BEGIN:VCALENDAR
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BEGIN:VEVENT
SUMMARY:Yi Li (John Jay College\, CUNY)
DTSTART:20200515T180000Z
DTEND:20200515T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/1
DESCRIPTION:Title: Monotone properties of the eigenfunction of Neumann problem
s\nby Yi Li (John Jay College\, CUNY) as part of CUNY Harmonic Analysi
s and PDE's Seminar\n\n\nAbstract\nIn this talk\, we present recent progre
ss on the eigenvalue problem\n\n\n$\\Delta u + \\mu u = 0$ in $\\Omega$\,
$\\frac{\\partial{u}}{\\partial{\\nu}} = 0$ on $\\partial{\\Omega}$\, (1)
\n\nwhere $\\Omega$ is a domain in $\\mathbb{R}^n$\, $\\frac{\\partial{u}}
{\\partial{ν}} := \\nabla u \\cdot \\nu$\, $\\nu$ denotes the outward uni
t normal vector on $\\partial{\\Omega}$. \n\n• “Hot spots” conjectur
e given by Rauch: the second Neumann eigenfunction attains its maximum and
minimum values only on the boundary of the domain.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Lacey (Georgia Institute of Technology)
DTSTART:20200522T180000Z
DTEND:20200522T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/2
DESCRIPTION:Title: Discrete improving inequalities\nby Michael Lacey (Geor
gia Institute of Technology) as part of CUNY Harmonic Analysis and PDE's S
eminar\n\n\nAbstract\nAverages improve functions\, even if averaging over
lower dimensional\nsurfaces\, most famously the sphere. Remarkably there a
re discrete analogs\nof these inequalities. Their emerging theory compleme
nts and extends\nthe more well known theories associated to discrete maxim
al functions of\nBourgain\, and discrete Radon transforms of Stein\, Waing
er and Ionescu.\nI will survey some recent results\, and point out some of
the fascinating\ncomplications that arise in the proofs.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bingyang Hu (University of Wisconsin\, Madison)
DTSTART:20200529T180000Z
DTEND:20200529T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/3
DESCRIPTION:Title: On the general dyadic system in Euclidean spaces\nby Bi
ngyang Hu (University of Wisconsin\, Madison) as part of CUNY Harmonic Ana
lysis and PDE's Seminar\n\n\nAbstract\nAdjacent dyadic systems are pivotal
in analysis and related fields to study continuous objects via collection
of dyadic ones. In this talk\, we will first give a complete characteriza
tion of the adjacent dyadic systems on the real line\, and then we will ge
neralize it to higher dimension. The first part of this talk (real line ca
se) is joint with Tess Anderson\, Liwei Jiang\, Cornor Olson and Zeyu Wei
under an REU project at Summer\, 2018 at UW-Madison\; while the second the
part (general case) is a joint work with Tess Anderson.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joris Roos (University of Wisconsin-Madison)
DTSTART:20200724T180000Z
DTEND:20200724T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/4
DESCRIPTION:Title: Spherical maximal functions and fractal dimensions of dilat
ion sets\nby Joris Roos (University of Wisconsin-Madison) as part of C
UNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nThe talk is about s
harp $L^p$ improving properties of Euclidean spherical maximal operators i
n two and higher dimensions with a supremum taken over a given set of radi
i $E$ in $[1\,2]$. We will discuss a characterization of the closed convex
sets which can occur as closure of the sharp $L^p$ improving region of th
is operator. The region depends not only on the Minkowski dimension of $E$
\, but also other properties of the fractal geometry such as the Assouad s
pectrum. This is joint work with Andreas Seeger\, extending earlier joint
work with Tess Anderson\, Kevin Hughes and Andreas Seeger.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura De Carli (Florida International University)
DTSTART:20200717T180000Z
DTEND:20200717T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/5
DESCRIPTION:Title: Sufficient conditions for the existence of exponential base
s on domains of $\\mathbb{R}^d$\nby Laura De Carli (Florida Internatio
nal University) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nA
bstract\nWe first consider the following general problem:\nGiven an orthon
ormal basis $V$ in a separable Hilbert space and a set of unit vectors $B
$\, we consider the sets $B_N$ obtained by replacing the first $N$ vectors
of $V$ with the first $N$ vectors of $B$. We show necessary\nand sufficie
nt conditions that ensure that the sets $B_N$ are Riesz bases of $H$ and w
e estimate the frame constants of these bases. Then\, we show conditions t
hat ensure that $B$ is a Riesz basis of $H$. We apply our results to prove
sufficient conditions for the existence of exponential bases on domains
of $\\mathbb{R}^d$. (joint work with Julian Edward\, FIU)\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalia Accomazzo Scotti (University of the Basque Country - Unive
rsity of British Columbia)
DTSTART:20200605T143000Z
DTEND:20200605T153000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/6
DESCRIPTION:Title: Maximal directional singular integrals\nby Natalia Acco
mazzo Scotti (University of the Basque Country - University of British Col
umbia) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\n
Maximal directional operators are formed by taking a one-dimensional opera
tor acting along a line\, and then studying the maximal value as the line
changes through a set of directions. One important example of this type of
operators is when we consider the one arising from the maximal function\,
which we can find has been broadly studied in the literature. In this tal
k we will review a little bit the history of these operators and we will g
ive some new results on the operator that arises from a singular integral.
\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bradley Currey (Saint Louis University)
DTSTART:20200703T180000Z
DTEND:20200703T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/7
DESCRIPTION:Title: Linear independence of translate systems\nby Bradley Cu
rrey (Saint Louis University) as part of CUNY Harmonic Analysis and PDE's
Seminar\n\n\nAbstract\nLet $G$ be a locally compact group and $f$ a comple
x function on $G$. For $x ∈ G$ define $L_xf(y) = f(s^{−1}y)$. We say $
f$ has independent translates if $\\{L_xf : x ∈ E\\}$ is linearly indepe
ndent for all finite subsets $E$ of $G$. The general problem is to determi
ne classes of functions on $G$ that have independent translates. We recall
known results for $G$ abelian\, and present a few new results for the cas
e where $G$ is the Heisenberg group.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Theresa C. Anderson (Purdue University)
DTSTART:20200619T180000Z
DTEND:20200619T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/8
DESCRIPTION:Title: Discrete maximal functions over surfaces of higher codimens
ion\nby Theresa C. Anderson (Purdue University) as part of CUNY Harmon
ic Analysis and PDE's Seminar\n\n\nAbstract\nConsidering discrete (integra
l) variants of continuous operators\, and\, separately\, continuous operat
ors that involve integration over surfaces of intermediate codimension\, h
ave been two challenging areas of investigation in analysis. Here we unit
e these themes\, providing an interesting interplay of harmonic analysis\,
analytic number theory and discrete geometry. We will describe the key f
eatures of our technique as well as history and background to put this pro
gram into context. This is joint work with Eyvi Palsson and Angel Kumchev
.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bochen Liu (National Center for Theoretical Sciences\, Taiwan)
DTSTART:20200612T140000Z
DTEND:20200612T150000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/9
DESCRIPTION:Title: Fourier frames for surface-carried measures\nby Bochen
Liu (National Center for Theoretical Sciences\, Taiwan) as part of CUNY Ha
rmonic Analysis and PDE's Seminar\n\n\nAbstract\nWe show that the surface
measure on a sphere does not admit a Fourier frame. On the other hand\, su
rface measure on the boundary of a polytope always admits a Fourier frame.
