Florida State)
DTSTART;VALUE=DATE-TIME:20201007T140000Z
DTEND;VALUE=DATE-TIME:20201007T150000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/27
DESCRIPTION:Title: Asymptotic properties of short-range interaction
functionals\nby Alex Vlasiuk (Florida State) as part of Point Distribu
tions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract\nSh
ort-range interactions\, such as the hypersingular Riesz energies\, are kn
own to be amenable to asymptotic analysis\, which allows to obtain for the
m the distribution of minimizers and asymptotics of the minima. We extract
the properties making such analysis possible into a standalone framework.
This allows us to give a unified treatment of hypersingular Riesz energie
s and optimal quantizers. We further obtain new results about the scale-in
variant nearest neighbor interactions\, such as the k-nearest neighbor tru
ncated Riesz energy. The suggested approach has applications to common met
hods for generating distributions with prescribed density: Riesz energies\
, centroidal Voronoi tessellations\, and popular meshing algorithms due to
Persson-Strang and Shimada-Gossard. It naturally generalizes from 2-body
to k-body interactions.\n\nBased on joint work with Douglas Hardin and Ed
Saff.\n\nThis talk will be recorded and posted on the webinar homepage. Sl
ides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Leopardi (NCI Australia)
DTSTART;VALUE=DATE-TIME:20201209T210000Z
DTEND;VALUE=DATE-TIME:20201209T220000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/29
DESCRIPTION:Title: Diameter bounded equal measure partitions of Ahlf
ors regular metric measure spaces\nby Paul Leopardi (NCI Australia) as
part of Point Distributions Webinar\n\nLecture held in Zoom\, password: 6
00Cell.\n\nAbstract\nThe algorithm devised by Feige and Schechtman for par
titioning higher dimensional spheres into regions of equal measure and sma
ll diameter is combined with David and Christ's construction of dyadic cub
es to yield a partition algorithm suitable to any connected Ahlfors regula
r metric measure space of finite measure. \n\nThis is joint work with Giac
omo Gigante of the University of Bergamo.\n\nThis talk will be recorded an
d posted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laurent Bétermin (U of Vienna)
DTSTART;VALUE=DATE-TIME:20201216T150000Z
DTEND;VALUE=DATE-TIME:20201216T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/30
DESCRIPTION:Title: Theta functions\, ionic crystal energies and opti
mal lattices\nby Laurent Bétermin (U of Vienna) as part of Point Dist
ributions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstract
\nThe determination of minimizing structures for pairwise interaction ener
gies is a very challenging crystallization problem. The goal of this talk
is to present recent optimality results among charges and lattice structur
es obtained with Markus Faulhuber (University of Vienna) and Hans Knüpfer
(University of Heidelberg). The central object of these works is the heat
kernel associated to a lattice\, also called lattice theta function. Seve
ral connections will be showed between interaction energies and theta func
tions in order to study the following problems:\n- Born’s Conjecture: ho
w to distribute charges on a fixed lattice in order to minimize the associ
ated Coulombian energy? In the simple cubic case\, Max Born conjectured th
at the alternation of charges +1 and -1 (i.e. the rock-salt structure of N
aCl) is optimal. The proof of this conjecture obtained with Hans Knüpfer
will be briefly discussed as well as its generalization to other lattices
and energies.\n- stability of the rock-salt structure: what could be condi
tions on interaction potentials such that the minimal energies among charg
es and lattices has a rock-salt structure? Many results\, both theoretical
and numerical and obtained with Markus Faulhuber and Hans Knüpfer\, will
be presented.\n- maximality of the triangular lattices among lattices wit
h alternation of charges: we will present this new universal optimality am
ong lattices obtained with Markus Faulhuber.\n\nThis talk will be recorded
and posted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Barg (University of Maryland)
DTSTART;VALUE=DATE-TIME:20201202T150000Z
DTEND;VALUE=DATE-TIME:20201202T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/31
DESCRIPTION:Title: Stolarsky's invariance principle for the Hamming
space\nby Alexander Barg (University of Maryland) as part of Point Dis
tributions Webinar\n\nLecture held in Zoom\, password: 600Cell.\n\nAbstrac
t\nStolarsky's invariance principle has enjoyed considerable\nattention in
the literature in the last decade. In this talk we study an\nanalog of St
olarsky's identity in finite metric spaces with an emphasis on\nthe Hammin
g space. We prove several bounds on the spherical discrepancy of\nbinary c
odes and identify some discrepancy minimizing configurations. We\nalso com
ment on the connection between the problem of minimizing the\ndiscrepancy
and the general question of locating minimum-energy\nconfigurations in the
space. The talk is based on arXiv:2005.12995 and\narXiv:2007.09721 (joint
with Maxim Skriganov).\n\nThis talk will be recorded and posted on the we
binar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David García-Zelada (Aix-Marseille University)
DTSTART;VALUE=DATE-TIME:20210203T160000Z
DTEND;VALUE=DATE-TIME:20210203T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/32
DESCRIPTION:Title: A large deviation principle for empirical measure
s\nby David García-Zelada (Aix-Marseille University) as part of Point
Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nThe main obje
ct of this talk will be a model of n interacting particles at equilibrium.
\nI will describe its macroscopic behavior as n grows to in\nnity by showi
ng a Laplace prin-\nciple or\, equivalently\, a large deviation principle.
