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SUMMARY:School "Complex analysis and Operator Theory"
DTSTART;VALUE=DATE-TIME:20211129T135000Z
DTEND;VALUE=DATE-TIME:20211203T172500Z
DTSTAMP;VALUE=DATE-TIME:20240725T134606Z
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/
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