\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Zhang (University of Rochester)
DTSTART:20200710T180000Z
DTEND:20200710T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/10
DESCRIPTION:Title: Sharp endpoint estimates for eigenfunctions restricted to
submanifolds of codimension 2\nby Cheng Zhang (University of Rochester
) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nBurq-
Gérard-Tzvetkov and Hu established $L^p$ estimates ($2\\le p\\le \\infty$
) for the restriction of eigenfunctions to submanifolds. The estimates are
sharp\, except for the log loss at the endpoint $L^2$ estimates for subma
nifolds of codimension 2. It has long been believed that the log loss at t
he endpoint can be removed in general\, while the problem is still open. S
o in this talk we will talk about the study of sharp endpoint restriction
estimates for eigenfunctions in this case. Recall that Chen-Sogge removed
the log loss for the geodesics on 3-dimensional manifolds. In a joint work
with Xing Wang\, we generalize their result to higher dimensions and prov
e that the log loss can be removed for totally geodesic submanifolds of co
dimension 2. Moreover\, on 3-dimensional manifolds\, we can remove the log
loss for curves with nonvanishing geodesic curvatures\, and more general
finite type curves. The problem in 3D is essentially related to Hilbert tr
ansforms along curves in the plane and a class of singular oscillatory int
egrals studied by Phong-Stein\, Ricci-Stein\, Pan\, Seeger\, Carbery-Pére
z. (Reference: arXiv:1606.09346v2)\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Krause (University of California\, Los Angeles)
DTSTART:20200731T180000Z
DTEND:20200731T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/11
DESCRIPTION:Title: Pointwise Ergodic Theorems for Non-conventional Bilinear P
olynomial Averages\nby Ben Krause (University of California\, Los Ange
les) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nWe
establish convergence in norm and pointwise almost everywhere for the non
-conventional bilinear polynomial ergodic averages\n\\[ A_N(f\,g)(x) := \\
frac{1}{N} \\sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\\]\nas $N \\to \\infty$\, w
here $T \\colon X \\to X$ is a measure-preserving transformation of a $\\s
igma$-finite measure space $(X\,\\mu)$\, $P(n) \\in \\mathbb{Z}[n]$ is a p
olynomial of degree $d \\geq 2$\, and $f \\in L^{p_1}(X)\, \\ g \\in L^{p_
2}(X)$ for some $p_1\,p_2 > 1$ with $\\frac{1}{p_1} + \\frac{1}{p_2} \\le
q 1$.\n\n(Joint with Mariusz Mirek (Rutgers)\, and Terry Tao (UCLA).)\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Máté Vizer (Alfréd Rényi Institute of Mathematics\, Hungary)
DTSTART:20200918T180000Z
DTEND:20200918T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/12
DESCRIPTION:Title: Recent developments in the discrete Fuglede conjecture
\nby Máté Vizer (Alfréd Rényi Institute of Mathematics\, Hungary) as p
art of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nFuglede's c
onjecture was stated in 1974\, and connects an analytic with a geometric p
roperty of a given bounded measurable subset $T \\subset \\mathbf{R}^d$. I
t states that $T$ accepts a complete orthogonal basis of exponential funct
ions if and only if it tiles $\\mathbf{R}^d$ by translations. This conject
ure has been disproved by Tao in 2004. The counterexample has been achieve
d by lifting counterexamples of Fuglede's conjecture in finite abelian gro
ups to counterexamples in Euclidean spaces.\n\n \nThis connection motivate
d researchers to pay more attention to the discrete version of Fuglede's c
onjecture. I would like to briefly overview recent progress on this conjec
ture\, including some work of the speaker\, that is joint with Kiss\, Mali
kiosis ans Somlai.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Felipe Negreira (University of the Republic (Uruguay))
DTSTART:20201211T190000Z
DTEND:20201211T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/13
DESCRIPTION:Title: Wavelets decompositions and its applications on regular me
tric spaces\nby Felipe Negreira (University of the Republic (Uruguay))
as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nIn thi
s talk we will first survey the techniques used to reproduce the Littlewoo
d-Paley analysis and the subsequent wavelet expansions for Riemannian mani
folds and then for the more general setting of regular metric spaces. Next
\, within the same framework of regular metric spaces\, we will show how o
ne can use these decompositions to characterize various function spaces -i
n particular Besov- and obtain different results such as sampling inequali
ties\, trace theorems and multifractal analysis type estimates.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincent R. Martinez (CUNY-Hunter College)
DTSTART:20201002T180000Z
DTEND:20201002T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/14
DESCRIPTION:Title: Data Assimilation & PDEs (Part I): An Overview of Recent
Rigorous Results\nby Vincent R. Martinez (CUNY-Hunter College) as part
of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nThis talk will
be an introduction to a series of six talks in the study of Data Assimila
tion for PDEs. In this first talk\, we will introduce the concept of data
assimilation\, survey recent results\, introduce some analytical tools and
techniques involved in establishing these rigorous results\, and lastly d
iscuss open avenues to explore.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gergely Kiss (Alfréd Rényi Institute of Mathematics\, Hungary)
DTSTART:20201023T180000Z
DTEND:20201023T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/15
DESCRIPTION:Title: On the Fuglede conjecture for the product of elementary ab
elian groups over prime fields\nby Gergely Kiss (Alfréd Rényi Instit
ute of Mathematics\, Hungary) as part of CUNY Harmonic Analysis and PDE's
Seminar\n\n\nAbstract\nFuglede in 1974 conjectured that a bounded domain $
S \\subset \\mathbb{R}^d$\ntiles the $d$-dimensional\nEuclidean space if a
nd only if the set of functions in $L^2(S)$ admits an orthogonal\nbasis of
exponential functions.\n\nIn my talk we will focus on the discrete versio
n of Fuglede’s conjecture that can be formulated as follows. Let $G$ be
a finite Abelian group $G$ and $\\widehat{G}$ the set of characters of $G$
\, indexed by the elements of $G$. Then $S\\subset G$ is spectral i
f and only if there exists a $\\Lambda\\in G$ such that ($\\chi_l)_{l\\in
\\Lambda}$ is an orthogonal base of complex valued functions defined on $S
$.\nFor a finite group $G$ and a subset $S$ of $G$ we say that $S$ is a
tile of $G$ if there is a $T \\subset G$ such that $S+T=G$ and $|S|\\
cdot |T|=|G|$. The discrete version Fuglede conjecture states that for an
abelian group $G$ a subset $S$ is spectral if and only if $S$ is a tile.\n
\nI will talk about the Fuglede conjecture for the product of elementary a
belian groups over prime fields. The importance of this particular case ca
n be illustrated with the fact that Fuglede's original conjecture were dis
proved first by Tao and his proof is based on a counterexample for the dis
crete Fuglede conjecture on elementary abelian $p$-groups.\n\nFirst I will
summarize the known results concerning the product of elementary abelian
groups over prime fields. In the second part of my talk I will present our
recent result which shows that the discrete Fuglede conjecture holds on\n
$\\mathbb{Z}_p^2 \\times \\mathbb{Z}_q$\, where $p$ and $q$ are different
primes.\n\n(joint work with Gábor Somlai)\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael S. Jolly (Indiana University-Bloomington)
DTSTART:20201030T180000Z
DTEND:20201030T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/16
DESCRIPTION:Title: Data Assimilation & PDEs (Part IV): Data assimilation for
the 2D Navier-Stokes equations using local observables\nby Michael S.