This implies\, in some cases\, an almost\nsure convergence to a determini
stic probability measure. Among the main motivating\nexamples we may \nnd
Coulomb gases on Riemannian manifolds\, the eigenvalue distri-\nbution of
Gaussian random matrices and the roots of Gaussian random polynomials.\nTh
is talk is based on arXiv:1703.02680.\n\nThis talk will be recorded and po
sted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arno Kuijlaars (Katholieke Universiteit Leuven)
DTSTART;VALUE=DATE-TIME:20210210T160000Z
DTEND;VALUE=DATE-TIME:20210210T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/33
DESCRIPTION:Title: The spherical ensemble with external sources\
nby Arno Kuijlaars (Katholieke Universiteit Leuven) as part of Point Distr
ibutions Webinar\n\nLecture held in Zoom.\n\nAbstract\nWe study a model of
a large number of points on the unit sphere under \nthe influence of a fi
nite number of fixed repelling charges. \nIn the large n limit the points
fill a region that is known as \nthe droplet. For small external charges t
he droplet is \nthe complement of the union of a number of spherical caps\
, one around \neach of the external charges. When the external charges gro
w\,\nthe spherical caps will start to overlap and the droplet ondergoes a
non-trivial\ndeformation.\n\nWe explicitly describe the transition for the
case of equal external\ncharges that are symmetrically located around the
north pole. In our\napproach we first identify a motherbody that\, due to
the symmetry in the problem\,\nwill be located on a number of meridians c
onnecting the north and south poles.\nAfter projecting onto the complex pl
ane\, and undoing the symmetry\, we\ncharacterize the motherbody by means
of the solution of a vector equilibrium\nproblem from logarithmic potentia
l theory.\n\nThis talk will be recorded and posted on the webinar homepage
. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alex Iosevich (University of Rochester)
DTSTART;VALUE=DATE-TIME:20210217T160000Z
DTEND;VALUE=DATE-TIME:20210217T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/34
DESCRIPTION:Title: Finite point configurations and frame theory\
nby Alex Iosevich (University of Rochester) as part of Point Distributions
Webinar\n\nLecture held in Zoom.\n\nAbstract\nWe are going to discuss som
e recent and not so recent applications of analytic and combinatorial resu
lts on finite point configurations to problems of existence of exponential
and Gabor frames and bases.\n\nThis talk will be recorded and posted on t
he webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kasso Okoudjou (Tufts University)
DTSTART;VALUE=DATE-TIME:20210224T160000Z
DTEND;VALUE=DATE-TIME:20210224T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/35
DESCRIPTION:Title: Completeness of Weyl-Heisenberg POVMs\nby Kas
so Okoudjou (Tufts University) as part of Point Distributions Webinar\n\nL
ecture held in Zoom.\n\nAbstract\nThe finite Gabor (or Weyl-Heisenberg) sy
stem generated by a unit-norm vector $g\\in \\mathbb{C}^d$ is the set of
vectors $$\\big\\{g_{k\,\\ell}=e^{2\\pi i k\\cdot}g(\\cdot - \\ell)\\big\
\}_{k\, \\ell =0}^{d-1}.$$ It is know that every such system forms a finit
e unit norm tight frame (FUNTF) for $\\C^d$\, i.e.\, $$d^3 \\|x\\|^2=\\su
m_{k\, \\ell=0}^{d-1}|\\langle x\, g_{k\,\\ell}\\rangle |^2\\quad \\forall
\\\, x\\in \\mathbb{C}^d.$$ Furthermore\, the Zauner conjecture asserts t
hat for each $d\\geq 2$\, there exist unit-norm vectors $g \\in \\mathbb{
C}^d$ such that this FUNTF is equiangular\, that is\, $|\\langle g\, g_{k\
, \\ell}\\rangle |^2= \\tfrac{1}{d+1}.$ \nAssuming the existence of a unit
-vector $g$ that positively answers Zauner's conjecture\, one can show tha
t the set of rank-one matrices $$\\big\\{\\pi_{k\,\\ell}=\\langle \\cdot\,
g_{k\,\\ell}\\rangle g_{k\, \\ell}\\big\\}_{k \\ell=0}^{d-1}$$ is complet
e in the space of $d\\times d$ matrices. Consequently\, $\\big\\{\\pi_{k\,
\\ell}\\big\\}_{k \\ell=0}^{d-1}$ forms a symmetric informationally comple
te positive operator-valued measure (SIC-POVM). \n\nIn fact\, it is known
that given a unit-norm vector $g\\in \\mathbb{C}^d$\, the POVM $\\big\\{\
\pi_{k\,\\ell}\\big\\}_{k \\ell=0}^{d-1}$ is informationally complete (IC)
if and only if $\\langle g\, g_{k\, \\ell} \\rangle \\neq 0$ for all $(k
\,\\ell)\\neq (0\,0)$. \nIn this talk\, we give a different proof of the
characterization of the IC-POVMs. We then focus on investigating non-info
rmationally complete POVMs. We will present some preliminary results perta
ining to the dimensions of the linear spaces spanned by these rank-one mat
rices. (This talk is based on on-going joint work with S.~Kang and A.~Gold
berger.)\n\nThis talk will be recorded and posted on the webinar homepage.
Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mircea Petrache (PUC Chile)
DTSTART;VALUE=DATE-TIME:20210303T160000Z
DTEND;VALUE=DATE-TIME:20210303T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/36
DESCRIPTION:Title: Sharp isoperimetric inequality\, discrete PDEs an
d Semidiscrete optimal transport\nby Mircea Petrache (PUC Chile) as pa
rt of Point Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nCo
nsider the following basic model of finite crystal cluster\nformation: in
a periodic graph G with vertices in R^d (representing\npossible molecular
bonds) a subset (of atoms) must be chosen\, so that\nthe total number of b
onds between a point in X and one outside X is\nminimized. These bonds for
m the edge-perimeter of X\, denoted \\partial\nX.\nIf the graph is periodi
c and locally finite\, any X satisfies an\ninequality of the form |X|^{d-1
} \\leq C |partial X|^d\, where the\noptimal C depends on the graph. How c
an we determine the structure of\nsets X realizing equality in the above\,
based on the geometry and of\nG?\nIf we take the continuum limit of G\, t
hen the classical Wulff shape\ntheory describes optimal limit shapes\, and
at least two proofs of\nisoperimetric inequality apply\, one based on PDE
and calibration\nideas\, and the other based on Optimal Transport ideas.
We focus on\nusing the heuristic coming from the continuum analogue\, to a
nswer the\nabove question in some cases\, in the discrete case. This appro
ach\nhighlights the tight connection between discrete PDEs and semidiscret
e\nOptimal Transport\, and a link to the Minkowski theorem for convex\npol
yhedra.\n\nThis talk will be recorded and posted on the webinar homepage.
Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yeli Niu (U of Alberta)
DTSTART;VALUE=DATE-TIME:20210310T153000Z
DTEND;VALUE=DATE-TIME:20210310T163000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/37
DESCRIPTION:Title: Discretization of integrals on compact metric
measure spaces\nby Yeli Niu (U of Alberta) as part of Point Distributi
ons Webinar\n\nLecture held in Zoom.\n\nAbstract\nLet $\\mu$ be a Borel
probability measure on a compact path-connected metric space $(X\, \\rh
o)$ for which there exist constants $c\,\\be\\ge 1$ such that $\\mu(B) \\
ge c r^{\\be}$ \n for every open ball $B\\subset X$ of radius $r>0$. For
a class\n of Lipschitz functions $\\Phi:[0\,\\infty)\\to\\RR$ that
are\n piecewise within a finite-dimensional subspace of\n
continuous functions\, we prove under certain mild conditions\n
on the metric $\\rho$ and the measure $\\mu$ that for each\n
positive integer $N\\ge 2$\, and each $g\\in L^\\infty(X\,\n d
\\mu)$ with $\\|g\\|_\\infty=1$\, there exist points $y_1\,\n
\\ldots\, y_{ N}\\in X$ and real \n numbers $\\lambda_1\, \\ldots\, \\l
ambda_{ N}$ such that for any $x\\in X$\, \n \\begin{align*}\n & \\le
ft| \\int_X \\Phi (\\rho (x\, y)) g(y) \\\,\\dd \\mu (y) - \\sum_{j =\n
1}^{ N} \\lambda_j \\Phi (\\rho (x\, y_j)) \\right| \\leqslant C N^{- \
\frac{1}{2} - \\frac{3}{2\\be}} \\sqrt{\\log N}\,\n \\end{align*}\n
where the constant $C>0$ is independent of $N$ and $g$. In the case whe
n $X$ is the unit sphere $\\sph$ of $\\RR^{d+1}$ with the ususal geodesic
distance\, we also prove that the constant $C$ here is independent of the
dimension $d$. Our estimates are better than those obtained from the
standard Monte Carlo methods\, which typically yield a weaker upper
bound $N^{-\\f12}\\sqrt{\\log N}$.\n\nThis talk will be recorded and post
ed on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xuemei Chen (UNC Wilmington)
DTSTART;VALUE=DATE-TIME:20210317T160000Z
DTEND;VALUE=DATE-TIME:20210317T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/38
DESCRIPTION:Title: Frame Design Using Projective Riesz Energy\nb
y Xuemei Chen (UNC Wilmington) as part of Point Distributions Webinar\n\nL
ecture held in Zoom.\n\nAbstract\nTight and well-separated frames are desi
rable in many signal \nprocessing applications. We introduce a projective
Riesz kernel for the \nunit sphere and investigate properties of N-point e
nergy minimizing \nconfigurations for such a kernel. We show that these mi
nimizing \nconfigurations\, for N sufficiently large\, form frames that ar
e \nwell-separated (have low coherence) and are nearly tight. We will also
\nshow some numerical experiments. This is joint work with Doug Hardin an
d \nEd Saff.\n\nThis talk will be recorded and posted on the webinar homep
age. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ruiwen Shu (U of Maryland)
DTSTART;VALUE=DATE-TIME:20210324T150000Z
DTEND;VALUE=DATE-TIME:20210324T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/39
DESCRIPTION:Title: Dynamics of Particles on a Curve with Pairwise Hy
per-singular Repulsion\nby Ruiwen Shu (U of Maryland) as part of Point
Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nWe investigat
e the large time behavior of $N$ particles restricted to a smooth closed c
urve in $\\mathbb{R}^d$ and subject to a gradient flow with respect to Euc
lidean hyper-singular repulsive Riesz $s$-energy with $s>1$. We show that
regardless of their initial positions\, for all $N$ and time $t$ large\, t
heir normalized Riesz $s$-energy will be close to the $N$-point minimal po
ssible energy. Furthermore\, the distribution of such particles will be cl
ose to uniform with respect to arclength measure along the curve.\n\nThis
talk will be recorded and posted on the webinar homepage. Slides will be a
vailable too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Musin (U of Texas Rio Grande Valley)
DTSTART;VALUE=DATE-TIME:20210331T150000Z
DTEND;VALUE=DATE-TIME:20210331T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/40
DESCRIPTION:Title: Majorization\, discrete energy on spheres and f-d
esigns\nby Oleg Musin (U of Texas Rio Grande Valley) as part of Point
Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nWe consider th
e majorization (Karamata) inequality and minimums of the majorization (M-s
ets) for f-energy potentials of m-point configurations in a sphere. We dis
cuss the optimality of regular simplexes\, describe M-sets with a small nu
mber of points\, define spherical f-designs and study their properties. Th
en we consider relations between the notions of f-designs and M-sets\, \n$
\\tau$\n-designs\, and two-distance sets\n\nThis talk will be recorded and
posted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Woden Kusner (U of Georgia)
DTSTART;VALUE=DATE-TIME:20210407T150000Z
DTEND;VALUE=DATE-TIME:20210407T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/41
DESCRIPTION:Title: Measuring chirality with the wind\nby Woden K
usner (U of Georgia) as part of Point Distributions Webinar\n\nLecture hel
d in Zoom.\n\nAbstract\nThe question of measuring "handedness" is of some
significance in both mathematics and in the real world. Propellors and scr
ews\, proteins and DNA\, in fact *almost everything* is chiral. Can we qua
ntify chirality? Or can we perhaps answer the question:\n"Are your shoes m
ore left-or-right handed than a potato?"\nWe can begin with the hydrodynam
ic principle that chiral objects rotate when placed in a collimated flow (
or wind). This intuition naturally leads to a trace-free tensorial chirali
ty measure for space curves and surfaces\, with a clear physical interpret
ation measuring twist. As a consequence\, the "average handedness" of an o
bject with respect to this measure will always be 0. This also strongly su
ggests that a posited construction of Lord Kelvin--the isotropic helicoid-
-can not exist.\njoint with Giovanni Dietler\, Rob Kusner\, Eric Rawdon an
d Piotr Szymczak\n\nThis talk will be recorded and posted on the webinar h
omepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Dragnev (Purdue Fort Wayne)
DTSTART;VALUE=DATE-TIME:20210414T150000Z
DTEND;VALUE=DATE-TIME:20210414T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/42
DESCRIPTION:Title: Bounds for Spherical Codes: The Levenshtein Frame
work Lifted\nby Peter Dragnev (Purdue Fort Wayne) as part of Point Dis
tributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nBased on the Dels
arte-Yudin linear programming approach\, we extend Levenshtein’s framewo
rk to obtain lower bounds for the minimum henergy of spherical codes of pr
escribed dimension and cardinality\, and upper bounds on the maximal cardi
nality of spherical codes of prescribed dimension and minimum separation.
These bounds are universal in the sense that they hold for a large class o
f potentials h and in the sense of Levenshtein. Moreover\, codes attaining
the bounds are universally optimal in the sense of Cohn-Kumar. Referring
to Levenshtein bounds and the energy bounds of the authors as “first lev
el”\, our results can be considered as “next level” universal bounds
as they have the same general nature and imply necessary and sufficient c
onditions for their local and global optimality. For this purpose\, we int
roduce the notion of Universal Lower Bound space (ULB-space)\, a space tha
t satisfies certain quadrature and interpolation properties. While there a
re numerous cases for which our method applies\, we will emphasize the mod
el examples of 24 points (24-cell) and 120 points (600-cell) on \nS\n3\n.
In particular\, we provide a new proof that the 600-cell is universally op
timal\, and in so doing\, we derive optimality of the 600-cell on a class
larger than the absolutely monotone potentials considered by Cohn-Kumar.\n
\nThis talk will be recorded and posted on the webinar homepage. Slides wi
ll be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Doug Hardin (Vanderbilt U)
DTSTART;VALUE=DATE-TIME:20210421T150000Z
DTEND;VALUE=DATE-TIME:20210421T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/43
DESCRIPTION:Title: Asymptotics of periodic minimal discrete energy p
roblems\nby Doug Hardin (Vanderbilt U) as part of Point Distributions
Webinar\n\nLecture held in Zoom.\n\nAbstract\nFor $s>0$ and a lattice $L$
in $R^d$\, we consider the\nasymptotics of $N$-point configurations mini
mizing the $L$-periodic Riesz\n$s$-energy as the number of points $N$ goes
to infinity. In particular\, we\nfocus on the case $0On the rank of non-informationally complete Gabor
POVMs\nby Shujie Kang (UT Arlington) as part of Point Distributions W
ebinar\n\nLecture held in Zoom.\n\nAbstract\nWe investigate Positive Opera
tor Valued Measures (POVMs) generated by Gabor frames in $\\mathbb{C}^d$.
