The Riemann zeta function:
\zeta(s):=\sum_{n=1}^{\infty}\frac{1}{n^s}
was first defined by L. Euler and happens to be one of the central functions in mathematics for centuries. Remarkable that Euler managed to compute the values of this function (even zeta values) at all even numbers:
\zeta(2)=\frac{\pi^2}{6},\quad \zeta(4)=\frac{\pi^4}{90}, \quad \zeta(6)=\frac{\pi^6}{945},\dots
While these nice formulas (proved in 1741) tell us a lot about the structure of even zeta values (all even zeta values are rational multiples of \pi and hence all \zeta(2k) are transcendental numbers) the properties of odd zeta values remain completely mysterious. So far we even do not know if \zeta(3) is transcendental or not!
In my talk, I am going to explain the fundamental Transcendence conjecture which (roughly) states that all odd zeta values are distinct and transcendental numbers and the refinement of this conjecture by D. Zagier. Then following P. Deligne, V. Drinfeld, M. Kontsevich, I will explain how algebraic geometry helps to handle this conjecture and the recent progress of F. Brown towards the proof of it.