School "Complex analysis and Operator Theory"

29 Nov 2021, 16:50 → 3 Dec 2021, 20:25 Europe/Moscow

Online

‎Complex Analysis, Combinatorics and Operator Theory

November 29 – December 3, 2021

An 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 mathematical specialty.

The lecture notes and exercises are below on this page

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Lecturers:

Ilya Shkredov Institute of mathematics, RAS

Yuriy Tomilov Institute of mathematics, PAN

Alexander Volberg Michigan State University

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Courses:

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Additive combinatorics and analysis (Ilya Shkredov)

Additive combinatorics is an actively developing area of math, that lies at the intersection of Number theory and Combinatorics. It also involves extensive use of techniques from Analysis, Graph theory, Ergodic theory, Probability, Algebraic geometry, Topology and Geometry of numbers.

In this course we discuss classic applications (Lectures 1, 2, 4) of functional-theoretic methods to additive combinatorics (setting aside arithmetic combinatorics and non-commutative approaches), and, vice versa, application of additive combinatorics to some questions from function theory (Lecture 3).

 

Lecture 1: Additive combinatorics: discrete Fourier transform, Roth theorem, sets without solutions of linear equations

Lecture 2: Additive combinatorics: Gowers uniformity norms, corners problem, transference principle (for primes and Sidon sets)

Lecture 3: Additive combinatorics: integer sets with small Wiener norm on integers, Littlewood problem, additive energies, additive dimension and dissociativity

Lecture 4: Additive combinatorics: almost periodicity of convolutions, Croot-Sisask theorem, applications to linear equations and sets with small doubling

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Bounded functional calculi for unbounded operators (Yuriy Tomilov)

Functional calculi is a classical subject of operator theory, unifying such distant areas as complex analysis, harmonic analysis and PDE. Several breakthroughs in analysis, e.g.  solving the famous Kato square root problem or obtaining the L^2-boundedness of Cauchy integrals on Lipschitz curves, depend on the functional calculi theory. Moreover, the theory became indispensable nowadays for a number of applied problems, stemming mostly from evolution equations.

 

Roughly, 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 function theory. The major issue is how to provide a good ``control'' of f(A) given A and f,  and we just touch this problem putting aside a number of modern developments around it. 

 

We'll start the course with a short overview of the theory of  functional calculi, putting an emphasis on generators of strongly continuous operator semigroups. This class of operators is of major importance for the study of partial differential equations, and it will be the main object of our studies as well. Then we'll proceed with introducing the so-called B-calculus for generators of Hilbert space semigroups and generators of holomorphic Banach space semigroups developed  recently in collaboration with C. Batty (Oxford) and A. Gomilko (Torun). We'll explain its background, underline its attractive features and  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 will be one of the main points of the course.

If time permits, we'll discuss a very recent generalisation of the B-calculus due to L. Arnold and  C. Le Merdy, and mention a couple of related open problems.

 

Lecture 1: A panorama of functional calculi from the bird's eye view.

Lecture 2: Basics of operator semigroups, their generators and related matters.

Lecture 3: Introduction to the B-calculus: motivation, properties and related function theory.

Lecture 4: Some applications of the B-calculus and its further perspectives.

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Buffon needle (Alexander Volberg)

The probability of Buffon needle to land near Cantor set. Quantitative Besicovitch theorem. 

One of classical theorems of Besicovitch claims that all self-similar Cantor sets of dimension 1 on the plane are  irregular in the sense of Besicovitch.  In particular, this means that the probability of Buffon needle to intersect such a set (conditioned to event that it intersected a disc contains this Cantor set) is zero. This immediately  implies that  the probability of Buffon needle to intersect the $\delta$-neighborhood of such Cantor set tends to zero when $\delta$ tends to zero. But how fast?

The attempts to clarify this (undertaken with Michael Bateman, Matt Bond, Izabella Łaba, Fedor Nazarov and Yuval Peres in various combinations) reveal the unexpected relations of this question to Fourier and Complex analysis, Combinatorics (including a bit of additive combinatorics), Algebra, Diophantine equations and Number theory in the form of Gelfond—Baker theory initiated by Hilbert’s 7th problem.

I hope to show this plethora of methods that we aimed to proving what we called ``a power estimate’’, namely that the above probability is at most $|\log\delta|^{-p}$  with some positive $p$. However, we managed 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 special algebraic properties. It is still much better than previous estimates of Peres—Solomyak. The problem turned out to be related to such combinatorial 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 Brujin, Schoenberg, Mann, Lam, Leung).

