We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. We also discuss some possible extension to maps of higher genus.
This is joint work with Gwendal Collet and Lukas Klausner[-]

All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao.[-]

Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a random matrix model and by function field considerations.[-]

For a non-principal Dirichlet character $\chi$ modulo $q$, the classical Pólya-Vinogradov inequality asserts that
$M (\chi) := \underset{x}{max}$$| \sum_{n \leq x}$$\chi(n)| = O (\sqrt{q} log$ $q)$.
This was improved to $\sqrt{q} log$ $log$ $q$ by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, this is known to be optimal, owing to an unconditional omega result due to Paley. In this talk, we shall present recent results on higher order character sums. In the first part, we discuss even order characters, in which case we obtain optimal omega results for $M(\chi)$, extending and refining Paley's construction. The second part, joint with Alexander Mangerel, will be devoted to the more interesting case of odd order characters, where we build on previous works of Granville and Soundararajan and of Goldmakher to provide further improvements of the Pólya-Vinogradov and Montgomery-Vaughan bounds in this case. In particular, assuming GRH, we are able to determine the order of magnitude of the maximum of $M(\chi)$, when $\chi$ has odd order $g \geq 3$ and conductor $q$, up to a power of $log_4 q$ (where $log_4$ is the fourth iterated logarithm).[-]

Let $\alpha$ $\epsilon$ $\mathbb{R}^d$ be a vector whose entries $\alpha_1, . . . , \alpha_d$ and $1$ are linearly independent over the rationals. We say that $S \subset \mathbb{T}^d$ is a bounded remainder set for the sequence of irrational rotations $\lbrace n\alpha\rbrace_{n\geqslant1}$ if the discrepancy
$ \sum_{k=1}^{N}1_S (\lbrace k\alpha\rbrace) - N$ $mes(S)$
is bounded in absolute value as $N \to \infty$. In one dimension, Hecke, Ostrowski and Kesten characterized the intervals with this property.
We will discuss the bounded remainder property for sets in higher dimensions. In particular, we will see that parallelotopes spanned by vectors in $\mathbb{Z}\alpha + \mathbb{Z}^d$ have bounded remainder. Moreover, we show that this condition can be established by exploiting a connection between irrational rotation on $\mathbb{T}^d$ and certain cut-and-project sets. If time allows, we will discuss bounded remainder sets for the continuous irrational rotation $\lbrace t \alpha : t$ $\epsilon$ $\mathbb{R}^+\rbrace$ in two dimensions.[-]

It is well known that the every letter $\alpha$ of an automatic sequence $a(n)$ has a logarithmic density -- and it can be decided when this logarithmic density is actually adensity. For example, the letters $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both frequences $1/2$. The purpose of this talk is to present a corresponding result for subsequences of general automatic sequences along primes and squares. This is a far reaching of two breakthroughresults of Mauduit and Rivat from 2009 and 2010, where they solved two conjectures by Gelfond on the densities of $0$ and $1$ of $t(p_n)$ and $t(n^2)$ (where $p_n$ denotes thesequence of primes). More technically, one has to develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. Then asan application one can deduce that the logarithmic densities of any automatic sequence along squares $(n^2){n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, if densities exist then they are (usually) rational. [-]

Automatic sequences and their number theoretic properties have been intensively studied during the last 20 or 30 years. Since automatic sequences are quite regular (they just have linear subword complexity) they are definitely no "quasi-random" sequences. However, the situation changes drastically when one uses proper subsequences, for example the subsequence along primes or squares. It is conjectured that the resulting sequences are normal sequences which could be already proved for the Thue-Morse sequence along the subsequence of squares.
This kind of research is very challenging and was mainly motivated by the Gelfond problems for the sum-of-digits function. In particular during the last few years there was a spectacular progress due to the Fourier analytic method by Mauduit and Rivat. In this talk we survey some of these recent developments. In particular we present a new result on subsequences along primes of so-called invertible automatic sequences.[-]

Given a subset A of an additive group, how small can the sumset $A+A = \lbrace a+a' : a, a' \epsilon$ $A \rbrace$ be ? And what can be said about the structure of $A$ when $A + A$ is very close to the smallest possible size ? The aim of this talk is to partially answer these two questions when A is either a subset of $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{T}$ and to explain how in this problem discrete and continuous setting are linked. This should also illustrate two important principles in additive combinatorics : reduction and rectification.
This talk is partially based on some joint work with Pablo Candela and some other work with Paul Péringuey.[-]

Discrepancy is a measure of equidistribution for sequences of points. We consider here discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts, from a topological dynamics viewpoint. A bounded remainder set is a set which yields bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. Bounded discrepancy provides particularly strong convergence properties of ergodic sums. It is also closely related to the notions of balance in word combinatorics.[-]