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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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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:20210514T201100Z
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 $0~~On 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:20210514T201100Z
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/
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