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SUMMARY:Online Mini-Courses in Spectral Theory and Mathematical Physics
DTSTART;VALUE=DATE-TIME:20201103T100000Z
DTEND;VALUE=DATE-TIME:20201203T160000Z
DTSTAMP;VALUE=DATE-TIME:20260714T064645Z
UID:indico-event-124@indico.eimi.ru
DESCRIPTION:Online Mini-Courses in\nSpectral Theory and Mathematical Physi
 cs\n\n3–26 November 2020\n\nThese are warm-up mini-courses for the EIM
 I thematic program on Spectral Theory and Mathematical Physics to be held 
 in St. Petersburg in 2021 (rescheduled from 2020 due to COVID-19 pandemic)
 . The target audience includes graduate\, master and senior bachelor stud
 ents of any mathematical speciality\; senior researchers are also welcome
 .\n\nThe meetings will be held on Zoom\, and registration is required to o
 btain the link. Click here to register.\n\nAlso see the timetable page for
  the schedule.\n\nFind the video recordings here.\n\nAlexander Pushnitski 
 (King's College London\, UK)\nSchmidt subspaces of Hankel operators\n\n3
 –5 November\, 2020\n\nLet Γ be a compact Hankel operator acting on the 
 Hardy class H² over the unit circle. The purpose of the lectures is to di
 scuss the structure of the Schmidt spaces of Γ (i.e. the eigenspaces of 
 Γ ⃰ Γ) as a class of subspaces of H².\n\nIt turns out that the Schmid
 t spaces of Γ are the images of model spaces under the action of isometri
 c multipliers. The action of Γ on the Schmidt spaces can also be explicit
 ly described. All of these notions will be introduced and discussed in det
 ail in the lectures. If time permits\, an inverse spectral problem for Γ 
 will be briefly described.\n\nThe lectures are based on recent joint work 
 of the author with Patrick Gérard (Orsay).\n\nLecture notes are available
  here\, and video recordings\, here.\n\nAlexander Sobolev (University Coll
 ege London\, UK)\nRecent analytic and spectral results for the multi-parti
 cle Schrödinger operator\n\n17–19 November 2020\n\nThe lectures focus o
 n the properties of the one-particle density matrix γ(x\, y)\, x\, y 
 ∈ ℝ³. This is one of the key objects in the quantum-mechanical appro
 ximation schemes. The aim is to present the following results obtained rec
 ently:\n\n\n	the asymptotic formula for the eigenvalues of the self-adjoin
 t operator Γ with the kernel γ(x\, y)\,\n	real analyticity of the funct
 ion γ(x\, y).\n\n\nLecture 1: Background and results (slides\, video)\nL
 ecture 2: Compact operators (slides\, video)\nLecture 3: Spectrum of the o
 ne-particle density matrix (slides\, video)\nLecture 4: Real analyticity o
 f the density matrix (slides\, video)\n\nAlexander Its (IUPUI\, USA & St. 
 Petersburg University\, Russia)\nToeplitz determinants and Painlevé trans
 cendents. A Riemann–Hilbert point of view\n\n2–3 December\, 2020\n\nSt
 arting with Onsager's celebrated solution of the two-dimensional Ising mod
 el in the 1940's\, Toeplitz and Hankel determinants have been one of the p
 rincipal analytic tools in modern mathematical physics\; specifically\, in
  the theory of exactly solvable statistical mechanics\, and quantum field 
 models\, and in the theory of random matrices.\n\nThe main analytical issu
 e of the theory of Toeplitz determinants is their large size asymptotic be
 havior. There are  two complementary approaches to study this question. T
 he (historically) first approach is based on the general operator techniqu
 es\, and it has been used in the theory of Toeplitz and Hankel determinant
 s since the classical works of Szego and Widom. The second approach  is y
 ounger\, and it is based on the Riemann-Hilbert method of the theory of in
 tegrable systems.\n\nIn this mini course\, the essence of the Riemann-Hilb
 ert method in the theory of Topelitz determinants will be presented. The f
 ocus will be on  the use of the method to obtain the Painlevé type desc
 ription of the transition asymptotics of Toeplitz determinants. The Rieman
 n-Hilbert view on the Painlevé functions will be also explained.\n\nIf t
 ime permits\, some most recent results related to the so-called bordered T
 oeplitz determinants and Toeplitz + Hankel determinants will be discussed.
 \n\nSlides are available here\, and video recordings\, here.\n\n\n\n\nhttp
 s://indico.eimi.ru/event/124/
LOCATION:via Zoom
URL:https://indico.eimi.ru/event/124/
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