Lecture 1: Estimate from below for the Buffon needle probability of the four-corner Cantor set

Lecture 2: Fourier analysis for the estimate from above for the Buffon needle probability of the Cantor-type set

Lecture 3: Complex analysis, Riesz products for the Buffon needle's quasi singular directions

Lecture 4: Combinatorics: why the measure of quasi singularity directions is rather small

Lecture 5: Algebra and Number theory: cyclotomic polynomials and Linear Multi-Polygon Relations, Gelfond-Baker theorem and Hilbert's 7th problem

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Institutions participating in the organization of the event:

• Leonhard Euler International Mathematical Institute in Saint Petersburg

All lectures, cloud link Friday lectures, temporary zoom link. Passcode: V.h4ZGZZ Monday lectures, temporary zoom link. Passcode: C#b4t.x$ Thursday talks and lectures, temporary zoom link, Passcode: g55%ZPQ$ Tuesday lectures, temporary zoom link. Passcode: hr.Rj4a+ Wednesday lectures, temporary zoom link. Passcode: Tbv3nW#9

Buffon needle probability of intersecting a small neighborhood of Cantor set

• Needle Lecture 1.pdf
• Needle Lecture 2 (2).pdf
• Needle Lecture 3.pdf
• Needle Lecture 4.pdf
• Needle Lecture 5 (3).pdf

Monday, 29 November

16:50 → 17:00 Connection to the school’s zoom and a greeting word from organizers

17:00 → 18:00 Additive combinatorics and function theory

Additive combinatorics: discrete Fourier transform, Roth theorem, sets without solutions of linear equations

Speaker: Prof. Ilya Shkredov

18:05 → 19:05 Bounded functional calculi for unbounded operators

A panorama of functional calculi from the bird's eye view.

Speaker: Prof. Yuriy Tomilov

19:10 → 20:10 Buffon needle probability of intersecting a small neighborhood of Cantor set

Estimate from below for the Buffon needle probability of the four-corner Cantor set

Speaker: Prof. Alexander Volberg

Tuesday, 30 November

17:00 → 18:00 Additive combinatorics and function theory

Additive combinatorics: Gowers uniformity norms, corners problem, transference principle (for primes and Sidon sets)

Speaker: Prof. Ilya Shkredov

18:05 → 19:05 Bounded functional calculi for unbounded operators

Basics of operator semigroups, their generators and related matters.

Speaker: Prof. Yuriy Tomilov

19:10 → 20:10 Buffon needle probability of intersecting a small neighborhood of Cantor set

Fourier analysis for the estimate from above for the Buffon needle probability of the four-corner Cantor set

Speaker: Prof. Alexander Volberg

Wednesday, 1 December

17:00 → 18:00 Additive combinatorics and function theory

Additive combinatorics: integer sets with small Wiener norm on integers, Littlewood problem, additive energies, additive dimension and dissociativity

Speaker: Prof. Ilya Shkredov

18:05 → 19:05 Bounded functional calculi for unbounded operators

Introduction to the B-calculus: motivation, properties and related function theory.

Speaker: Prof. Yuriy Tomilov

19:10 → 20:10 Buffon needle probability of intersecting a small neighborhood of Cantor set

Complex analysis, Riesz products for the Buffon needle's quasi singular directions

Speaker: Prof. Alexander Volberg

Thursday, 2 December

17:00 → 17:55 Winston Heap : Conditional mean values of long Dirichlet polynomials.

Mean values of Dirichlet polynomials play an important role in analytic number theory. When the length of the sum exceeds the length of the integral they pose a significant challenge requiring techniques from number theory and harmonic analysis. The mean values of these "long" Dirichlet polynomials appear in many important open problems in number theory. In this talk we survey some of these problems and then describe how, conditionally on the Riemann hypothesis, one can compute asymptotics for the moments of long Dirichlet polynomials over primes provided they are sufficiently weighted. We also give some applications to large deviations.

Speaker: Winston Heap

18:05 → 18:50 Pavel Gubkin: On asymptotic boundary behaviour of orthogonal polynomials and Dirac's eigenfunctions

The talk will consist of two parts. We start with the discussion of an asymptotic boundary behaviour of orthogonal polynomials on the unit circle and formulate a theorem by A. Mate, P. Nevai and V. Totik on average almost everywhere convergence. Then we discuss how the techniques from this area can be used in studying the differential operators of Dirac type.

Speaker: Pavel Gubkin

19:10 → 20:10 Buffon needle probability of intersecting a small neighborhood of Cantor set

Combinatorics: why the measure of quasi singularity directions is rather small

Speaker: Prof. Alexander Volberg

Friday, 3 December

17:00 → 18:00 Additive combinatorics and function theory

Additive combinatorics: almost periodicity of convolutions, Croot-Sisask theorem, applications to linear equations and sets with small doubling

Speaker: Prof. Ilya Shkredov

18:05 → 19:05 Bounded functional calculi for unbounded operators

Some applications of the B-calculus and its further perspectives.

Speaker: Prof. Yuriy Tomilov

19:10 → 20:25 Buffon needle probability of intersecting a small neighborhood of Cantor set

Algebra and Number theory: cyclotomic polynomials and Linear Multi-Polygon Relations, Gelfond--Baker theorem and Hilbert's 7th problem

Speaker: Prof. Alexander Volberg