A complete (Gabor) POVM is one that spans the space $\\mathbb{C}^{d^{2}}$
of $d\\times d$ matrices. It turns out that being a complete Gabor POVM is
a generic property. As a result\, the focus of this talk will be on non-c
omplete Gabor POVMs. We will describe the possible ranks of these Gabor PO
VMs\, and derive various consequences for the underlying Gabor frames. In
particular\, we will give details in dimensions $4$ and $5$.\n\nThis talk
will be recorded and posted on the webinar homepage. Slides will be avail
able too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mario Ullrich (JKU Linz)
DTSTART;VALUE=DATE-TIME:20210505T141500Z
DTEND;VALUE=DATE-TIME:20210505T150000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/45
DESCRIPTION:Title: Random matrices and approximation using function
values\nby Mario Ullrich (JKU Linz) as part of Point Distributions Web
inar\n\nLecture held in Zoom.\n\nAbstract\nWe consider $L_2$-approximation
of functions using linear algorithms and want to compare the power of fun
ction values with the power of arbitrary linear information. Under mild as
sumptions on the class of functions\, we show that the minimal worst-case
errors based on function values decay at almost the same rate as those wit
h arbitrary info\, if the latter decay fast enough. Our results are to som
e extent best possible and\, in special cases\, improve upon well-studied
point constructions\, like sparse grids\, which were previously assumed to
be optimal. The proof is based on deep results on large random matrices\,
including the recent solution of the Kadison-Singer problem\, and reveals
that (classical) least-squares methods might be surprisingly powerful in
a general setting.\n\nThis talk will be recorded and posted on the webinar
homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Johann Brauchart (TU Graz)
DTSTART;VALUE=DATE-TIME:20210519T150000Z
DTEND;VALUE=DATE-TIME:20210519T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/46
DESCRIPTION:Title: Weighted $L^2$ -Norms of Gegenbauer Polynomials
— and more!\nby Johann Brauchart (TU Graz) as part of Point Distribu
tions Webinar\n\nLecture held in Zoom.\n\nAbstract\nAbstract: \nI discuss
integrals of the form\n\\begin{equation*}\n\\int_{-1}^1(C_n^{(\\lambda)}(x
))^2(1-x)^\\alpha (1+x)^\\beta\\dd x\,\n\\end{equation*}\nwhere $C_n^{(\\l
ambda)}$ denotes the Gegenbauer-polynomial of index $\\lambda>0$ and $\\al
pha\,\\beta>-1$. Such integrals for orthogonal polynomials involving\, in
particular\, a ``wrong'' weight function appear in physics applications an
d point distribution problems. \n\nI present exact formulas for the integr
als and their generating functions\, and give asymptotic formulas as $n\\
to\\infty$. \n\nThis is joint work with Peter Grabner also from TU Graz.\n
\nThis talk will be recorded and posted on the webinar homepage. Slides wi
ll be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:William Chen (Macquarie U)
DTSTART;VALUE=DATE-TIME:20210512T150000Z
DTEND;VALUE=DATE-TIME:20210512T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/47
DESCRIPTION:Title: The Veech 2-circle problem and non-integrable fla
t dynamical systems\nby William Chen (Macquarie U) as part of Point Di
stributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nWe are motivated
by an interesting problem studied more than 50 years ago by Veech and whi
ch can be considered a parity\, or mod 2\, version of the classical equidi
stribution problem concerning the irrational rotation sequence. The Veech
discrete 2-circle problem can also be visualized as a continuous flat dyna
mical system\, in the form of 1-direction geodesic flow on a surface obtai
ned by modifying the surface comprising two side-by-side unit squares by t
he inclusion of barriers and gates on the vertical edges\, with appropriat
e modification of the edge identifications. A famous result of Gutkin and
Veech says that 1-direction geodesic flow on any flat finite polysquare tr
anslation surface exhibits optimal behavior\, in the form of an elegant un
iform-periodic dichotomy. Here the modified surface in question is no long
er such a surface\, and there are vastly different outcomes depending on t
he values of certain parameters.\n\nThis talk will be recorded and posted
on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Austin Anderson and Alex White (Florida State)
DTSTART;VALUE=DATE-TIME:20210602T153000Z
DTEND;VALUE=DATE-TIME:20210602T163000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/48
DESCRIPTION:Title: Asymptotics of Best Packing and Best Covering
\nby Austin Anderson and Alex White (Florida State) as part of Point Distr
ibutions Webinar\n\nLecture held in Zoom.\n\nAbstract\nWe discuss recent p
rogress on asymptotics for the dual problems of best packing and best cove
ring in Euclidean space. For future investigations\, we highlight their re
lation to large parameter limits of minimal Riesz s-energy and Riesz s-pol
arization\, respectively. Next\, we address a weak-separation argument for
coverings and its flexibility as compared to similar arguments for polari
zation and packing. Finally\, we examine how a recent non-existence proof
for the asymptotics of best packing on dependent fractals is adapted to bo
th constrained and unconstrained covering- the second case owing largely t
o weak separation of coverings. This is joint work with Oleksandr Vlasiuk
and Alexander Reznikov of Florida State University.\n\nThis talk will be r
ecorded and posted on the webinar homepage. Slides will be available too.\
n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alan Legg (Purdue U Fort Wayne)
DTSTART;VALUE=DATE-TIME:20210609T150000Z
DTEND;VALUE=DATE-TIME:20210609T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/49
DESCRIPTION:Title: Logarithmic Equilibrium on the Sphere in the Pres
ence of Multiple Point Charges\nby Alan Legg (Purdue U Fort Wayne) as
part of Point Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\n
(Joint work with Peter Dragnev) We consider the problem of finding the equ
ilibrium measure on the unit sphere in R^3 using logarithmic potentials\,
in the presence of external fields made up of a finite number of point cha
rges on the sphere. For any such external field\, the complement of the eq
uilibrium measure turns out to be the stereographic preimage from the plan
e of a union of classical quadrature domains.\n\nThis talk will be recorde
d and posted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Assaf Goldberger (Tel Aviv U)
DTSTART;VALUE=DATE-TIME:20210616T150000Z
DTEND;VALUE=DATE-TIME:20210616T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/50
DESCRIPTION:Title: Configurations\, Automorphisms and Cohomology
\nby Assaf Goldberger (Tel Aviv U) as part of Point Distributions Webinar\
n\nLecture held in Zoom.\n\nAbstract\nPoint configurations on finite dimen
sional real or complex spaces\, typically on the unit sphere\, are importa
nt in Physics\, Coding Theory\, Classical and Quantum Information Theory\,
Geometry\, Number Theory and more. The Automorphism group a of point conf
iguration is a tool to study it\, and to generate new ones. In this talk w
e will show how to generate automorphism groups from group-theoretic consi
derations\, and how to construct configurations that satisfy the group. Th
e main tool in use is Group Cohomology. We show that there is a spectral s
equence which captures all possible solutions and all obstructions to the
construction of a solution. In addition this sequence captures the Galois
structure of algebraic configurations. Galois structures were discovered r
ecently in the case of Zauner SIC-POVMs. This point of view can be general
ized to a much broader framework\, e.g. higher tensors as replacements of
the Gramian matrix\, perfect squares are seen to be dual to configuration
Gramians when one uses homology instead of cohomology\, and there are some
connections to Number Theory\, such as the theory of Brauer Groups. Other