Jolly (Indiana University-Bloomington) as part of CUNY Harmonic Analysis a
nd PDE's Seminar\n\n\nAbstract\nWe will discuss an approximate\, global da
ta assimilation/synchronization algorithm based on purely local observatio
ns for the two-dimensional Navier-Stokes equations on the torus. We will p
resent a rigorous result stating that\, for any error threshold\, if the r
eference flow is analytic with sufficiently large analyticity radius\, the
n it can be recovered within that threshold. We will then show the result
of numerical tests of the effectiveness of this approach\, as well as vari
ants with data on moving subdomains. In particular\, computations demonstr
ate that machine precision synchronization is achieved for mobile data col
lected from a small fraction of the domain. This is joint work with Animik
h Biswas and Zachary Bradshaw.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tural Sadigov (Hamilton College)
DTSTART:20201009T180000Z
DTEND:20201009T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/18
DESCRIPTION:Title: Data Assimilation & PDEs (Part II): A Computational Study
of a data assimilation algorithm for the damped\, driven Korteweg - de Vri
es equation - Fourier Modes\, Nodes\, and Volume Elements cases\nby Tu
ral Sadigov (Hamilton College) as part of CUNY Harmonic Analysis and PDE's
Seminar\n\n\nAbstract\nIn this talk\, we describe a continuous data assim
ilation algorithm for damped\, driven Korteweg - de Vries equation\, and s
ummarize analytical results regarding the Fourier modes case. Then we desc
ribe the numerical method we use to solve the Korteweg - de Vries equation
and confirm these analytical results in the case of a particular force an
d damping parameter that create an interesting chaotic solution. We numeri
cally show that\, even with a chaotic solution nearby a potential saddle p
oint in the attractor\, the data assimilation algorithm is robust enough t
o lock on to the reference solution with the right parameters in the assim
ilation algorithm. We also present various numerical results regarding two
other cases in the context of the same equation: nodal case and finite vo
lume elements case. This work is sponsored by the U.S. Air Force under MOU
FA8750-15-3-6000.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aseel Farhat (Florida State University)
DTSTART:20201016T180000Z
DTEND:20201016T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/19
DESCRIPTION:Title: Data Assimilation & PDEs (Part III): Data Assimilation alg
orithms for the Rayleigh-Bénard Convection problem.\nby Aseel Farhat
(Florida State University) as part of CUNY Harmonic Analysis and PDE's Sem
inar\n\n\nAbstract\nAnalyzing the validity and success of a data assimilat
ion algorithm when some state variable observations are not available is a
n important problem meteorology and engineering. In this talk\, we will pr
esent various continuous data assimilation (downscaling) algorithms for th
e Rayleigh-Bénard problem that do not require observations of all evolvin
g state variables of the system. For the 2D incompressible Bénard convect
ion problem\, for example\, our algorithm uses only velocity measuremen
ts (temperature measurements are not necessary). We rigorously identif
y conditions that guarantee synchronization between the observed system an
d the model\, then confirm the applicability of these results via numerica
l simulations.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Collin Victor (University of Nebraska-Lincoln)
DTSTART:20201113T190000Z
DTEND:20201113T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/20
DESCRIPTION:Title: Data Assimilation & PDEs (Part VI): Continuous Data Assimi
lation Enhanced by Mobile Observers\nby Collin Victor (University of N
ebraska-Lincoln) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\n
Abstract\nIn this talk we examine computationally a continuous data assimi
lation algorithm using observers that move continuously in time. Specifica
lly\, we look at using the Azouani-Olson-Titi data assimilation algorithm
in the context of measurement devices which move in time\, such as satelli
tes or drones. We find that\, in the context of the 1D Allen-Cahn equation
s\, by moving the sampling points dynamically\, we can greatly reduce the
number of sampling points required\, while achieving better accuracy. Addi
tionally we look at the adaptation of this algorithm to the 2D Incompressi
ble Navier-Stokes equations using observers that move according to various
regimes\, where we obtain similar results.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mahya Ghandehari (University of Delaware)
DTSTART:20201120T190000Z
DTEND:20201120T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/21
DESCRIPTION:Title: Fourier algebras of the group of ${\\mathbb R}$-affine tra
nsformations and a dual convolution\nby Mahya Ghandehari (University o
f Delaware) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstr
act\nA major trend in Non-commutative Harmonic Analysis is to investigate
function spaces related to Fourier analysis (and representation theory) of
non-abelian groups.\nThe Fourier algebra\, which is associated with the l
eft regular representation of the ambient group\, is an important example
of such function spaces. This function algebra encodes the properties of t
he group in various ways\; for instance the existence of derivations on th
is algebra translates into information about the commutativity of the grou
p itself. \n\nIn this talk\, we investigate the Fourier algebra of the gro
up of ${\\mathbb R}$-affine transformations. In particular\, we discuss t
he non-commutative Fourier transform for this group\, and provide an expl
icit formula for the convolution product on the "dual side" of this transf
orm. As an application of this new dual convolution product\, we show an e
asy dual formulation for (the only known) symmetric derivative on the Four
ier algebra of the group. \n\nThis talk is mainly based on joint articles
with Y. Choi.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter A. Linnell (Virginia Tech)
DTSTART:20201218T190000Z
DTEND:20201218T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/22
DESCRIPTION:Title: The discrete Pompeiu problem\nby Peter A. Linnell (Vir
ginia Tech) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstr
act\nLet K be a compact subset of R^n with nonzero Lebesgue measure. The
Pompeiu problem asks if f=0 is the only continuous function such that the
integral of f over s(K) is 0 for all rigid motions s of R^n. We will cons
ider a version of the Pompeiu problem for discrete groups. We shall also d
escribe U(G) and its role in this problem. This is joint work with Mike P
uls.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Luli (University of California\, Davis)
DTSTART:20201204T190000Z
DTEND:20201204T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/23
DESCRIPTION:Title: Smooth Nonnegative Interpolation\nby Kevin Luli (Unive
rsity of California\, Davis) as part of CUNY Harmonic Analysis and PDE's S
eminar\n\n\nAbstract\nSuppose $E$ is an arbitrary subset of $R^n$. Let $f:
E \\rightarrow [0\, \\infty)$. How can we decide if $f$ extends to a no
nnegative $C^m$ function $F$ defined on all of $R^n$? Suppose $E$ is finit
e. Can we compute a nonnegative $C^m$ function $F$ on $R^n$ that agrees wi
th $f$ on $E$ with the least possible $C^m$ norm? How many computer operat
ions does this take? In this talk\, I will explain recent results on these
problems. Non-negativity is one of the most important shape preserving pr
operties for interpolants. In real life applications\, the range of the in
terpolant is imposed by nature. For example\, probability density\, the am
ount of snow\, rain\, humidity\, chemical concentration are all nonnegativ
e quantities and are of interest in natural sciences. Even in one dimensio
n\, the existing techniques can only handle nonnegative interpolation unde
r special assumptions on the data set. Our results work without any assump
tions on the data sets.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marcin Bownik (University of Oregon)
DTSTART:20210212T190000Z
DTEND:20210212T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/24
DESCRIPTION:Title: Parseval wavelet frames on Riemaniann manifolds\nby Ma
rcin Bownik (University of Oregon) as part of CUNY Harmonic Analysis and P
DE's Seminar\n\n\nAbstract\nAbstract:\nIn this talk we discuss how to cons
truct Parseval wavelet frames in $L^2(M)$ for a general Riemannian manifol
d $M$. We also show the existence of wavelet unconditional frames in $L^p(
M)$ for $1 < p <\\infty$. This construction is made possible thanks to smo
oth orthogonal projection decomposition of the identity operator on $L^2(M
)$\, which is an operator version of a smooth partition of unity. We also
show some applications such as a characterization of Triebel-Lizorkin $F_{
p\,q}^s(M)$ and Besov $B_{p\,q}^s(M)$ spaces on compact manifolds in terms
of magnitudes of coefficients of Parseval wavelet frames. This talk is ba
sed on a joint work with Dziedziul and Kamont.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emily J. King (Colorado State University)
DTSTART:20210514T180000Z
DTEND:20210514T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/25
DESCRIPTION:Title: Mathematical analysis of neural networks\nby Emily J.
