BEGIN:VCALENDAR VERSION:2.0 PRODID:-//CERN//INDICO//EN BEGIN:VEVENT SUMMARY:School-conference in analysis in Sirius DTSTART;VALUE=DATE-TIME:20231013T070000Z DTEND;VALUE=DATE-TIME:20231017T160000Z DTSTAMP;VALUE=DATE-TIME:20260614T040609Z 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/ END:VEVENT END:VCALENDAR