Uncharted waters

Mathematics is a very lively science, and the blood that animates it are the problems whose solution we still do not know. This uncharted waters have a very diverse geography: some regions are located very deep inside mathematics themselves, in their purest territories, but they also border on other sciences, or on technology – the Internet, for instance, is today an important source of problems for the world of mathematics. We will dedicate this section to recent mathematical discoveries, and also to exploring the uncharted waters of the problems that are still to be solved.

Painlevé conjecture

28 May, 2021 Juan Arias de Reyna0

There was a time when Newtonian mechanics was an absolute truth. Its greatest success, the explanation of planetary motions. It is when the hypothesis of a providential God is dropped that the natural question arises: […]

We are happy being poor: Erdös-Faber-Lovász’s Conjecture

30 April, 2021 Juan Arias de Reyna0

Combinatorial problems, as Number Theory ones, are easy to understand by everyone. They can even be more direct. Paul Erdös was very fond of combinatorial problems. He had a special gift for posing problems. Today […]

Mind the gap: the Mochizuki controversy

6 April, 2021 José María Tornero2

If we were to carry out a survey about what is most characteristic of Mathematics, what distinguishes it most precisely from other branches of human knowledge, we would most probably find that the concept of […]

A twelve-second flight: towards Langlands’ conjectures

12 March, 2021 José María Tornero0

Laurent Fargues (CNRS, France) and Peter Scholze (University of Bonn, Germany) made a preprint entitled Geometrization of the local Langlands correspondence available to the mathematical community on 26 February. This title announces what had been […]

Floating bodies

1 March, 2021 Juan Arias de Reyna0

Archimedes on floating bodies One of Archimedes’ works deals with floating bodies. It came as a great surprise to me when I read the second proposition of this work of Archimedes: Proposition 2. The surface […]

2020, a great year

29 January, 2021 Juan Arias de Reyna0

I mean for mathematics. I have been reading arXiv entries related to my activity on a daily basis for many years. This year I have noticed an explosion of articles. Confinement has made mathematicians work […]

The squaring of the circle

11 December, 2020 Juan Arias de Reyna0

Problemas clásicos How envious of the Greeks of classical times! Their gods posed mathematical problems, as when the Delphic oracle asked them to double the cubic altar to appease the plagues. We have here a […]

Frogs, birds… and Riemann Hypothesis

13 November, 2020 Juan Arias de Reyna4

Quasicrystals Do quasicrystals, the physics of the atomic nucleus, the zeros of the zeta function, crystalline measurements and the Fourier transform have anything in common? Well, a lot, as you will find out if you […]

Sums and products of different colours

16 October, 2020 Juan Arias de Reyna0

Can we colourize the natural numbers with a finite number of colours such that, for all \((a,b)\ne(2,2)\), the colour of the sum \(a+b\) is different from the colour of the product \(ab\)? (The exception is […]

Torsion in elliptic curves on numerical fields (explained for you)

14 September, 2020 José María Tornero0

Elliptic curves are, without doubt, one of the most captivating objects in the mathematical universe. They combine two features that rarely appear together: they are simple as objects (that is, they are simple to understand… […]

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