King (Colorado State University) as part of CUNY Harmonic Analysis and PDE
's Seminar\n\n\nAbstract\nNeural networks have proven themselves to be use
ful in a wide range of applications but operate more-or-less as black boxe
s. Mathematicians have an opportunity to crack the mystery. In this talk\,
after a gentle introduction to neural networks\, three approaches to math
ematically analyze neural networks will be presented. First\, singular va
lues will be generalized to better understand the inherent rank of weight
matrices in a certain type of neural network. Second\, a tool from high-di
mensional geometry\, Gaussian mean width\, will be shown empirically to di
stinguish between correctly and incorrectly classified data as they travel
through a neural network. Finally\, we will analyze approximation proper
ties of spaces of neural networks of a fixed architecture.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Senger (Missouri State University)
DTSTART:20210226T190000Z
DTEND:20210226T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/26
DESCRIPTION:Title: Sharpness of Falconer-type estimates for dot products\
nby Steven Senger (Missouri State University) as part of CUNY Harmonic Ana
lysis and PDE's Seminar\n\n\nAbstract\nIn 1985\, Falconer conjectured that
if a subset has Hausdorff dimension sufficiently high with respect to the
ambient dimension\, then the Lebesgue measure of the set of distances it
determines should be positive. His first partial result toward this end hi
nged on a lemma showing that if a measure satisfies an energy condition re
lated to the ambient dimension\, then its support must not have a high con
centration of points separated by any given distance. Since then\, this le
mma has been explored in other contexts\, and serves as a bit of a litmus
test on how much we can say about various functionals under similar circum
stances. We employ techniques from discrete geometry to construct a family
of sharpness examples showing that this energy condition is sharp in the
analogous problem for dot products.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Greenfeld (University of California\, Los Angeles)
DTSTART:20210312T190000Z
DTEND:20210312T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/27
DESCRIPTION:Title: Translational tilings in lattices\nby Rachel Greenfeld
(University of California\, Los Angeles) as part of CUNY Harmonic Analysi
s and PDE's Seminar\n\n\nAbstract\nLet $F$ be a finite subset of $\\mathbb
{Z}^d$. We say that $F$ is a translational tile of $\\mathbb{Z}^d$ if it i
s possible to cover $\\mathbb{Z}^d$ by translates of $F$ without any overl
aps.\nThe periodic tiling conjecture\, which is perhaps the most well-know
n conjecture in the area\, suggests that any translational tile admits at
least one periodic tiling. In the talk\, we will motivate and discuss the
study of this conjecture. We will also present some new results\, joint wi
th Terence Tao\, on the structure of translational tilings in lattices and
introduce some applications.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pooja Rao (The State University of New York at Stony Brook)
DTSTART:20210319T180000Z
DTEND:20210319T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/28
DESCRIPTION:Title: Modeling mixing accurately in numerical simulations of int
erfacial instabilities\nby Pooja Rao (The State University of New York
at Stony Brook) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\n
Abstract\nHydrodynamic instabilities\, such as the shock-driven Richtmyer-
Meshkov instability (RMI) and the gravity-driven Rayleigh-Taylor instabili
ty (RTI)\, occur in unstable configurations where the density differs bet
ween two fluids. The physical interface between two fluids is imperfect a
nd the localized perturbations give rise to the growth of these instabilit
ies when accelerated either via shock or gravity.\n\nThese instabilities p
lay a critical role in numerous applications ranging from performance degr
adation in inertial confinement fusion (ICF) capsules to supernova explosi
ons. Oftentimes\, they occur in tandem\, such as in the ICF experiments.\n
\nTo accurately model the instability growth requires special treatment of
the discrete representation of the interface between the two fluids. Usin
g one such numerical approach\, Front-tracking\, we investigate a simplifi
ed representation of the instability growth in inertial confinement experi
ments\, focusing on the growth profile of a Rayleigh-Taylor instability wh
ich is seeded by a Richtmyer-Meshkov instability.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Animikh Biswas (University of Maryland\, Baltimore County)
DTSTART:20210326T180000Z
DTEND:20210326T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/29
DESCRIPTION:Title: Determining Functionals and Maps\, Data Assimilation and a
n Observable Regularity Criterion for Three-Dimensional Hydrodynamical Equ
ations\nby Animikh Biswas (University of Maryland\, Baltimore County)
as part of CUNY Harmonic Analysis and PDE's Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zachary Bradshaw (University of Arkansas)
DTSTART:20210409T180000Z
DTEND:20210409T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/30
DESCRIPTION:Title: Non-decaying solutions to the critical surface quasi-geost
rophic equations with symmetries\nby Zachary Bradshaw (University of A
rkansas) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract
\nWe develop a theory of self-similar solutions to the critical surface qu
asi-geostrophic equations. We construct self-similar solutions for arbitra
rily large data in various regularity classes and demonstrate\, in the sma
ll data regime\, uniqueness and global asymptotic stability. These solutio
ns are non-decaying in space which leads to ambiguity in the drift velocit
y. This ambiguity is corrected by imposing m-fold rotational symmetry. The
self-similar solutions exhibited here lie just beyond the known well-pose
dness theory and are expected to shed light on potential non-uniqueness\,
due to symmetry-breaking bifurcations\, in analogy with work of Jia and Sv
erak on the Navier-Stokes equations. This is joint work with Dallas Albrit
ton of Courant Institute\, NYU.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Sondak (Harvard University)
DTSTART:20210423T180000Z
DTEND:20210423T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/32
DESCRIPTION:Title: An Autoencoder and Reduced Basis for Dynamical Systems
\nby David Sondak (Harvard University) as part of CUNY Harmonic Analysis a
nd PDE's Seminar\n\n\nAbstract\nThe governing equations of nature are near
ly always nonlinear and often exhibit chaos. Nevertheless\, scientists and
engineers must still make informative predictions based on solutions to t
he governing equations. One promising approach for making predictions in r
egimes of scientific interest is to develop reliable reduced models of the
physics. These models should be fast and easy to compute while respecting
the\nunderlying dynamics of the system. The present work leverages the ex
pressibility of modern machine learning models to learn a basis for the re
duced space of dynamical systems. An autoencoder with a sparsity-promoting
latent space penalization is trained on data from the periodic Kuramoto-S
ivashinsky (K-S) equation and the damped and undamped KdV equations. It is
shown that the dimension of the learned reduced space is consistent with
that of the inertial manifold for the K-S equation. From here\, a nonlinea
r basis is determined for the K-S equation from the trained autoencoder mo
del. For the KdV equations\, it is shown\, over a large range of damping c
oefficients\, that the autoencoder learns a reduced manifold and that the
dimension of this manifold\ndecreases with increasing damping coefficient.
\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ruxi Shi (Mathematical Institute of the Polish Academy of Sciences
)
DTSTART:20210219T190000Z
DTEND:20210219T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/33
DESCRIPTION:Title: Spectral sets and spectral measures on groups with one pri
me factor\nby Ruxi Shi (Mathematical Institute of the Polish Academy o
f Sciences) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstr
act\nA Borel set $\\Omega$ (resp. a Borel probability measure $\\mu$) on a
locally compact group is called a spectral set (resp. a spectral measure)
if there exists a subset of continuous group characters that forms an ort
hogonal basis of the Hilbert space $L^2(\\Omega)$ (resp. $L^2(\\mu)$). In
this talk\, I will consider locally compact abelian groups with one prime
factor\, say $p$\, for example\, $\\mathbb{Z}/p^n\\mathbb{Z}$\, $(\\mathbb
{Z}/p\\mathbb{Z})^d$\, $\\mathbb{Q}_p$\, etc. I will discuss the propertie
s and characterization of spectral sets and spectral measures on these gro
ups.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ali Pakzad (Indiana University Bloomington)
DTSTART:20210430T180000Z
DTEND:20210430T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/34
DESCRIPTION:Title: The role of energy dissipation in modeling turbulence\
nby Ali Pakzad (Indiana University Bloomington) as part of CUNY Harmonic A
nalysis and PDE's Seminar\n\n\nAbstract\nA distinguishing feature of turbu
lent flows is the emergence of complicated chaotic structures involving a
wide range of length scales. Simulating all these scales is infeasible for
practical problems\, such as simulating storm fronts and hurricanes using
direct numerical simulation. To overcome this difficulty\, turbulence mod
els\, which model the universal effect of small scales on large scales\, a
re introduced to account for sub-mesh scale effects on a coarse mesh. Key
to getting a good approximation for a turbulence model is to correctly cal
ibrate the energy dissipation in the model on a coarse mesh. Energy dissip
ation rates of various turbulence models have been analyzed but assuming a
super-fine mesh.\n\nIn this talk\, I present a calculation of the energy
dissipation in a turbulence model discretized on a coarse mesh. Motivated
by this result\, I will show how the Smagorinsky model\, a common turbulen
ce model used in Large Eddy Simulation\, can be modified for better perfor
mance. If time allows\, I would describe some recent results in this direc
tion for shear driven turbulence with noise at the boundary.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrea Giorgini (Indiana University Bloomington)
DTSTART:20210507T180000Z
DTEND:20210507T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/35
DESCRIPTION:Title: Diffuse Interface modeling for two-phase flows: from the m
odel H to the AGG model\nby Andrea Giorgini (Indiana University Bloomi
ngton) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\n
In the last decades\, the Diffuse Interface theory (also known as Phase Fi
eld theory) has made significant progresses in the description of multi-ph
ase flows from modeling to numerical simulations. A particularly active to
pic has been the development of thermodynamically consistent models extend
ing the well-known Model H in the case of unmatched fluid densities. In th
is talk\, I will focus on the AGG model proposed by H. Abels\, H. Garcke a
nd G. Grün in 2012. The model consists of a Navier-Stokes-Cahn-Hilliard s
ystem characterized by a concentration-dependent density and an additional
flux term due to interface diffusion. Using the method of matched asympto
tic expansions\, the sharp interface limit of the AGG model corresponds to
the two-phase Navier-Stokes equations. In the literature\, the analysis o
f the AGG system has only been focused on the existence of weak solutions.
