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SUMMARY:School-conference in analysis in Sirius
DTSTART;VALUE=DATE-TIME:20231013T070000Z
DTEND;VALUE=DATE-TIME:20231017T160000Z
DTSTAMP;VALUE=DATE-TIME:20240721T164106Z
UID:indico-event-1362@indico.eimi.ru
DESCRIPTION:L. Euler Institute School in Analysis in Sirius\n\n\n\nOcto
ber 13 – October 17\, 2023\n\nL. Euler Institute (EIMI SPbU) presents an
advanced school for young researchers featuring five minicourses in moder
n analysis\, such as wavelets\, discrete and continuous duality with respe
ct to Dirac and Schrödinger operators\, singular integral operators\, Pad
é approximations\, orthogonal polynomials. The target audience includes g
raduate\, master and senior bachelor students of any mathematical specialt
y.\n\nThe lecture notes and exercises are below on this page\n\nThe progr
am is here.\n\nZOOM streaming is here.\n\n\nLecturers:\n\n \n\n\n \n \n
\n Maria Skopina\n Saint Petersburg university\n \n \n Mark Ma
lamud\n Peoples' Friendship university\n \n \n Roman Bessonov\n
Saint Petersburg university\n \n \n Dmitry Stolyarov\n Saint Peter
sburg university\n \n \n Aleksandr Komlov\n Institute of mathemati
cs\, 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 repres
entation systems consisting of dilations and translations of one or severa
l functions. An interest in these investigations was connected with an en
gineering aspect (that is not typical for the end of XX-th century)\, be
cause such systems are in great demand in their applications to sign
al 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. Dau
bechies\, A. Cohen\, W. Lawton and others. A general method for constructi
ng wavelet systems\, based on the notion of multiresolution analysis\, was
developed by Y. Meyer and S. Mallat. One of the main goals was constructi
on of compactly supported smooth wavelets. This problem was solved due to
I. Daubechies. Later the wavelet theory was extended in different directi
ons\, in particular\, wavelet frames\, multivariate wavelets with matrix
dilations\, wavelet on the groups and other structures were actively stud
ied up to now.\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 Dir
ac operators with point interactions and quantum graphs (Mark Malamud)\n\n
\n\nThe lectures will be devoted to the duality of certain spectral prop
erties of the operators mentioned above and their discrete counterparts
. Among others we will discuss self-adjointness\, semiboundedness\, discr
eteness and absolutely continuous properties\, compactness and finitenes
s of negative parts of these operators\, etc.\n\n \n\nLecture 1: TBA\n\nL
ecture 2: TBA\n\nLecture 3: TBA\n\n\nEntropy function in the theory of ort
hogonal polynomials (Roman Bessonov)\n\n \n\nThis minicourse is devoted t
o recent results in the theory of orthogonal polynomials on the unit
circle. They were obtained via a new method based on the usage of an entr
opy function of the underlying orthogonality measure. The main goal of t
he course is to present a detailed description of this method and discuss
related open problems. The basics of the theory of orthogonal polynomi
als will be given along the way. \n\n \n\nLecture 1: TBA\n\nLecture 2: T
BA\n\nLecture 3: TBA\n\n\nEstimates of differential operators in L^1 and r
elated questions (Dmitriy Stolyarov)\n\nIn 1938 S.L. Sobolev proved his fa
mous embedding theorem: the Sobolev space W_p^1 embeds continuously into t
he L^q space\, provided 1/p - 1/q = 1/d\, d being the dimension of the und
erlying space\, 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. Brezis observed that there exist similar estimates\, which are rel
atively 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 attribu
ted to the vectorial nature of the differential operators in question (as\
, for instance\, the fact that the gradient is a vector function\, and not
a scalar one). Nowadays\, there is a certain change of perspective regard
ing these topics: such inequalities are believed to be interesting not onl
y in a hermetic sense -- as a challenge to one's analytical prowess\, -- b
ut also as a way to establish deep connections to the geometric measure th
eory. We will describe the (by now) classical Bourgain--Brezis theory\, hi
ghlight the aforementioned geometric connections\, and explain how simple
tricks from harmonic analysis help in this business.\n\nLecture 1: TBA\n\n
Lecture 2: TBA\n\nLecture 3: TBA\n\n\nPadé approximations\, their general
izations and related problems (Aleksandr Komlov)\n\n \n\nPadé approximan
ts are the best rational approximations of a given power series. We show t
hat Padé approximants are closely related to orthogonal polynomials. This
mini-course will cover the basics of the classical Stahl theory of conver
gence of Padé approximants of multivalued analytic functions. To do this\
, we will touch on the potential theory on the complex plane and Riemann s
phere. We also consider such generalizations of Padé polynomials as Hermi
te-Padé polynomials of types I and II. For them\, there is no general con
vergence theory analogous to the Stahl theory. We discuss the cases in whi
ch it is possible to describe their asymptotic behavior\, how to use them
for asymptotic recovery of the values of multivalued analytic functions\,
and formulate new problems arising here.\n\n \n\nLecture 1: TBA\n\nLectur
e 2: TBA\n\nLecture 3: TBA\n\nLecture 4: TBA\n\n\n\nInstitutions participa
ting in the organization of the event:\n\n\n Leonhard Euler International
Mathematical Institute in Saint Petersburg\n Probability techniques in Ana
lysis laboratory at Saint Petersburg university\n\n\nhttps://indico.eimi.r
u/event/1362/
LOCATION:Sochi\, Sirius Math Centre
URL:https://indico.eimi.ru/event/1362/
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