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.