During the seminar\, I will present the first results concerning the exis
tence\, uniqueness and stability of strong solutions for the AGG model in
two dimensions.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anuj Kumar (Indiana University Bloomington)
DTSTART:20211001T170000Z
DTEND:20211001T181500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/36
DESCRIPTION:Title: On well-posedness and smoothing of solutions to the genera
lized SQG equations in critical Sobolev spaces\nby Anuj Kumar (Indiana
University Bloomington) as part of CUNY Harmonic Analysis and PDE's Semin
ar\n\n\nAbstract\nThis talk is based on recent works in which we study the
dissipative generalized surface quasi-geostrophic equations in a supercri
tical regime where the order of the dissipation is small relative to order
of the velocity\, and the velocities are less regular than the advected s
calar by up to one order of derivative. The existence and uniqueness theor
y of these equations in the borderline Sobolev spaces is addressed\, as we
ll as the instantaneous Gevrey class smoothing of their corresponding solu
tions. These results appear to be the first of its kind for a quasilinear
parabolic equation whose coefficients are of higher order than its linear
term. The main tool is the use of an approximation scheme suitably adapted
to preserve the underlying commutator structure. We also study a family o
f inviscid generalized SQG equations where the velocities have been mildly
regularized\, for instance\, logarithmically. The well-posedness of these
equations in borderline Sobolev spaces is addressed.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bing-Ying Lu (University of Bremen)
DTSTART:20211015T170000Z
DTEND:20211015T181500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/37
DESCRIPTION:Title: Universality near the gradient catastrophe point in the se
miclassical sine-Gordon equation\nby Bing-Ying Lu (University of Breme
n) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nWe s
tudy the semiclassical limit of the sine-Gordon (sG) equation with below t
hreshold pure impulse initial data of Klaus-Shaw type. The Whitham average
d approximation of this system exhibits a gradient catastrophe in finite t
ime. In accordance with a conjecture of Dubrovin\, Grava and Klein\, we fo
und that in an O(ε4/5) neighborhood near the gradient catastrophe point\,
the asymptotics of the sG solution are universally described by the Painl
evé I tritronquée solution. A linear map can be explicitly made from the
tritronquée solution to this neighborhood. Under this map: away from the
tritronquée poles\, the first correction of sG is universally given by t
he real part of the Hamiltonian of the tritronquée solution\; localized d
efects appear at locations mapped from the poles of tritronquée solution\
; the defects are proved universally to be a two parameter family of speci
al localized solutions on a periodic background for the sG equation. We ar
e able to characterize the solution in detail. Our approach is the rigorou
s steepest descent method for matrix Riemann--Hilbert problems\, substanti
ally generalizing Bertola and Tovbis's results on the nonlinear Schröding
er equation to establish universality beyond the context of solutions of a
single equation. This is joint work with Peter D. Miller.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Suen Chun Kit Anthony (The Education University of Hong Kong)
DTSTART:20211022T170000Z
DTEND:20211022T181500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/38
DESCRIPTION:Title: Vanishing parameter limit for a class of active scalar equ
ations\nby Suen Chun Kit Anthony (The Education University of Hong Kon
g) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nIn t
his talk\, we study an abtract class of active scalar equations which depe
nd on some viscosity parameters κ and ν. We examine the wellposedness of
the equations in different scenarios and address the convergence of solut
ions as κ or ν vanishes. We further discuss some physical applications o
f the general results obtained from such abtract class of active scalar eq
uations.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hau-Tieng Wu (Duke University)
DTSTART:20211029T170000Z
DTEND:20211029T181500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/39
DESCRIPTION:Title: Some recent progress in diffusion based manifold learning<
/a>\nby Hau-Tieng Wu (Duke University) as part of CUNY Harmonic Analysis a
nd PDE's Seminar\n\n\nAbstract\nDiffusion based manifold learning has been
actively developed and applied in past decades. However\, there are still
many interesting practical and theoretical challenges. I will share some
recent progress in this direction\, particularly from the angle of robustn
ess and scalability and the associated theoretical support under the manif
old setup. If time permits\, its clinical application will be discussed.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincent R. Martinez (CUNY-Hunter College)
DTSTART:20211008T170000Z
DTEND:20211008T181500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/40
DESCRIPTION:Title: On well-posedness and smoothing of solutions to the genera
lized SQG equations in critical Sobolev spaces\, Part II\nby Vincent R
. Martinez (CUNY-Hunter College) as part of CUNY Harmonic Analysis and PDE
's Seminar\n\n\nAbstract\nWe will continue the discussion about the issue
of well-posedness at critical regularity for a family for active scalar eq
uations with increasingly singular constitutive law.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Deniz Bilman (University of Cincinnati)
DTSTART:20211105T170000Z
DTEND:20211105T181500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/41
DESCRIPTION:Title: High-Order Rogue Waves and Solitons\, and Solutions Interp
olating Between Them\nby Deniz Bilman (University of Cincinnati) as pa
rt of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nIt is known
from our recent work that both fundamental rogue wave solutions (with Pete
r Miller and Liming Ling) and multi-pole soliton solutions (with R. Buckin
gham) of the nonlinear Schrödinger (NLS) equation exhibit the same asympt
otic behavior in the limit of large order in a shrinking region near the p
eak amplitude point\, despite the quite different boundary conditions thes
e solutions satisfy at infinity. We show how rogue waves and solitons of a
rbitrary orders can be placed within a common analytical framework in whic
h the ''order'' becomes a continuous parameter\, allowing one to tune cont
inuously between types of solutions satisfying different boundary conditio
ns. In this scheme\, solitons and rogue waves of increasing integer orders
alternate as the continuous order parameter increases. We show that in a
bounded region of the space-time of size proportional to the order\, these
solutions all appear to be the same when the order is large. However\, in
the unbounded complementary region one sees qualitatively different asymp
totic behavior along different sequences. In this talk we focus on the beh
avior in this exterior region. The asymptotic behavior is most interesting
for solutions that are neither rogue waves nor solitons. This is joint wo
rk with Peter Miller.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Götz Pfander (Katholische Universität Eichstätt-Ingolstadt)
DTSTART:20211119T180000Z
DTEND:20211119T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/42
DESCRIPTION:Title: Exponential bases for partitions of intervals\nby Göt
z Pfander (Katholische Universität Eichstätt-Ingolstadt) as part of CUNY
Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nFourier series form a
cornerstone of analysis\; it allows the expansion of a complex valued 1-pe
riodic function in the orthogonal basis of integer frequency exponentials
(for the space of square integrable functions on the unit interval). A sim
ple rescaling argument shows that by splitting the integers into evens and
odds\, we obtain orthogonal bases for functions defined on the first\, re
spectively the second half of the unit interval. We shall generalize this
curiosity and show that\, given any finite partition of the unit interval
into subintervals\, we can split the integers into subsets\, each of which
forms a basis (not necessarily orthogonal) for functions on the respectiv
e subinterval. In addition\, novel fundamental results in the theory of Fo
urier series will be discussed.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tao Zhang (Guangzhou University)
DTSTART:20211203T180000Z
DTEND:20211203T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/43
DESCRIPTION:Title: Fuglede's conjecture holds in $\\mathbb{Z}_p\\times\\mathb
b{Z}_{p^n}$\nby Tao Zhang (Guangzhou University) as part of CUNY Harmo
nic Analysis and PDE's Seminar\n\n\nAbstract\nFuglede's conjecture states
that for a subset $\\Omega$ of a locally compact abelian group $G$ with po
sitive and finite Haar measure\, there exists a subset of the dual group o
f $G$ which is an orthogonal basis of $L^2(\\Omega)$ if and only if it til
es the group by translation. In this talk\, we consider the Fuglede's conj
ecture in the group $\\mathbb{Z}_p\\times\\mathbb{Z}_{p^n}$. I will talk a
bout the main idea of our proof.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tuan Pham (Brigham Young University)
DTSTART:20211210T180000Z
DTEND:20211210T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/44
DESCRIPTION:Title: Conservation of frequencies and applications to the well-p
osedness problem of the Navier-Stokes equations\nby Tuan Pham (Brigham
Young University) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n
\nAbstract\nA well-known conserved quantity of the Navier-Stokes equations
is the total energy. This conservation law has been used extensively in t
