Mathematics: “substance is in the new methods and views"
Six questions to Miguel Walsh, one of the most promising young mathematicians worldwide.
Mathematics: “substance is in the new methods and views"
Miguel Walsh, 26 years old, has achieved great success in the world of mathematics. As a fellow of CONICET, when he was 24 years old he obtained his Ph.D. at the University of Buenos Aires. His thesis is about discrete inverse theorems. Walsh is currently investigating at the University of Oxford, where he works on two areas: the ergodic theory and the number theory.
In early 2014 the young scientist obtained the Clay Research Fellowship, a distinction of high international recognition supporting the most influential young mathematicians in the world. The works for which Walsh earned the fellowship were conducted entirely in Argentina as part of her studies as a fellow of the Council.
In 2013 he had already been recognized by the Mathematical Congress of the Americas, held in Mexico. Walsh was one of the five winners of the 2013 MCA Award which distinguishes the scientific work of the most outstanding mathematicians of the continent under forty years old.
Besides having published articles of its sole authorship in Annals of Mathematics, Duke Mathematical Journal, and Geometric and Functional Analysis  three prestigious journals  Walsh received praise from colleagues around the world, highlighting both his progress and his innovative methodological proposals, including Terence Tao, winner of the Fields Medal in 2006.
What sparked your initial interest in mathematics?
My original motivation came from trying to solve an open problem whose formulation I was able to understand. Mathematics had never caused me any interest until that moment, and I found the problem in a completely casual manner. After the experience of solving that problem, so different of what I may have suspected in terms of imagining and creating, I found fascinating the possibility of this being my occupation.
What area of research do you work in?What questions do you seek to answer in general terms?
An example of an issue that I have studied is the inverse problem in the number theory. If one chooses a random set of numbers and applies among them the basic arithmetic operations such as addition and multiplication, there are certain properties that are intuitively expected to have all the results, but sometimes it does not occur. Notably, despite being a very natural problem, it has been recently understood that in these situations, and in other more general, all sets for which these exceptions are given have a common structure, which leads to very interesting consequences. Moreover, if other critical arithmetic operation, called modular reduction were considered, it is also possible to find a common structure for all exceptional sets. Here is where I could contribute in this area, demonstrating a general result of this nature. The attractive feature of this phenomenon is that it provides a new perspective regarding basic objects of mathematics, extending the range of situations in which we apply them.
You have received recognition from various institutions and academics worldwide, what would you say is the most prominent aspect of your published work?
Another subject on which I have worked on is the ergodic theory, which broadly examines how systems evolve over time. For this, comes from a very general definition of what is meant by a system, and seeks to understand on the one hand the characteristics of different types of systems that may exist, and on the other hand what are their properties in common. In particular, an important tool to measure how the properties that interest us in these systems evolve is called ergodic average. The ergodic averages give a numerical answer for those properties that we want to measure, and there was considerable interest in understanding for what systems these responses converge to a definite value, instead of indefinitely varying over time. A gratifying aspect of my research was to give a general answer to this question.
Will these advances have a particular application in the future?
It is not easy to determine, because the way in which the purest part of mathematics contributes applications is very unique. Sometimes results are obtained with immediate applications, often ideas are introduced and materialized into concrete progress decades later, but mainly it is a distillation process seeking to understand indepth the most fundamental objects in the logical world, properties and right ways of thinking about them. In mathematics open problems indicate what we do not know, and usually the substance is in the new methods and viewpoints introduced to try to solve them. Finally, it is the evolution in the ways of thinking which ends up being more influential, and its consequences are unpredictable.
What are your next challenges?
Actually, it is difficult to say what is going to be thinking, sometimes even in the short term, because you can not predict what will arouse curiosity. This is particularly true when working on problems connected with different areas. Even if you have defined the problem to be studied, the way to resolve it usually leads to very distant subjects to the original.
What advice would you give to a youngster who wants to start his career as a mathematician?
Perhaps the most important is to find topics you really love and work on them, being the least possible influenced by other circumstances. Nowadays it is a favorable situation thanks to the internet access to most of the research articles, as well as to lecture notes, blogs and video conferences.

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