The millennium problems. Challenges to further mathematics

Abstract

Although it may seem surprising, the field of mathematics experiences constant advances. Each year, new results are published using original concepts and methods that have a relevant impact on the following years. We must note that some of these new results are obtained with the solution to previously raised issues. Some of these problems are studied for many years, so many, in fact, that they end up becoming notorious. For instance, we can highlight three well-known ruler-and-compass constructions from ancient times (doubling the cube, trisecting the angle and squaring the circle); we have only known they are unsolvable since the nineteenth century.