he local regularity theory\, especially since the groundbreaking work of C
affarelli-Kohn-Nirenberg in 1982. In the Fourier space\, the Navier-Stokes
equations are naturally associated with a stochastic cascade as noted by
Le Jan and Sznitman in 1997. In the dynamic of this stochastic cascade\, t
he initial frequency is conserved. In this talk\, I will explain how the c
onservation of frequencies can be used to study the well-posedness problem
of the Navier-Stokes equations. A notable application is that any initial
data in L2 whose Fourier transform is supported in the half-space produce
s a unique global mild solution.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Balazs (Austrian Academy of Sciences)
DTSTART:20211217T180000Z
DTEND:20211217T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/45
DESCRIPTION:Title: Continuous Frames and Reproducing Kernels\nby Peter Ba
lazs (Austrian Academy of Sciences) as part of CUNY Harmonic Analysis and
PDE's Seminar\n\n\nAbstract\nFrame theory has become a tremendously active
research field\, with connection to many mathematical disciplines but als
o applications. In short\, frames are a sequence of elements that allow st
able representation of elements in a Hilbert space. One generalization of
the original definition considers not a sequence indexed by a discrete set
\, but a function indexed by a continuous set. This will be the topic of t
his talk\, in particular\, how closely this concept is intertwined with re
producing kernel Hilbert spaces (RKHS). We start with a short motivation w
hy frame theory is important\, also for applications. We introduce the bas
ic definitions. We show recent developments which focus on the various fac
ets of the interplay of continuous frames and RKHS. In particular\, we ana
lyze the structure of the reproducing kernel of a RKHS using frames and re
producing pairs. We show that finite redundancy of a continuous frame impl
ies atomic structure of the underlying measure space. This implies that al
l the attempts to extend the notion of Riesz basis to general measure spac
es are fruitless since every such family can be identified with a discrete
Riesz basis\, by using the RKHS structure of the range of the analysis op
erator. This can also be used to formulate a result that classifies all du
al functions to a given continuous frame. Finally\, we will give general k
ernel theorems for operators acting between coorbit spaces\, which are Ban
ach spaces associated to continous frame representations and contain most
of the usual function spaces (Besov spaces\, modulation spaces\, etc.). Th
is collects work with K. Gröchenig\, M. Speckbacher and N. Teofanov.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Magsino (Ohio State University)
DTSTART:20220304T190000Z
DTEND:20220304T201500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/46
DESCRIPTION:Title: Singular Values of Random Subensembles of Frame Vectors\nby Mark Magsino (Ohio State University) as part of CUNY Harmonic Analys
is and PDE's Seminar\n\n\nAbstract\nFrame theory studies redundant represe
ntations in a Hilbert space. In finite dimensions\, this is simply a spann
ing set but there are many interesting and useful frames in these settings
. One application involves compressed sensing\, which is a method for effi
cient acquisition and reconstruction of signals using underrepresented sys
tems. However\, verifying the key property of compressed sensing frames is
NP-hard which makes constructing them difficult. One way around this is t
o examine random subensembles of these frames and try to control their sin
gular values. We will show that the singular values of random subensembles
of so-called equiangular tight frames are closely linked to the Kesten-Mc
Kay distribution. This is joint work with Dustin Mixon and Hans Parshall.\
n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gareth Speight (University of Cincinnati)
DTSTART:20220311T190000Z
DTEND:20220311T201500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/47
DESCRIPTION:Title: Whitney Extension and Lusin Approximation in Carnot Groups
\nby Gareth Speight (University of Cincinnati) as part of CUNY Harmoni
c Analysis and PDE's Seminar\n\n\nAbstract\nLusin's theorem states that an
y measurable function can be approximated by a continuous function\, excep
t on a set of small measure. Analogous results for higher smoothness give
conditions under which a function may admit a Lusin type approximation by
C^m functions. Such results can often be obtained as a consequence of a su
itable Whitney extension theorem. We review what is known in the Euclidean
setting then describe some recent extensions to Carnot groups\, a family
of non-Euclidean spaces that nevertheless have a rich geometric structure.
Based on joint work with Marco Capolli\, Andrea Pinamonti\, and Scott Zim
merman.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Weilin Li (New York University)
DTSTART:20220325T180000Z
DTEND:20220325T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/48
DESCRIPTION:Title: Function approximation with one-bit Bernstein and one-bit
neural networks\nby Weilin Li (New York University) as part of CUNY Ha
rmonic Analysis and PDE's Seminar\n\n\nAbstract\nThe celebrated universal
approximation theorems for neural networks (NNs) typically state that ever
y sufficiently nice function can be arbitrarily well approximated by a neu
ral network with carefully chosen real parameters. Motivated by recent que
stions regarding NN compression\, we ask whether it is possible to represe
nt any reasonable function with a quantized NN -- a NN whose parameters ar
e only allowed to be selected from a small set of allowable parameters. We
answer this question in the affirmative. Our main theorem shows that any
continuously differentiable multivariate function can be approximated by a
one-bit quadratic NN (a NN with quadratic activation whose nonzero weight
s and biases are only allowed to contain +1 or -1 entries) and the rate of
approximation of our scheme is able to exploit any additional smoothness
of the target function. A key component of our work is a novel approximati
on result by linear combinations of multivariate Bernstein polynomials\, w
ith only +1 and -1 coefficients. Joint work with Sinan Gunturk.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Itay Londner (Weizmann Institute of Science)
DTSTART:20220401T180000Z
DTEND:20220401T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/49
DESCRIPTION:Title: Tiling the integers with translates of one tile: the Coven
-Meyerowitz tiling conditions\nby Itay Londner (Weizmann Institute of
Science) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract
\nIt is well known that if a finite set of integers $A$ tiles the integers
by translations\, then the translation set must be periodic\, so that the
tiling is equivalent to a factorization $A+B=\\Z_M$ of a finite cyclic gr
oup. Coven and Meyerowitz (1998) proved that when the tiling period $M$ ha
s at most two distinct prime factors\, each of the sets $A$ and $B$ can be
replaced by a highly ordered "standard" tiling complement. It is not know
n whether this behaviour persists for all tilings with no restrictions on
the number of prime factors of $M$.\n\nIn joint work with Izabella Laba (U
BC)\, we proved that this is true for all sets tiling the integers with pe
riod $M=(pqr)^2$. In my talk I will discuss this problem and introduce som
e ideas from the proof.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shahaf Nitzan (Georgia Institute of Technology)
DTSTART:20220408T180000Z
DTEND:20220408T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/50
DESCRIPTION:Title: The uncertainty principle in finite dimensions\nby Sha
haf Nitzan (Georgia Institute of Technology) as part of CUNY Harmonic Anal
ysis and PDE's Seminar\n\n\nAbstract\nI will give a survey of some results
related to the talks title\, and discuss a couple of new observations in
the area. The talk is based on joint work with Jan-Fredrik Olsen and Micha
el Northington.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Weinan Wang (The University of Arizona)
DTSTART:20220429T180000Z
DTEND:20220429T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/51
DESCRIPTION:Title: Local well-posedness for the Boltzmann equation with very
soft potential and polynomially decaying initial data\nby Weinan Wang
(The University of Arizona) as part of CUNY Harmonic Analysis and PDE's Se
minar\n\n\nAbstract\nWe consider the local well-posedness of the spatially
inhomogeneous non-cutoff Boltzmann equation when the initial data decays
polynomially in the velocity variable. We consider the case of very soft p
otentials $\\gamma + 2s < 0$. Our main result completes the picture for lo
cal well-posedness in this decay class by removing the restriction $\\gamm
a + 2s > -3/2$ of previous works. It is based on the Carleman decompositio
n of the collision operator into a lower order term and an integro-differe
ntial operator similar to the fractional Laplacian.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arie Israel (The University of Texas at Austin)
DTSTART:20220513T180000Z
DTEND:20220513T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/52
DESCRIPTION:Title: The norm of linear extension operators for $C^m(R^n)$\
nby Arie Israel (The University of Texas at Austin) as part of CUNY Harmon
ic Analysis and PDE's Seminar\n\n\nAbstract\nWe will describe a new proof
of the finiteness principle for Whitney's extension problem. As a byproduc
t\, we obtain the existence of linear extension operators with an improved
bound on the norm of the operator. We discuss connections to the algorith
mic problem of interpolation of data.\n\nThis is joint work with Jacob Car
