Alternatively have a look at the program.

## Elementary theory of prime numbers

Chebychev's Theorem states that the order of magnitude of the prime counting function $\pi(x)$ is $x/\log x$. This result was "the state of the art" until the proof of the Prime Number Theorem by Hadamard and de la Vallée-Poussin in 1895. We shall outline a proof of Chebychev's estimates and deduce from his estimate Mertens' Theorem which provides an asymptotic formula for $\sum_{p\le x}p^{-1}$ as $x\rightarrow\infty$.

## Artin’s conjecture and the ternary Goldbach problem

Fix 3 non-square integers a_1,a_2,a_3, none of which being -1.

I will talk about the problem of representating a large odd integer as a sum of 3 primes p_1,p_2,p_3

such that every p_i has a_i as a primitive root.

Solutions of Diophantine equations with all variables being primes with prescribed primitive roots have not been studied before.

## Square-free values of a polynomial (and the analytic rank of an elliptic curve)

Let f be a primitive separable polynomial of positive degree in Z[x]. It is conjectured that a positive proportion of the

values are square-free. This is known as the square-free conjecture. In 1953, Erdös showed that for cubics there are

infinitely many square-free values, and in 1967, Hooley gave the result about positive density when $\deg=3$. Although

we know that is suffices to check the conjecture for the factors, it is still not know for irreducible polynomials of degree

greater than $3$. In $h$ variables, Greaves proves the conjecture for degree $\leq 3h$.

## An introduction to the Hardy--Littlewood method

We will go through the conditional (under the generalised Riemann hypothesis) proof of Hardy and Littlewood

that every sufficiently large odd integer is the sum of three primes.

## Alfred Tarski’s great algorithm. Decidability of elementary algebra and geometry

On of the most important decidability results is Tarski Theorem. It allows us to determine whether a closed first-order

formula with real variables is true or not. As a corollary we get the decidability of elementary geometry. The original

proof given by Tarski was very involved. After that it was simplified by many researchers. In my talk I am to present

a version of this algorithm which is

- easy to understand;
- easy to implement on a computer;
- very inefficient (compared to modern advanced algorithms).

## p-adic interpolation of modular forms and Galois representations

In a seminal paper, Serre studied p-adic limits of modular forms with the aim of constructing

p-adic L-functions for totally real number fields. Since then, the theme of p-adic interpolation

of automorphic forms has played an important role in many achievements of contemporary

number theory, such as Wiles's proof of Fermat's Last Theorem. I will give a brief overview

of Serre's method and of the results of Hida, Coleman-Mazur, Buzzard and Chenevier in the

construction of p-adic families of modular forms and of their associated Galois representations.

## On the Liouville function in short intervals

We present a brilliant proof due to Matomäki-Radziwiłł of the asymptotic behaviour of Liouville function in short intervals. We also sketch a proof of a more general result concerning a similar problem for a real-valued multiplicative function.

## The Erdős–Kac theorem

the normal distribution with average loglog(n) and variance loglog(n).A proof via the Central Limit

Theorem will be given.

## Schinzel's hypothesis in many variables -- Cancelled --

We will present a generalisation of Schinzel's hypothesis and of the Bateman-Horn's conjecture concerning

prime values of a system of polynomials in one variable to the case of a integer form in many variables.

The proof will rely on Birch's circle method and will be achieved in 25% fewer variables than in the classical

Birch setting. We hope to be able to apply our result to derive the Hasse principle and weak approximation

in cases that are not accessible in the classical Birch's setting. This is joint work with Efthymios Sofos.

## Prime and squarefree values of polynomials in moderately many variables

We will present a generalisation of Schinzel's hypothesis and of the Bateman-Horn's conjecture concerning

prime values of a system of polynomials in one variable to the case of a integer form in many variables.

In particular, we will establish in this talk that a polynomial in moderately many variables takes infinitely

many prime (but also squarefree) values under some necessary assumptions. The proof will rely on Birch's

circle method and will be achieved in 50% fewer variables than in the classical Birch setting. Moreover it

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