applications are the generation of Hadamard and Weighing matrices. We wil
l discuss these extensions as time permits. This is joint work with Giora
Dula.\n\nThis talk will be recorded and posted on the webinar homepage. Sl
ides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert McCann (University of Toronto)
DTSTART;VALUE=DATE-TIME:20210630T150000Z
DTEND;VALUE=DATE-TIME:20210630T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/51
DESCRIPTION:Title: Maximizing the sum of angles between pairs of lin
es in Euclidean space\nby Robert McCann (University of Toronto) as par
t of Point Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nCho
ose $N$ unoriented lines through the origin of $\\R^{d+1}$. \n Suppose ea
ch pair of lines repel each other with a force {whose strength is} \n inde
pendent of the (acute) angle\nbetween them\, so that they prefer to be or
thogonal to each other. However\, unless $N \\le d+1$\,\nit is impossibl
e for all pairs of lines to be orthogonal. What then are their stable con
figurations?\nAn unsolved conjecture of Fejes T\\' oth (1959) asserts that
the lines should be equidistributed as evenly as possible over a standard
\nbasis in $\\R^{d+1}$. By modifying the force to make it increase as a p
ower of the distance\, we show the analogous\nclaim to be true for all po
sitive powers if we are only interested in local stability\, and for suff
iciently large powers if we require global stability.\n\nThese results rep
resent joint work with Tongseok Lim (of Purdue's Krannert School of Manage
ment).\n\nThis talk will be recorded and posted on the webinar homepage. S
lides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giuseppe Negro (University of Birmingham)
DTSTART;VALUE=DATE-TIME:20210707T150000Z
DTEND;VALUE=DATE-TIME:20210707T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/52
DESCRIPTION:Title: Intermittent symmetry breaking for the maximizers
to the Agmon-Hörmander estimate on the sphere\nby Giuseppe Negro (Un
iversity of Birmingham) as part of Point Distributions Webinar\n\nLecture
held in Zoom.\n\nAbstract\nThe $L^2$ norm of a function on Euclidean space
equals the $L^2$ norm of its Fourier transform\; this is the theorem of
Plancherel. This is true for functions\, but it fails for measures\, such
as densities on a sphere. In 1976\, Agmon and Hörmander observed that it
is possible to recover a kind of Plancherel theorem in this case\, by loca
lizing on balls\; this turns out to be the most basic example of a "Fourie
r restriction estimate"\, relevant both to analysis and to PDE. In this ta
lk\, we will explicitly determine the densities that maximize such estimat
e\, discovering that they break the rotational symmetry depending on the r
adius of the localizing ball. This is joint work with Diogo Oliveira e Sil
va.\n\nThis talk will be recorded and posted on the webinar homepage. Slid
es will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander McDonald (University of Rochester)
DTSTART;VALUE=DATE-TIME:20210714T150000Z
DTEND;VALUE=DATE-TIME:20210714T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/53
DESCRIPTION:Title: Volumes spanned by k-point configurations in $\\m
athbb{R}^d$\nby Alexander McDonald (University of Rochester) as part o
f Point Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nWe con
sider a Falconer type problem concerning volumes determined by point confi
gurations in \n$\\mathbb{R}^d$\, and prove that a set with sufficiently la
rge Hausdorff dimension determines a positive measure worth of volumes. Th
e strategy for proving the result is to study the group action of the spec
ial linear group on the space of configurations.\n\nThis talk will be reco
rded and posted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Passant (U of Rochester)
DTSTART;VALUE=DATE-TIME:20210721T150000Z
DTEND;VALUE=DATE-TIME:20210721T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/54
DESCRIPTION:Title: Configurations and Erdős style distance problems
\nby Jonathan Passant (U of Rochester) as part of Point Distributions
Webinar\n\nLecture held in Zoom.\n\nAbstract\nI will discuss point large c
onfigurations in real space and how incidence geometry results of Guth-Kat
z and Rudnev can help generalise the results of Solymosi-Tardos and Rudnev
on the number of congruent and similar triangles.\n\nThis talk will be re
corded and posted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damir Ferizović (TU Graz)
DTSTART;VALUE=DATE-TIME:20210818T150000Z
DTEND;VALUE=DATE-TIME:20210818T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/55
DESCRIPTION:Title: The spherical cap discrepancy of HEALPix points
a>\nby Damir Ferizović (TU Graz) as part of Point Distributions Webinar\n
\nLecture held in Zoom.\n\nAbstract\nIn this talk I will present an algori
thm well known in the Astrophysics and\nCosmology community: HEALPix\, sh
ort for "Hierarchical\, Equal Area and\niso-Latitude Pixelation\," which
divides the two dimensional sphere\n$\\mathbb{S}^2$ into 12 rectangular s
hapes (base pixel) of equal area\, and\nallows for further subdivision of
each pixel into four smaller\, equal area\nsubpixel mimicking the simpli
city of the unit square in many ways. This\nalgorithm\, introduced by Gó
rski et al.\, also comes with a projection to the\nplane that up to a co
nstant preserves area.\n\nHEALPix also distributes $N$-many points on $\\m
athbb{S}^2$ by placing them\nat centers of pixel of the current level of
subdivision\, i.e. first $N=12$\,\nthen $N=12\\cdot 4$\, $N=12\\cdot 4^2\
, \\ldots\, N=12\\cdot 4^k$\, etc. The\nspherical cap discrepancy of thes
e points will be proven to be of order\n$N^{-1/2}$\, via recycling method
s introduced by Aistleitner\, Brauchart and\nDick.\\\\\n\n\nThis is a joi
nt work with Julian Hofstadler and Michelle Mastrianni.\n\nThis talk will
be recorded and posted on the webinar homepage. Slides will be available t
oo.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aicke Hinrichs (JKU Linz)
DTSTART;VALUE=DATE-TIME:20210825T150000Z
DTEND;VALUE=DATE-TIME:20210825T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/56
DESCRIPTION:Title: Dispersion - a survey of recent results and appli
cations\nby Aicke Hinrichs (JKU Linz) as part of Point Distributions W
ebinar\n\nLecture held in Zoom.\n\nAbstract\nThe dispersion of a point set
\, which is the volume of the largest axis-parallel box in the unit cube t
hat does not intersect the point set\, is an alternative to the discrepanc
y as a measure for certain (uniform) distribution properties. The computat
ion of the dispersion\, or even the best possible dispersion\, in dimensio
n two has a long history in computational geometry and computational compl
exity theory. Given the prominence of the problem\, it is quite surprising
that\, until recently\, very little was known about the size of the large
st empty box in higher dimensions. This changed in the last five years. In
this survey talk we focus on recent developments and new applications of
dispersion outside the area of computational geometry.\n\nThis talk will b
e recorded and posted on the webinar homepage.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Friedrich Pillichshammer (JKU Linz)
DTSTART;VALUE=DATE-TIME:20210901T150000Z
DTEND;VALUE=DATE-TIME:20210901T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/57
DESCRIPTION:Title: $L_{2}$--star\, extreme and periodic discrepancy<
/a>\nby Friedrich Pillichshammer (JKU Linz) as part of Point Distributions
Webinar\n\nLecture held in Zoom.\n\nAbstract\nThis talk is devoted to thr
ee notions of discrepancies with respect to the $L_{2}$ norm and a variet
y of test sets. The $L_{2}$ -star discrepancy uses as test sets the class
of axis-parallel boxes anchored in the origin\, the $L_{2}$ extreme discre
pancy uses arbitrary axis-parallel boxes and the \n$L_{2}$ periodic discr
epancy uses so-called periodic intervals which range over the whole torus.