ruth and Abraham Frei-Pearson.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Laura De Carli (Florida International University)
DTSTART:20220930T180000Z
DTEND:20220930T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/53
DESCRIPTION:Title: Weaving Riesz bases\, and piecewise weighted frames\
nby Laura De Carli (Florida International University) as part of CUNY Harm
onic Analysis and PDE's Seminar\n\n\nAbstract\nThis talk consists of two p
arts loosely connected to one another. In the first part we discuss the pr
operties of a family of Riesz bases on a separable Hilbert space $H$ obtai
ned in the following way: For every $N>1$ we let $B_N=\\{w_j\\}_{j=1}^N\\
bigcup\\{v_j\\}_{j=N+1}^\\infty$\, \nwhere $\\{v_j\\}_{j=1}^\\infty$ is a
Riesz basis of $H$ and $B=\\{w_j\\}_{j=1}^\\infty$ is a set of unit vecto
rs. We find necessary and sufficient conditions that ensure that the $B
_N$ and $B$ are Riesz bases\, and we apply our results to the construc
tion of exponential bases on domains of $L^2$.\n\nIn the second part of th
e talk we present results on weighted Riesz bases and frames in finite or
infinite-dimensional Hilbert spaces\, with piecewise constant weights. We
use our results to construct tight frames in finite-dimensional Hilbert s
paces.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Ginsberg (Princeton University)
DTSTART:20221104T180000Z
DTEND:20221104T191500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/54
DESCRIPTION:Title: The stability of model shocks and the Landau law of decay<
/a>\nby Daniel Ginsberg (Princeton University) as part of CUNY Harmonic An
alysis and PDE's Seminar\n\n\nAbstract\nIt is well-known that in three spa
ce dimensions\, smooth solutions to the equations describing a compressibl
e gas can break down in finite time. One type of singularity which can ari
se is known as a shock\, which is a hypersurface of discontinuity across w
hich the integral forms of conservation of mass and momentum hold and thro
ugh which there is nonzero mass flux. One can find approximate solutions t
o the equations of motion which describe expanding spherical shocks. We us
e these model solutions to construct global-in-time solutions to the irrot
ational compressible Euler equations with shocks. This is joint work with
Igor Rodnianski.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleh Oleg Asipchuk (Florida International University)
DTSTART:20221111T190000Z
DTEND:20221111T201500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/55
DESCRIPTION:Title: Construction of exponential bases on split intervals\n
by Aleh Oleg Asipchuk (Florida International University) as part of CUNY H
armonic Analysis and PDE's Seminar\n\n\nAbstract\nLet $I$ be a union of in
tervals of total length 1. It is well known that exponential bases exist o
n $L^2(I)$\, but explicit expressions for such bases are only known in spe
cial cases. In this work\, we construct exponential Riesz bases on $L^2(I)
$ with some mild assumptions on the gaps between the intervals. We also ge
neralize Kadec's stability theorem in some special and significant cases.\
n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Goluskin
DTSTART:20221118T200000Z
DTEND:20221118T211500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/56
DESCRIPTION:Title: Verifying global stability of fluid flows despite transien
t growth of energy (Special Time)\nby David Goluskin as part of CUNY H
armonic Analysis and PDE's Seminar\n\n\nAbstract\nVerifying nonlinear stab
ility of a laminar fluid flow against all perturbations is a classic chall
enge in fluid dynamics. All past results rely on monotonic decrease of a p
erturbation energy or a similar quadratic generalized energy. This energy
method cannot show global stability of any flow in which perturbation ener
gy may grow transiently. For the many flows that allow transient energy gr
owth but seem to be globally stable (e.g. pipe flow and other parallel she
ar flows at certain Reynolds numbers) there has been no way to mathematica
lly verify global stability. After explaining why the energy method was th
e only way to verify global stability of fluid flows for over 100 years\,
I will describe a different approach that is broadly applicable but more t
echnical. This approach\, proposed in 2012 by Goulart and Chernyshenko\, u
ses sum-of-squares polynomials to computationally construct non-quadratic
Lyapunov functions that decrease monotonically for all flow perturbations.
I will present a computational implementation of this approach for the ex
ample of 2D plane Couette flow\, where we have verified global stability a
t Reynolds numbers above the energy stability threshold. This energy stabi
lity result for 2D Couette flow had not been improved upon since being fou
nd by Orr in 1907. The results I will present are the first verification o
f global stability - for any fluid flow - that surpasses the energy method
. This is joint work with Federico Fuentes (Universidad Catolica de Chile)
and Sergei Chernyshenko (Imperial College London).\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amir Sagiv (Columbia University)
DTSTART:20221202T190000Z
DTEND:20221202T201500Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/57
DESCRIPTION:Title: Floquet Hamiltonians - effective gaps and resonant decay\nby Amir Sagiv (Columbia University) as part of CUNY Harmonic Analysis
and PDE's Seminar\n\n\nAbstract\nFloquet topological insulators are an eme
rging category of materials whose properties are transformed by time-perio
dic forcing. Can their properties be understood from their first-principle
s continuum PDE models? Experimentally\, graphene is known to transform in
to an insulator under a time-periodic driving. A spectral gap\, however\,
is conjectured to not form. How do we reconcile these two facts? We show t
hat the original Schrodinger equation has an effective gap- a new and phys
ically-relevant relaxation of a spectral gap. Next\, we challenge the prev
ailing notion of Floquet edge modes\; due to resonance\, localized modes i
n periodically-forced media are only metasable. Sufficiently rapid forcing
couples the localized mode to the bulk\, and so energy eventually leaks a
way from the localized edge/defect\, in the spirit of the Fermi Golden Rul
e.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin D. Stubbs (Institute of Pure and Applied Mathematics\, UCLA)
DTSTART:20230203T190000Z
DTEND:20230203T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/59
DESCRIPTION:Title: A Mathematical Invitation to Wannier Functions\nby Kev
in D. Stubbs (Institute of Pure and Applied Mathematics\, UCLA) as part of
CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nWannier functions
\, first proposed in the 1930s\, have had a long history in computational
chemistry as a practical means to speed up calculations. Stated in a mathe
matical language\, Wannier functions are an orthonormal basis for certain
types of spectral subspaces which are generated by the action of a transla
tion group. In the 1980s however\, it was realized that there is an intima
te connection between Wannier functions and topology. In particular\, Wann
ier functions with fast spatial decay exist if and only if a certain vecto
r bundle is topologically trivial. Materials with non-trivial topology hos
t a number of remarkable properties which are robust to physical imperfect
ions. In this talk\, I will give a brief introduction to topological mater
ials and Wannier functions in periodic systems. I will then discuss my wor
k on extending these results to systems without any underlying periodicity
.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raghavendra Venkatraman (courant institute of mathematical science
s)
DTSTART:20230217T190000Z
DTEND:20230217T200000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/61
DESCRIPTION:Title: Homogenization questions inspired by machine learning and
the semi-supervised learning problem\nby Raghavendra Venkatraman (cour
ant institute of mathematical sciences) as part of CUNY Harmonic Analysis
and PDE's Seminar\n\n\nAbstract\nThis talk comprises two parts. In the fir
st part\, we revisit the problem of pointwise semi-supervised learning (SS
L). Working on random geometric graphs (a.k.a point clouds) with few "labe
led points"\, our task is to propagate these labels to the rest of the poi
nt cloud. Algorithms that are based on the graph Laplacian often perform p
oorly in such pointwise learning tasks since minimizers develop localized
spikes near labeled data. We introduce a class of graph-based higher order
fractional Sobolev spaces (H^s) and establish their consistency in the la
rge data limit\, along with applications to the SSL problem. A crucial too
l is recent convergence results for the spectrum of the graph Laplacian to
that of the continuum.\nObtaining optimal convergence rates for such spec
tra is an open question in stochastic homogenization. In the rest of the t
alk\, we'll discuss how to get state-of-the-art and optimal rates of conve
rgence for the spectrum\, using tools from stochastic homogenization.\nThe
first half is joint work with Dejan Slepcev (CMU)\, and the second half i
s joint work with Scott Armstrong (Courant).\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fushuai Jiang (Univeristy of Maryland)
DTSTART:20230310T180000Z
DTEND:20230310T190000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/63
DESCRIPTION:Title: Quasi-optimal $C^2(R^n)$ Interpolation with Range Restrict
ion\nby Fushuai Jiang (Univeristy of Maryland) as part of CUNY Harmoni
c Analysis and PDE's Seminar\n\n\nAbstract\nExperimental data often have r
ange or shape constraints imposed by nature. For example\, probability den
sity or chemical concentration are non-negative quantities\, and the traje
ctory design through an obstacle course may need to avoid two boundaries.
