Beukers' proofs of irrationality of $\zeta(3)$ and Brown's program of irrationality proofs for zeta values.

In 1978, Roger Apery produced a sensation when he proved the irrationality of $\zeta(3)$. Later, Frits Beukers gave two reinterprations of Apery's proof; the first one uses iterated integrals and Legendre polynomials and the second one uses modular forms. In this talk, we shall present these two reinterpretations and we also give an account into Brown's program of irrationality proofs for zeta values. Brown's main idea is a common geometric framework where period integrals play a special role in understanding these irrationality proofs. It was proved by Brown that these integrals are $\mathbb{Q}$-linear combinations of multiple zeta values of a given weight.