All three geometrical notions of $L_{2}$ -discrepancy can be interpreted
as worst-case error for quasi-Monte Carlo integration in corresponding fun
ction spaces. We compare these notions of discrepancy\, discuss some relat
ionships and present results for typical QMC point sets such as lattice po
int sets and digital nets. We turn our attention also to the dependence on
the dimension $d$ and examine whether these $L_{2}$ discrepancies satisf
y some tractability properties or suffer from the curse of dimensionality.
\n\nThis talk will be recorded and posted on the webinar homepage. Slides
will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Vybíral (Czech Technical University)
DTSTART;VALUE=DATE-TIME:20210908T150000Z
DTEND;VALUE=DATE-TIME:20210908T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/58
DESCRIPTION:Title: Dispersion of point sets in high dimensions\n
by Jan Vybíral (Czech Technical University) as part of Point Distribution
s Webinar\n\nLecture held in Zoom.\n\nAbstract\nWe will discuss the disper
sion of a point set\, which is simply the volume of the largest box not in
tersecting the given point set. We shall present several recent result abo
ut this notion\, including estimates of its high-dimensional asymptotic an
d deterministic constructions. If time permits\, we sketch the most import
ant parts of the proofs.\n\nThis talk will be recorded and posted on the w
ebinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fátima Lizarte (U of Cantabria)
DTSTART;VALUE=DATE-TIME:20210728T150000Z
DTEND;VALUE=DATE-TIME:20210728T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/59
DESCRIPTION:Title: A sequence of well conditioned polynomials\nb
y Fátima Lizarte (U of Cantabria) as part of Point Distributions Webinar\
n\nLecture held in Zoom.\n\nAbstract\nIn 1993\, Shub and Smale posed the p
roblem of finding a sequence of\nunivariate polynomials $P_N$ of degree $N
$ with condition number bounded\nabove by $N$. In this talk\, we show the
origin of this problem\, previous\nknowledge until this work\, its relati
on to other interesting mathematical\nproblems such as Smale's 7th problem
\, and our main result obtained: a simple\nand direct answer to the Shub a
nd Smale problem for $N=4M^2$\, with $M$ a\npositive integer\, as well as
comments on its proof.\\\\\n\nThis is a joint work with Carlos Beltr\\'an.
\n\nThis talk will be recorded and posted on the webinar homepage. Slides
will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jordi Marzo (U of Cantabria)
DTSTART;VALUE=DATE-TIME:20210804T150000Z
DTEND;VALUE=DATE-TIME:20210804T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/60
DESCRIPTION:Title: Quadrature rules\, Riesz energies\, discrepancies
and elliptic polynomials\nby Jordi Marzo (U of Cantabria) as part of
Point Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nI will t
alk about the relation between optimal quadratures\, Riesz (or logarithmic
) energies and minimal discrepancy configurations. In particular I will di
scuss the use of the zeros of elliptic (or Kostlan-Shub-Smale) polynomials
\, among other configurations\, as quadrature nodes for Sobolev spaces on
the sphere. There will be many open problems throughout the talk.\n\nThis
talk will be recorded and posted on the webinar homepage. Slides will be a
vailable too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Rudolf (University of Göttingen)
DTSTART;VALUE=DATE-TIME:20210922T150000Z
DTEND;VALUE=DATE-TIME:20210922T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/61
DESCRIPTION:Title: On the spherical dispersion\nby Daniel Rudolf
(University of Göttingen) as part of Point Distributions Webinar\n\nLect
ure held in Zoom.\n\nAbstract\nIn the seminar we provide upper and lower b
ounds on the minimal spherical dispersion. In particular\, we see that the
inverse of the minimal spherical dispersion behaves linearly in the dimen
sion. We also talk about upper and lower bounds of the expected dispersion
for points chosen independently and uniformly at random from the Euclidea
n unit sphere. \n\nThe content of the talk is partially based on https://a
rxiv.org/abs/2103.11701.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kateryna Pozharska (Institute of Mathematics\, NAS of Ukraine)
DTSTART;VALUE=DATE-TIME:20210929T150000Z
DTEND;VALUE=DATE-TIME:20210929T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/62
DESCRIPTION:Title: Sampling recovery of functions from reproducing k
ernel Hilbert spaces in the uniform norm\nby Kateryna Pozharska (Insti
tute of Mathematics\, NAS of Ukraine) as part of Point Distributions Webin
ar\n\nLecture held in Zoom.\n\nAbstract\nWe study the recovery of multivar
iate functions from reproducing kernel Hilbert spaces in the uniform norm.