In this talk\, we investigate the theory of multivariate smooth interpolat
ion with range restriction from the perspective of Whitney Extension Probl
ems. Given a function defined on a finite set with no underlying geometric
assumption\, I will describe an $O(N(log N)^{-n})$ procedure to compute a
twice continuously differentiable interpolant that preserves a prescribed
shape (e.g. nonnegativity) and whose second derivatives are as small as p
ossible up to a constant factor (i.e.\, quasi-optimal). I will also provid
e explicit numerical results in one dimension. This is based on the joint
works with Charles Fefferman (Princeton)\, Chen Liang (UC Davis)\, Yutong
Liang (former UC Davis)\, and Kevin Luli (UC Davis).\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arthur A. Danielyan (University of South Florida)
DTSTART:20230317T170000Z
DTEND:20230317T180000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/64
DESCRIPTION:Title: On a converse of Fatou's theorem\nby Arthur A. Daniely
an (University of South Florida) as part of CUNY Harmonic Analysis and PDE
's Seminar\n\n\nAbstract\nFatou's theorem states that a bounded analytic f
unction in the unit disc has radial limits a.e. on the unit circle $T$. Th
is talk presents the following new theorem in the converse direction. \n\n
Theorem 1. Let $E$ be a subset on $T$. There exists a bounded analytic fun
ction in the open unit disc which has no radial limits on $E$ but has unre
stricted limits at each point of $T\\backslash E$ if and only if $E$ is an
$F_\\sigma$ set of measure zero.\n\nThe sufficiency part of this theorem
immediately implies a well-known theorem of Lohwater and Piranian the proo
f of which is complicated enough. However\, the proof of Theorem 1 only us
es the Fatou's interpolation theorem\, for which too the author has recent
ly suggested a new simple proof.\n\nIt turns out that for the Blaschke pro
ducts\, a well-known subclass of bounded analytic functions\, Theorem 1 ta
kes the following form. \n\nTheorem 2. Let $E$ be a subset on the unit cir
cle $T$. There exists a Blaschke product which has no radial limits on $E$
but has unrestricted limits at each point of $T\\backslash E$ if and only
if $E$ is a closed set of measure zero. \n\nThe proof of the necessity pa
rt of Theorem 2 is completely elementary\, but it still contains some meth
odological novelty. The proof of the sufficiency uses Theorem 1 as well as
some known results on Blaschke products. (Theorem 2 is a joint result wit
h Spyros Pasias.)\n\nReferences.\n\n1. A. A. Danielyan\, On Fatou's theore
m\, Anal. Math. Phys. V. 10\, Paper no. 28\, 2020. \n\n2. A. A. Danielyan\
, A proof of Fatou's interpolation theorem\, J. Fourier Anal. Appl.\, V. 2
8\, Paper no. 45\, 2022. \n\n3. A. A. Danielyan and S. Pasias\, On a bound
ary property of Blaschke products\, to appear in Anal. Mathematica\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael A. Perlmutter (University of California\, Los Angeles)
DTSTART:20230331T170000Z
DTEND:20230331T180000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/65
DESCRIPTION:Title: Geometric Scattering on Measure Spaces\nby Michael A.
Perlmutter (University of California\, Los Angeles) as part of CUNY Harmon
ic Analysis and PDE's Seminar\n\n\nAbstract\nGeometric Deep Learning is an
emerging field of research that aims to extend the success of convolution
al neural networks (CNNs) to data with non-Euclidean geometric structure.
Despite being in its relative infancy\, this field has already found great
success in many applications such as recommender systems\, computer graph
ics\, and traffic navigation. In order to improve our understanding of the
networks used in this new field\, several works have proposed novel versi
ons of the scattering transform\, a wavelet-based model of CNNs for graphs
\, manifolds\, and more general measure spaces. In a similar spirit to the
original Euclidean scattering transform\, these geometric scattering tran
sforms provide a mathematically rigorous framework for understanding the s
tability and invariance of the networks used in geometric deep learning. A
dditionally\, they also have many interesting applications such as drug di
scovery\, solving combinatorial optimization problems\, and predicting pat
ient outcomes from single-cell data. In particular\, motivated by these ap
plications to single-cell data\, I will also discuss recent work proposing
a diffusion maps style algorithm with quantitative convergence guarantees
for implementing the manifold scattering transform from finitely many sam
ples of an unknown manifold.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nabil T. Fadai (University of Nottingham)
DTSTART:20230414T170000Z
DTEND:20230414T180000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/66
DESCRIPTION:Title: Semi-infinite travelling waves arising in moving-boundary
reaction-diffusion equations\nby Nabil T. Fadai (University of Notting
ham) as part of CUNY Harmonic Analysis and PDE's Seminar\n\n\nAbstract\nTr
avelling waves arise in a wide variety of biological applications\, from t
he healing of wounds to the migration of populations. Such biological phen
omena are often modelled mathematically via reaction-diffusion equations\;
however\, the resulting travelling wave fronts often lack the key feature
of a sharp edge. In this talk\, we will examine how the incorporation of
a moving boundary condition in reaction-diffusion models gives rise to a v
ariety of sharp-fronted travelling waves for a range of wave speeds. In pa
rticular\, we will consider common reaction-diffusion models arising in bi
ology and explore the key qualitative features of the resulting travelling
wave fronts.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Geet Varma (RMIT University)
DTSTART:20230324T170000Z
DTEND:20230324T180000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/67
DESCRIPTION:Title: Weaving Frames Linked with Fractal Convolutions\nby Ge
et Varma (RMIT University) as part of CUNY Harmonic Analysis and PDE's Sem
inar\n\n\nAbstract\nWeaving frames have been introduced to deal with some
problems in signal processing and wireless sensor networks. More recently\
, the notion of fractal operator and fractal convolutions have been linked
with perturbation theory of Schauder bases and frames. However\, the exis
ting literature has established limited connections between the theory of
fractals and frame expansions. In this paper we define Weaving frames gene
rated via fractal operators combined with fractal convolutions. The aim is
to demonstrate how partial fractal convolutions are associated to Riesz b
ases\, frames and the concept of Weaving frames. This current view point d
eals with ones sided convolutions i.e both left and right partial fractal
convolution operators on Lebesgue space $L^p$ for $1\\le p<\\infty$ . Some
applications via partial fractal convolutions with null function have bee
n obtained for the perturbation theory of bases and weaving frames.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lu Zhang (Columbia University)
DTSTART:20230421T170000Z
DTEND:20230421T180000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/68
DESCRIPTION:Title: Coupling physics-deep learning inversion\nby Lu Zhang
(Columbia University) as part of CUNY Harmonic Analysis and PDE's Seminar\
n\n\nAbstract\nIn recent years\, there is an increasing interest in applyi
ng deep learning to geophysical/medical data inversion. However\, direct a
pplication of end-to-end data-driven approaches to inversion have quickly
shown limitations in the practical implementation. Indeed\, due to the lac
k of prior knowledge on the objects of interest\, the trained deep learnin
g neural networks very often have limited generalization. In this talk\, w
e introduce a new methodology of coupling model-based inverse algorithms w
ith deep learning for two typical types of inversion problems. In the firs
t part\, we present an offline-online computational strategy of coupling c
lassical least-squares based computational inversion with modern deep lear
ning based approaches for full waveform inversion to achieve advantages th
at can not be achieved with only one of the components. In the second part
\, we present an integrated data-driven and model-based iterative reconstr
uction framework for joint inversion problems. The proposed method couples
the supplementary data with the partial differential equation model to ma
ke the data-driven modeling process consistent with the model-based recons
truction procedure. We also characterize the impact of learning uncertaint
y on the joint inversion results for one typical inverse problem.`1\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Chun-Kit Lai (San Francisco State University)
DTSTART:20230428T170000Z
DTEND:20230428T180000Z
DTSTAMP:20260422T215027Z
UID:HarmonicAnalysisandPDE/69
DESCRIPTION:Title: On measure and topological Erdos similarity problems\n
by Chun-Kit Lai (San Francisco State University) as part of CUNY Harmonic
Analysis and PDE's Seminar\n\n\nAbstract\nA pattern is called universal in
another collection of sets\, when every set in the collection contains so
me linear and translated copy of the original pattern. Paul Erdős propose
d a conjecture that no infinite set is universal in the collection of sets
with positive measure.\n\nIn this talk\, we explore an analogous problem
in the topological setting. Instead of sets with positive measure\, we inv
estigate the collection of dense sets and in the collection of generic se
ts (dense G-delta and complement has Lebesgue measure zero). We refer to
such pattern as topologically universal and generically universal respecti
vely. We will show that Cantor sets on $R^d$ are never topologically unive
rsal and Cantor sets with positive Newhouse thickness on $R^1$ are not ge
nerically universal. This gives a positive partial answer to a question by
Svetic concerning the Erdős similarity problem on Cantor sets. Moreover\
, we also obtain a higher dimensional generalization of the generic univer
sality problem.\n\nThis is a joint work with John Gallagher\, who was a Ma
ster student in SFSU\, and Eric Weber from Iowa State University.\n
LOCATION:https://researchseminars.org/talk/HarmonicAnalysisandPDE/69/
END:VEVENT
END:VCALENDAR