Adam Kanigowski, Sparse equidistribution in dynamical systems.
Lectures 1-3. Let \( (X, T) \) be a topological dynamical system, \( (a_n)_{n \in \mathbb{N}} \) a bounded sequence of real (or complex) numbers. For \( x \in X \) and \( f \in C(X) \) we consider: \[ \frac{1}{\sum_{n\leq N} |a_n|} \sum_{n\leq N} a_n \cdot f(T^n x).\] A classical example considered in dynamics is to take \( a_n = 1 \) in which case we are considering classical ergodic averages. In these talks we will be mostly focusing on the case where \( (a_n) \) comes from a sparse (but structured) subset of natural numbers. The main example is \( a_n = 1 \) if \( n \) is a prime number and \( a_n = 0 \) otherwise or, more generally, for \( k \geq 2 \), \( a_n = 1 \) if \( n \) is a product of at most \( k \) primes and \( a_n = 0 \) otherwise. We will also discuss the case of multiplicative sequences, i.e. when \( (a_n) \) is the Möbius function or the Liouville function. We will first discuss some classical results. Next we will focus on some recent progress – here has been a significant progress on these questions in the past 15 years. Finally we plan to discuss some open questions.

Mariusz Mirek, On recent developments in pointwise ergodic theory
Lectures 1-3. This will be a survey talk about recent progress on pointwise convergence problems for classical and multiple ergodic averages along polynomial orbits. Relations with number theory and additive combinatorics will be also discussed.

Slides: 1 2 3

Maksym Radziwiłł, Recent developments in analytic number theory
Lecture 1. Introduction to exponential sums and their applications in number theory. Specifically I will cover applications to Waring's problem and to counting points on varieties. I will also discuss applications to other problems.

Lecture 2. I will discuss the idea behind Vinogradov's method for exponential sums in the context of a toy problem. Specifically we will prove Karatsuba's result bounding exponential sums with phase \( \bar{x} / p \) where \( \bar{x} \) denotes the inverse of \( x \) modulo \( p \). As a result this will allow us to assert that the inverses of \( x \) modulo \( p \) become roughly equidistributed modulo \( p \) as soon as \( x \) traverses just a few values in any given interval.

Lecture 3. I will introduce Vinogradov's method in the context of polynomial phases (this is relevant to Waring's problem and to bounds for the Riemann zeta-function). I will show how the problem of bounding these exponential sums is reduced to a counting problem known as the Vinogradov mean-value theorem. I will begin the proof of the sharp bounds for Vinogradov's mean-value theorem following Bourgain, Demeter and Guth.

Lecture 4. I will conclude the proof of sharp bounds for Vinogradov's mean-value theorem. We will follow the exposition Guo-Li-Yung-Zorin-Kranich