Connection to the school’s zoom and a greeting word from organizers

Additive combinatorics and function theory

• Ilya Shkredov

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

Bounded functional calculi for unbounded operators

• Yuriy Tomilov

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

Buffon needle probability of intersecting a small neighborhood of Cantor set

• Alexander Volberg

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

Additive combinatorics and function theory

• Ilya Shkredov

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

Bounded functional calculi for unbounded operators

• Yuriy Tomilov

Basics of operator semigroups, their generators and related matters.

Buffon needle probability of intersecting a small neighborhood of Cantor set

• Alexander Volberg

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

Additive combinatorics and function theory

• Ilya Shkredov

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

Bounded functional calculi for unbounded operators

• Yuriy Tomilov

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

Buffon needle probability of intersecting a small neighborhood of Cantor set

• Alexander Volberg

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

Winston Heap : Conditional mean values of long Dirichlet polynomials.

• Winston Heap

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.

Pavel Gubkin: On asymptotic boundary behaviour of orthogonal polynomials and Dirac's eigenfunctions

• Pavel Gubkin

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.

Buffon needle probability of intersecting a small neighborhood of Cantor set

• Alexander Volberg

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

Additive combinatorics and function theory

• Ilya Shkredov

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

Bounded functional calculi for unbounded operators

• Yuriy Tomilov

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

Buffon needle probability of intersecting a small neighborhood of Cantor set

• Alexander Volberg

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