###### 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