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SUMMARY:School-conference in analysis in Sirius
DTSTART;VALUE=DATE-TIME:20231013T070000Z
DTEND;VALUE=DATE-TIME:20231017T150000Z
DTSTAMP;VALUE=DATE-TIME:20230922T082824Z
UID:indico-event-1362@indico.eimi.ru
DESCRIPTION:L. Euler Institute School in Analysis in Sirius\n\nOctober
13 – October 17\, 2023\n\nThis is an advanced school for young researche
rs featuring five minicourses in modern analysis\, such as wavelets\, disc
rete and continuous duality with respect to Dirac and Schrödinger operato
rs\, singular integral operators\, Padé approximations\, orthogonal polyn
omials. The target audience includes graduate\, master and senior bachelor
students of any mathematical specialty.\n\nThe lecture notes and exercis
es are below on this page\n\n\nLecturers:\n\n \n\n\n \n \n \n Maria
Skopina\n Saint Petersburg university\n \n \n Mark Malamud\n Peo
ples' Friendship university\n \n \n Roman Bessonov\n Saint Petersb
urg university\n \n \n Dmitry Stolyarov\n Saint Petersburg univers
ity\n \n \n Aleksandr Komlov\n Institute of mathematics\, RAS\n
\n \n \n\n\n\nCourses:\n\n\nWavelets (Maria Skopina)\n\nThe word "wavelet
" appeared about thirty years ago. At that time a new topic of analysis
arose due to the desire to study bases and other representation sys
tems consisting of dilations and translations of one or several functions.
An interest in these investigations was connected with an engineering as
pect (that is not typical for the end of XX-th century)\, because such s
ystems are in great demand in their applications to signal and image
processing. Wavelet analysis was formed in the late 1980s -- early 1990s
in the works of Y. Meyer\, S. Mallat\, P.J. Lemarier\, I. Daubechies\, A.
Cohen\, W. Lawton and others. A general method for constructing wavelet s
ystems\, based on the notion of multiresolution analysis\, was developed b
y Y. Meyer and S. Mallat. One of the main goals was construction of compac
tly supported smooth wavelets. This problem was solved due to I. Daubechie
s. Later the wavelet theory was extended in different directions\, in par
ticular\, wavelet frames\, multivariate wavelets with matrix dilations\,
wavelet on the groups and other structures were actively studied up to no
w.\n\n \n\nLecture 1: TBA\n\nLecture 2: TBA\n\nLecture 3: TBA\n\nLecture
4: TBA\n\n\nTo the spectral theory of 1-D Schrödinger and Dirac operators
with point interactions and quantum graphs (Mark Malamud)\n\n \n\nThe le
ctures will be devoted to the duality of certain spectral properties of th
e operators mentioned above and their discrete counterparts . Among othe
rs we will discuss self-adjointness\, semiboundedness\, discreteness and
absolutely continuous properties\, compactness and finiteness of negativ
e parts of these operators\, etc.\n\n \n\nLecture 1: TBA\n\nLecture 2: TB
A\n\nLecture 3: TBA\n\n\nEntropy function in the theory of orthogonal poly
nomials (Roman Bessonov)\n\n \n\nThis minicourse is devoted to recent res
ults in the theory of orthogonal polynomials on the unit circle. They
were obtained via a new method based on the usage of an entropy functio
n of the underlying orthogonality measure. The main goal of the course is
to present a detailed description of this method and discuss related open
problems. The basics of the theory of orthogonal polynomials will be
given along the way. \n\n \n\nLecture 1: TBA\n\nLecture 2: TBA\n\nLectur
e 3: TBA\n\n\nEstimates of differential operators in L^1 and related quest
ions (Dmitriy Stolyarov)\n\nIn 1938 S.L. Sobolev proved his famous embeddi
ng theorem: the Sobolev space W_p^1 embeds continuously into the L^q space
\, provided 1/p - 1/q = 1/d\, d being the dimension of the underlying spac
e\, and p > 1. This result was extended to the case p=1 twenty years later
by E. Gagliardo and L. Nirenberg. In early 2000's J. Bourgain and H. Brez
is observed that there exist similar estimates\, which are relatively easy
to obtain for p > 1\, but are much more involved (if they are even valid)
for p = 1. One of the key aspects in these matters is attributed to the v
ectorial nature of the differential operators in question (as\, for instan
ce\, the fact that the gradient is a vector function\, and not a scalar on
e). Nowadays\, there is a certain change of perspective regarding these to
pics: such inequalities are believed to be interesting not only in a herme
tic sense -- as a challenge to one's analytical prowess\, -- but also as a
way to establish deep connections to the geometric measure theory. We wil
l describe the (by now) classical Bourgain--Brezis theory\, highlight the
aforementioned geometric connections\, and explain how simple tricks from
harmonic analysis help in this business.\n\nLecture 1: TBA\n\nLecture 2: T
BA\n\nLecture 3: TBA\n\n\nPadé approximations\, their generalizations and
related problems (Aleksandr Komlov)\n\n \n\nPadé approximants are the b
est rational approximations of a given power series. We show that Padé ap
proximants are closely related to orthogonal polynomials\, Chebyshev (func
tional) continued fractions\, and Jacobi operators. This mini-course will
cover the basics of the classical Stahl theory of convergence of Padé app
roximants of multivalued analytic functions. To do this\, we will touch on
the potential theory on the complex plane and Riemann sphere. We also con
sider such generalizations of Padé polynomials as Hermite-Padé polynomia
ls of types I and II. For them\, there is no general convergence theory an
alogous to the Stahl theory. We discuss the cases in which it is possible
to describe their asymptotic behavior\, how to use them for asymptotic rec
overy of the values of multivalued analytic functions\, and formulate new
problems arising here.\n\n \n\nLecture 1: TBA\n\nLecture 2: TBA\n\nLectur
e 3: TBA\n\nLecture 4: TBA\n\n\nInstitutions participating in the organiza
tion of the event:\n\n\n Leonhard Euler International Mathematical Institu
te in Saint Petersburg\n Probability techniques in Analysis laboratory at
Saint Petersburg university\n\n\nhttps://indico.eimi.ru/event/1362/
LOCATION:Sochi\, Sirius Math Centre
URL:https://indico.eimi.ru/event/1362/
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