Surprisingly\, a certain weighted least squares recovery operator which u
ses random samples from a distribution\, depending on the spectral propert
ies of the corresponding embedding\, leads to near optimal results in seve
ral relevant situations. As an application we obtain new recovery guarante
es for Sobolev type spaces related to Jacobi type differential operators o
n the one hand and classical multivariate periodic Sobolev type spaces wit
h general smoothness weight on the other hand. By applying a recently intr
oduced sub-sampling technique related to Weaver's conjecture\, we further
reduce the sampling budget and improve on bounds for the corresponding sam
pling numbers.\nThis is a joint work with Tino Ullrich.\n\nThis talk will
be recorded and posted on the webinar homepage. Slides will be available t
oo.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nihar Gargava (EPFL)
DTSTART;VALUE=DATE-TIME:20210915T150000Z
DTEND;VALUE=DATE-TIME:20210915T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/63
DESCRIPTION:Title: Lattice packings through division algebras\nb
y Nihar Gargava (EPFL) as part of Point Distributions Webinar\n\nLecture h
eld in Zoom.\n\nAbstract\nWe will show the existence of lattice packings i
n a sparse family of dimensions. This construction will be a generalizatio
n of Venkatesh's lattice packing result of 2013. In our construction\, we
replace the appearance of the cyclotomic number field with a division alge
bra over the rationals. This improves the best known lower bounds on latti
ce packing problem in many dimensions. The talk will cover previously know
n bounds\, an overview of the new bounds and a live numerical simulation o
f Siegel's mean value theorem.\n\nThis talk will be recorded and posted on
the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giancarlo Travaglini (Università di Milano-Bicocca)
DTSTART;VALUE=DATE-TIME:20220126T160000Z
DTEND;VALUE=DATE-TIME:20220126T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/64
DESCRIPTION:Title: Irregularities of distribution and convex planar
sets\nby Giancarlo Travaglini (Università di Milano-Bicocca) as part
of Point Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nThe t
erm \\textit{Irregularities of Distribution} (often replaced with Geometr
ic Discrepancy) has been introduced by Klaus Roth in 1954 and refers to th
e question of how to choose a set of $N$ sampling points which can be use
d to approximate all the sets in a given reasonably large family inside th
e unit square.\nIn this talk\nwe consider a planar convex body $C$ and we
prove several analogs of Roth's\ntheorem. When $\\partial C$ is $\\mathcal
{C}%\n^{2}$ regardless of curvature\, we prove that for every set $\\mathc
al{P}_{N}$\nof $N$ points in $\\mathbb{T}^{2}$ we have the sharp bound%\n\
\[\n\\left\\{\\int_{0}^{1}\\int_{\\mathbb{T}^{2}}\\left\\vert \\mathrm{car
d}\\left(\n\\mathcal{P}_{N}\\mathcal{\\cap}\\left( \\lambda C+t\\right)
\\right) -\\lambda\n^{2}N\\left\\vert C\\right\\vert \\right\\vert ^{2}~d
td\\lambda\\right\\}^{1/2}\\geqslant cN^{1/4}\\\;.\n\\]\nWhen $\\partial C
$ is only piecewise $\\mathcal{C}^{2}$ and is not a polygon we\nprove the
sharp bound%\n\\[\n\\left\\{\\int_{0}^{1}\\int_{\\mathbb{T}^{2}}\\left\\ve
rt \\mathrm{card}\\left(\n\\mathcal{P}_{N}\\mathcal{\\cap}\\left( \\lambd
a C+t\\right) \\right) -\\lambda\n^{2}N\\left\\vert C\\right\\vert \\rig
ht\\vert ^{2}~dtd\\lambda\\right\\}^{1/2}\\geqslant cN^{1/5}.\n\\]\nWe als
o give a whole range of intermediate sharp results between $N^{1/5}$ and\n
$N^{1/4}$. Our proofs depend on a lemma of Cassels-Montgomery\, on ad hoc\
nconstructions of finite point sets\, and on a geometric type estimate for
the\naverage decay of the Fourier transform of the characteristic functio
n of $C$.\n\nThis talk will be recorded and posted on the webinar homepage
. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Beltrán (University of Cantabria)
DTSTART;VALUE=DATE-TIME:20220209T160000Z
DTEND;VALUE=DATE-TIME:20220209T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/65
DESCRIPTION:Title: Distributing many points in the complex Grassmann
ian an its application in communications\nby Carlos Beltrán (Universi
ty of Cantabria) as part of Point Distributions Webinar\n\nLecture held in
Zoom.\n\nAbstract\nI will discuss on a fundamental model for wireless com
munications (called Noncoherent Communications) that has a very simple mat
hematical explanation and modelling. It turns out that in order to achieve
optimal communication rate one must choose a finite collection of points
in the complex Grassmannian. In this talk I will present the problem from
scratch\, explaining the setting\, the model\, the road to the mathematica
l problem of point distribution… and a new approach to this last mathema
tical problem\, based on Riemannian optimization\, which outperforms all k
nown methods to the date for choosing well distributed codes in these clas
sical spaces. This is joint work with a team of engineers\, credits will b
e given during the talk.\n\nThis talk will be recorded and posted on the w
ebinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Galyna Livshyts (Georgia Tech)
DTSTART;VALUE=DATE-TIME:20220216T160000Z
DTEND;VALUE=DATE-TIME:20220216T170000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/66
DESCRIPTION:Title: An efficient net construction and applications to
random matrix theory\, and to minimal dispersion estimation\nby Galyn
a Livshyts (Georgia Tech) as part of Point Distributions Webinar\n\nLectur
e held in Zoom.\n\nAbstract\nWe explain a construction of an efficient net
on the sphere in a high-dimensional space\, and draw some applications in
random matrix theory (partly joint with Tikhomirov and Vershynin). One of
the steps in our construction is also relevant for estimating minimal dis
persion in the cube (joint with Litvak).\n\nThis talk will be recorded and
posted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Sloan (U of New South Wales)
DTSTART;VALUE=DATE-TIME:20220309T210000Z
DTEND;VALUE=DATE-TIME:20220309T220000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/67
DESCRIPTION:Title: Pros and cons of lattice points for high-dimensio
nal approximation\nby Ian Sloan (U of New South Wales) as part of Poin
t Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nIn this talk
\, based on recent joint work with Vesa Kaarnioja\, Yoshihito Kazashi\, Fr
ances Kuo and Fabio Nobile\, I describe a fast method for high-dimensional
approximation on the torus. A typical approximation scheme for a given fu
nction $f(\\bt)$ involves choosing a set of points $\\bt_1\, \\ldots\, \\b
t_n$ at which function values are to be given as inputs\, and a methodolog
y for constructing a more or less smooth approximation $f_n(\\bt)$. We sh
all see that lattice points have advantages\, but also limitations\, as sa
mple points. The method to be described is based on so-called kernels and
lattice points. It appears to offer considerable promise for practical h
igh-dimensional approximation.\n\nThis talk will be recorded and posted on
the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lenny Fukshansky (Claremont McKenna College)
DTSTART;VALUE=DATE-TIME:20220323T150000Z
DTEND;VALUE=DATE-TIME:20220323T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/68
DESCRIPTION:Title: Lattices from group frames and vertex transitive
graphs\nby Lenny Fukshansky (Claremont McKenna College) as part of Poi
nt Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nTight frame
s in Euclidean spaces are widely used convenient generalizations of orthon
ormal bases. A particularly nice class of such frames is generated as orbi
ts under irreducible actions of finite groups of orthogonal matrices: thes
e are called irreducible group frames. Integer spans of rational irreducib
le group frames form Euclidean lattices with some very nice geometric prop
erties\, called strongly eutactic lattices. We discuss this construction\,
focusing on an especially interesting infinite family in arbitrarily larg
e dimensions\, which comes from vertex transitive graphs. We demonstrate s
everal examples of such lattices from graphs that exhibit some rather fasc
inating properties. This is joint work with Deanna Needell\, Josiah Park a
nd Yuxin Xin.\n\nThis talk will be recorded and posted on the webinar home
page. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bence Borda (TU Graz)
DTSTART;VALUE=DATE-TIME:20220406T150000Z
DTEND;VALUE=DATE-TIME:20220406T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/69
DESCRIPTION:Title: A smoothing inequality for the Wasserstein metric
on compact manifolds\nby Bence Borda (TU Graz) as part of Point Distr
ibutions Webinar\n\nLecture held in Zoom.\n\nAbstract\nImproving a result
of Brown and Steinerberger\, we present a Berry-Esseen type smoothing ineq
uality for the quadratic Wasserstein metric on compact Riemannian manifold
s\, which estimates the distance between two probability measures in terms
of their Fourier transforms. The inequality is sharp\, and has a wide ran
ge of applications in probability theory and number theory. We discuss sha
rp convergence rates of the empirical measure of an i.i.d. or stationary w
eakly dependent sample\, complementing recent results of Bobkov and Ledoux
on the unit cube\, and Ambrosio\, Stra and Trevisan on compact manifolds.
