BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//CERN//INDICO//EN
BEGIN:VEVENT
SUMMARY:School "Complex analysis and Operator Theory"
DTSTART;VALUE=DATE-TIME:20211129T135000Z
DTEND;VALUE=DATE-TIME:20211203T172500Z
DTSTAMP;VALUE=DATE-TIME:20260311T230343Z
UID:indico-event-623@indico.eimi.ru
DESCRIPTION:‎Complex Analysis\, Combinatorics and Operator Theory\n\nNov
 ember 29 – December 3\, 2021\n\nAn advanced school for young researchers
  featuring three minicourses in vibrant areas of mathematics.  The target
  audience includes graduate\, master and senior bachelor students of any m
 athematical specialty.\n\nThe lecture notes and exercises are below on th
 is page\n\n\nLecturers:\n\n\n	\n		\n			\n\n			Ilya Shkredov\n			Institute 
 of mathematics\, RAS\n			\n			\n			Yuriy Tomilov\n			Institute of mathemat
 ics\, PAN\n			\n			\n			Alexander Volberg\n			Michigan State University\n	
 		\n		\n	\n\n\n\nCourses:\n\n\nAdditive combinatorics and analysis (Ilya S
 hkredov)\n\nAdditive combinatorics is an actively developing area of math\
 , that lies at the intersection of Number theory and Combinatorics. It als
 o involves extensive use of techniques from Analysis\, Graph theory\, Ergo
 dic theory\, Probability\, Algebraic geometry\, Topology and Geometry of n
 umbers.\n\nIn this course we discuss classic applications (Lectures 1\, 2\
 , 4) of functional-theoretic methods to additive combinatorics (setting as
 ide arithmetic combinatorics and non-commutative approaches)\, and\, vice 
 versa\, application of additive combinatorics to some questions from funct
 ion theory (Lecture 3).\n\n \n\nLecture 1: Additive combinatorics: discre
 te Fourier transform\, Roth theorem\, sets without solutions of linear equ
 ations\n\nLecture 2: Additive combinatorics: Gowers uniformity norms\, cor
 ners problem\, transference principle (for primes and Sidon sets)\n\nLectu
 re 3: Additive combinatorics: integer sets with small Wiener norm on integ
 ers\, Littlewood problem\, additive energies\, additive dimension and diss
 ociativity\n\nLecture 4: Additive combinatorics: almost periodicity of con
 volutions\, Croot-Sisask theorem\, applications to linear equations and se
 ts with small doubling\n\n\nBounded functional calculi for unbounded opera
 tors (Yuriy Tomilov)\n\nFunctional calculi is a classical subject of opera
 tor theory\, unifying such distant areas as complex analysis\, harmonic an
 alysis and PDE. Several breakthroughs in analysis\, e.g.  solving the fam
 ous Kato square root problem or obtaining the L^2-boundedness of Cauchy in
 tegrals on Lipschitz curves\, depend on the functional calculi theory. Mor
 eover\, the theory became indispensable nowadays for a number of applied p
 roblems\, stemming mostly from evolution equations.\n\n \n\nRoughly\, the
  functional calculi theory provides  a way to assign a linear operator A 
 on a Banach space a coherent family operators f(A)\, when f runs through a
  function algebra. So it can be considered as an operator version of funct
 ion theory. The major issue is how to provide a good ``control'' of f(A) g
 iven A and f\,  and we just touch this problem putting aside a number of 
 modern developments around it. \n\n \n\nWe'll start the course with a sh
 ort overview of the theory of  functional calculi\, putting an emphasis o
 n generators of strongly continuous operator semigroups. This class of ope
 rators is of major importance for the study of partial differential equati
 ons\, and it will be the main object of our studies as well. Then we'll pr
 oceed with introducing the so-called B-calculus for generators of Hilbert 
 space semigroups and generators of holomorphic Banach space semigroups dev
 eloped  recently in collaboration with C. Batty (Oxford) and A. Gomilko (
 Torun). We'll explain its background\, underline its attractive features a
 nd  provide several applications. Being a far-reaching generalisation of 
 the classical Hille-Phillips functional calculus\, the B-calculus appeared
  to be very handy for norm-estimates of various operator functions. This w
 ill be one of the main points of the course.\n\nIf time permits\, we'll di
 scuss a very recent generalisation of the B-calculus due to L. Arnold and
   C. Le Merdy\, and mention a couple of related open problems.\n\n \n\nL
 ecture 1:  A panorama of functional calculi from the bird's eye view. \n\n
 Lecture 2: Basics of operator semigroups\, their generators and related ma
 tters. \n\nLecture 3: Introduction to the B-calculus: motivation\, propert
 ies and related function theory.\n\nLecture 4: Some applications of the B-
 calculus and its further perspectives. \n\n\nBuffon needle (Alexander Volb
 erg)\n\nThe probability of Buffon needle to land near Cantor set. Quantita
 tive Besicovitch theorem. \n\nOne of classical theorems of Besicovitch cl
 aims that all self-similar Cantor sets of dimension 1 on the plane are  i
 rregular in the sense of Besicovitch.  In particular\, this means that th
 e probability of Buffon needle to intersect such a set (conditioned to eve
 nt that it intersected a disc contains this Cantor set) is zero. This imme
 diately  implies that  the probability of Buffon needle to intersect the
  $\\delta$-neighborhood of such Cantor set tends to zero when $\\delta$ te
 nds to zero. But how fast?\n\nThe attempts to clarify this (undertaken wit
 h Michael Bateman\, Matt Bond\, Izabella Łaba\, Fedor Nazarov and Yuval P
 eres in various combinations) reveal the unexpected relations of this ques
 tion to Fourier and Complex analysis\, Combinatorics (including a bit of a
 dditive combinatorics)\, Algebra\, Diophantine equations and Number theory
  in the form of Gelfond—Baker theory initiated by Hilbert’s 7th proble
 m.\n\nI hope to show this plethora of methods that we aimed to proving wha
 t we called ``a power estimate’’\, namely that the above probability i
 s at most $|\\log\\delta|^{-p}$  with some positive $p$. However\, we man
 aged to get such power estimate only if the base of Cantor set is 3 and 4\
 , and\, if the base is at least 5\,  for some Cantor sets with very speci
 al algebraic properties. It is still much better than previous estimates o
 f Peres—Solomyak. The problem turned out to be related to such combinato
 rial questions as: count the number of intersections of diagonals inside a
  regular polygon (Poonen—Rubinstein)\, and such algebraic questions as: 
 describe families of roots of unity\, whose sum vanishes (Rédei\, de Bruj
 in\, Schoenberg\, Mann\, Lam\, Leung).\n\nLecture 1: Estimate from below f
 or the Buffon needle probability of the four-corner Cantor set\n\nLecture 
 2: Fourier analysis for the estimate from above for the Buffon needle prob
 ability of the Cantor-type set\n\nLecture 3: Complex analysis\, Riesz prod
 ucts for the Buffon needle's quasi singular directions\n\nLecture 4: Combi
 natorics: why the measure of quasi singularity directions is rather small\
 n\nLecture 5: Algebra and Number theory: cyclotomic polynomials and Linear
  Multi-Polygon Relations\, Gelfond-Baker theorem and Hilbert's 7th problem
 \n\n\nInstitutions participating in the organization of the event:\n\n\n	L
 eonhard Euler International Mathematical Institute in Saint Petersburg\n\n
 \nhttps://indico.eimi.ru/event/623/
LOCATION:Online
URL:https://indico.eimi.ru/event/623/
END:VEVENT
END:VCALENDAR