## Gunther Cornelissen - Utrecht University, The Netherlands

### Arithmetic equivalence

An irreducible polynomial over the rational numbers can be reducible or not modulo prime numbers; think of $x^2+1$, which is reducible modulo an odd prime p exactly when 4 divides p-1. Around 1880, L. Kronecker studied the question in how far a polynomial with integer coefficients is determined by its splitting behaviour modulo prime numbers. In modern language, this is the question whether a number field (a finite field extension of the rational numbers) is characterized uniquely by splitting behaviour of primes, viz. by its Dedekind zeta function. The problem was studied by M. Bauer and F. Gassmann in the early 20th century. Gassmann used Galois theory to give a group theoretical characterisation of “arithmetic equivalence” of two number fields, meaning they have the same zeta function; thus allowing him to construct the first examples of such non-isomorphic fields.

In these lectures, we will recall of these results in modern language, and explain some more recent results that invoke more group representation theory to construct for every number field a Dirichlet L-series that uniquely characterizes it (joint work with de Smit, Smit, Li and Marcolli): the arithmetic equivalence problem is resolved by using moderate (i.e. only “abelian”) information.

Gassmann’s result experienced a striking revival when T. Sunada realised it could be used in Riemannian geometry to construct isospectral, non-isometric manifolds. Time permitting, we will also show this link, and what the analogue of L-series is, resolving the isospectrality problem (joint work with Peyerimhoff).

## Christian Elsholtz - Graz University of Technology, Austria

### Multiplicatively defined sets with additive structure.

The set of sums of two squares, $x^2+y^2$ (with $x,y$ integers), has a well known multiplicative description. There is an algebraic reason explaining the situation. Apart from such sets, an arbitrary sumset does not normally have a multiplicative description, and a multiplicatively defined set, such as the set of squares or the set of primes should not be a sumset. In the case of primes (even with finitely many deviations allowed) this is an open problem!

One can try to find large additive structures (so called Hilbert cubes) of type $a_0+{0,a_1}+ \cdots + {0,a_d}$ in mutiplicative sets. For the set of squares it is conjectured that the maximal dimnesion $d$ is bounded.

We will study such problems in the three lectures. The methods involved will include methods from additive combinatorics and analytic number theory, such as sumset growth and the large sieve.

## Vítězslav Kala - Charles University, Czech Republic

### Universal quadratic forms and indecomposables in number fields

A quadratic form with Z-coefficients is universal if it is positive definite and represents all positive integers. The most famous example is the sum of four squares that was shown to be universal by Lagrange around 1790. More than 200 years later (around 2005), Bhargava and Hanke proved the powerful 290-theorem that "a positive definite quadratic form is universal iff it represents 1, 2, 3, ..., 290". While this result mostly finished the study of universal forms over Z, the analogous questions over (totally real) number fields remain widely open.

In this series of lectures, I will explain the main recent results in the area along with the fundamental tools for proving them - as well as a number of open problems (some of which should be quite accessible and hopefully also solvable!).

Specifically, in real quadratic fields Q(sqrt D), we have a basic understanding of the required number of variables of a universal form. This number is related to the continued fraction for sqrt D via the key notion of an indecomposable algebraic integer. While we know much less about universal forms and indecomposables in higher degrees, trace 1 elements in the codifferent turned out to be a useful tool that we'll also discuss in the talks. Overall our tools will be primarily algebraic, but we will also encounter analytic objects such as the class number formula and Dedekind zeta function.

## Jakub Konieczny - University of Lyon, France

### Generalised polynomials and applications

Generalised polynomials are expressions built up from ordinary polynomials using the floor function, addition and multiplication, such as $\sqrt{2}n [\sqrt{3} n^2] + \sqrt{6}n^3$. In contrast to ordinary polynomials, generalised polynomials can be bounded or even finitely-valued without being constant. Sturmian sequences are a notable example of a class of {0,1}-valued generalised polynomial sequences. Another, less obvious, example is the characteristic function of the Fibonacci numbers. Various properties of bounded generalised polynomials have long been studied. In fact, one can interpret the classical equidistribution theorems of Kronecker and Weyl as statements about generalised polynomials of a special form. More recently, I. Håland Knutson obtained equidistribution results for several wide families of bounded generalised polynomials. Later, V. Bergelson and A. Leibman established a connection with dynamics on nilmanifolds which lead to a complete description of possible distributions of bounded generalised polynomial sequences, a resolution of the word problem (up to equality almost everywhere), as well as to other interesting results. In joint work with J. Byszewski, we studied sparse generalised polynomials, that is, generalised polynomials that vanish almost everywhere. We obtained a number of surprising examples as well as criteria which can be used to show that a given sequence is not given by a generalised polynomial formula. In the lecture series, we will discuss the results mentioned above, as well as some other developments. The techniques will involve diophantine approximation, ergodic theory and (semi-)algebraic geometry, as well as purely elementary methods.

## Bonus Lecture: Dominik Kwietniak - Jagiellonian University, Poland

### Sarnak’s conjecture implies the Chowla conjecture along a subsequence

Ergodic methods have proven remarkably fruitful in understanding some number theoretic and combinatorial problems, cf. the ergodic theoretic proof of Szemerédi’s theorem by Furstenberg, and more recent work of Frantzikinakis and Host. Sarnak was the first to note that one can study arithmetic properties of multiplicative functions through the so-called Furstenberg systems. In particular, ergodic theory helps us to deal with the Sarnak and Chowla conjectures. For example, Frantzikinakis-Host's theorem mentioned above establishes the validity of logarithmic Sarnak's conjecture for systems with not too many ergodic measures.

Let  $\lambda$ be the Liouville function, that is, $\lambda(n)$ is equal $1$ when $n$ is the product of an even number of primes, and $-1$ when $n$ is the product of an odd number of primes. Here, the number of primes is counted with multiplicity. The Chowla conjecture asserts that  the statistics of sign patterns appearing in the sequence $\lambda(n)$ is indistinguishable from a sign pattern of a typical random sequence with entries $\pm 1$. In recent years, it has been realised that one can make more progress on Chowla's conjecture if one works with the logarithmically averaged version of the conjecture. It is not difficult to show that the Chowla conjecture implies the logarithmically averaged Chowla's conjecture.

The main aim of my lecture will be the of the following result: If the logarithmically averaged Chowla conjecture is true, then there exists a sequence going to infinity such that the Chowla conjecture is true along that sequence. The result, obtained in a joint work with Mariusz Lemańczyk and Alexander Gomilko, is quite surprising in view of this emerging consensus that the logarithmically averaged Chowla conjecture is easier than the ordinary Chowla conjecture. The proof should be accessible to anyone with a modest knowledge of topology and basic measure theory.