We also estimate the convergence rate of random walks on compact groups t
o the Haar measure\, and establish the functional CLT and the functional L
IL for additive functionals. On compact semisimple Lie groups these hold e
ven without a spectral gap assumption. As an application to finite point s
ets arising in number theory\, we show that a classical construction of Lu
botzky\, Phillips and Sarnak on SU(2) and SO(3) achieves optimal rate in t
he quadratic Wasserstein metric.\n\nThis talk will be recorded and posted
on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laurent Bétermin (Université Claude Bernard Lyon 1)
DTSTART;VALUE=DATE-TIME:20220420T150000Z
DTEND;VALUE=DATE-TIME:20220420T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/70
DESCRIPTION:Title: Minimality results for the Embedded Atom Model\nby Laurent Bétermin (Université Claude Bernard Lyon 1) as part of Poi
nt Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nThe Embedde
d-Atom Model (EAM) provides a phenomenological description of atomic arran
gements in metallic systems. It consists of a configurational energy depen
ding on atomic positions and featuring the interplay of two-body atomic in
teractions and nonlocal effects due to the corresponding electronic clouds
. In this talk\, I will present minimality results for this system among l
attices in dimensions 2 and 3 as well as other aspects of the problem. Thi
s is a joint work with Manuel Friedrich (University of Erlangen) and Uliss
e Stefanelli (University of Vienna).\n\nThis talk will be recorded and pos
ted on the webinar homepage. Slides will be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Codina Cotar (University College London)
DTSTART;VALUE=DATE-TIME:20220511T150000Z
DTEND;VALUE=DATE-TIME:20220511T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/71
DESCRIPTION:Title: Equality of the Jellium and Uniform Electron Gas
next-order asymptotic terms for Coulomb and Riesz potentials\nby Codin
a Cotar (University College London) as part of Point Distributions Webinar
\n\nLecture held in Zoom.\n\nAbstract\nWe consider the sharp next-order as
ymptotics problems for: (1) the minimum energy for optimal N-point configu
rations\; (2) the N-Marginal Optimal Transport\; and (3) the Jellium probl
em for N-point configurations\, in all three cases with Riesz costs with i
nverse power-law long-range interactions. The first problem describes the
ground state of a Coulomb or Riesz gas\, the second appears as a semiclass
ical limit of DFT energy\, modelling a quantum version of the same system
(and is called Uniform Electron Gas in the physics literature)\, and the t
hird describes charges in a uniform negative background\, a rough model fo
r electrons in a metal. Recently the second-order terms in the large-N asy
mptotic expansions for power s in dimension d were shown for: (1) for \\ma
x(0\,d-2)\\le sGeneralized Erdős-Turán inequalities and stabil
ity of energy minimizers\nby Ruiwen Shu (University of Oxford) as part
of Point Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nThe
classical Erdős-Turán inequality on the distribution of roots for comple
x polynomials can be equivalently stated in a potential theoretic formulat
ion\, that is\, if the logarithmic potential generated by a probability me
asure on the unit circle is close to 0\, then this probability measure is
close to the uniform distribution. We generalize this classical inequality
from $d=1$ to higher dimensions $d>1$ with the class of Riesz potentials
which includes the logarithmic potential as a special case. In order to qu
antify how close a probability measure is to the uniform distribution in a
general space\, we use Wasserstein-infinity distance as a canonical exten
sion of the concept of discrepancy. Then we give a compact description of
this distance. Then for every dimension $d$\, we prove inequalities boundi
ng the Wasserstein-infinity distance between a probability measure $\\rho$
and the uniform distribution by the $L^p$-norm of the Riesz potentials ge
nerated by $\\rho$. Our inequalities are proven to be sharp up to the cons
tants for singular Riesz potentials. Our results indicate that the phenome
non discovered by Erdős and Turán about polynomials is much more univers
al than it seems. Finally we apply these inequalities to prove stability t
heorems for energy minimizers\, which provides a complementary perspective
on the recent construction of energy minimizers with clustering behavior\
n\nThis talk will be recorded and posted on the webinar homepage. Slides w
ill be available too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Damir Ferizović (KU Leuven)
DTSTART;VALUE=DATE-TIME:20220519T150000Z
DTEND;VALUE=DATE-TIME:20220519T160000Z
DTSTAMP;VALUE=DATE-TIME:20240329T155127Z
UID:PointDistributionsPotentialThry/73
DESCRIPTION:Title: Spherical cap discrepancy of perturbed lattices u
nder the Lambert projection\nby Damir Ferizović (KU Leuven) as part o
f Point Distributions Webinar\n\nLecture held in Zoom.\n\nAbstract\nGiven
any full rank lattice and a natural number N \, we regard the point set gi
ven by the scaled lattice intersected with the unit square under the Lambe
rt map to the unit sphere\, and show that its spherical cap discrepancy is
at most of order N \, with leading coefficient given explicitly and depen
ding on the lattice only. The proof is established using a lemma that boun
ds the amount of intersections of certain curves with fundamental domains
that tile R^2 \, and even allows for local perturbations of the lattice wi
thout affecting the bound\, proving to be stable for numerical application
s. A special case yields the smallest constant for the leading term of the
cap discrepancy for deterministic algorithms up to date.\n\nThis talk wil
l be recorded and posted on the webinar homepage. Slides will be available
too.\n
LOCATION:https://researchseminars.org/talk/PointDistributionsPotentialThry
/73/
END:VEVENT
END:VCALENDAR