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Enrique Zuazua Expert on Applied Mathematics
There are only 11 years to go but the matter seems complex in view of the eye-watering progress occurring in some fields of Mathematics.
Mathematics has its own challenges on the table, such as the problems highlighted by the Clay Foundation in 2000: the millennium problems. Recently a Russian named Perelman brilliantly solved one of them: the Poincaré conjecture, but others remain that will doubtlessly keep the best mathematicians busy for various generations. It is difficult to predict when and where one of them might be solved. Will it be the Riemann conjecture, or the oneness and regularity of the solutions for the Navier-Stokes equations for viscous fluids in 3 spatial dimensions?
However, independently of what might emerge from the work to solve these singular and emblematic problems, there are different lines of mathematics which are moving forward at a significant pace that will, doubtlessly, change the maths panorama in 2020.
On the one hand, Mathematics is interacting increasingly closely with other disciplines. So, alliances which were already well established in fields of engineering or Physics, for example, will be extended to other disciplines such as neurosciences, in an effort to understand the human brain: the brain as a great calculator and communications network. At the same time, progress made in understanding it will influence our way of using the increasingly impressive calculation capacities offered by supercomputers.
In fact, the emergence of these new gigantic machines will not only allow us to run larger and faster calculations, but also question how we understand computation. It is not enough to use existing programs, merely adapting them, to be able to make use of their full calculation capacity. It is therefore important to rethink our algorithms and this opens up a whole barrage of relevant, complexity-associated questions, which will doubtlessly occupy a major part of mathematicians’ energy in the next few years.
Mathematics will also tackle more and more questions and topics which have not been particularly associated with maths until now, such as Social Sciences. The recent and significant economic crisis will also contribute to this. The fact that, little by little, we will become increasingly aware of the need for a new social organisation, where elements such as justice or happiness take on greater value, also throws up interesting maths-related questions. What are the mathematical models and the analytical tools required to understand how opinion is formed in advanced societies? Which mechanisms determine the movement of human and animal collectives in different scenarios?
These are just some of the questions that will occupy mathematicians in the coming years. The great scientific challenges for humanity, such as understanding the structure of matter and the universes, the exploration of possibilities offered in nanoscale, new medical applications of robotic and control technologies, etc. will doubtlessly guide most of the work done by mathematicians.
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