In order to prove a model, you need more than just a computer: you need a mathematician. That’s at least if the researcher – whether engineer, chemist or physicist – wants to be 100% certain of the results. In short, mathematical research is essential to advance many other disciplines, including applied ones such as meteorology or studies into how to make objects invisible. The Aspen Italia website team discussed these issues with Alessio Figalli, full professor at the University of Texas at Austin, where he works in the field of mathematical analysis, involving research into “optimal transport”, “partial differential equations”, and “calculus of variations”.
One of your areas of research is optimal transport. What does that entail, and what applications could it have?
The issue of optimal transport is an easy one to formulate, even if it’s a far from obvious matter from a mathematical point of view. It consists of finding the most economical way to transport a certain mass distribution from one place to another. This field of research had military origins: Gaspard Monge was the first person to pursue it, during the French Revolution and the Napoleonic period, with the aim of finding the most efficient way to transport earth for fortifications.
The problem of optimal transport has been studied for a long time, in part because, from a mathematical perspective, a legitimate question arises as to whether there really is an optimal way of transporting a mass, but it is also particularly interesting because it has had a significant impact in many other areas. One of the various applications, for example, is in the study of weather models. Indeed, under certain circumstances, the distribution of moving cloud particles follows a sort of optimal transport. This enables a predictive model to be developed for the movement of clouds.
What role do mathematicians play in developing models in other disciplines?
Generally, someone – an engineer, chemist, or physicist, for example – gives us a model that we mathematicians study, and from which we draw outcomes. If the model is correct, the mathematical result guarantees what can be expected. This is because a mathematical analysis provides rigorous proof of it. Of course, a model can also be tested with a numerical simulation, that is, by entering the data into a computer. The risk, however, is that in a hundred instances the computer will give a particular response, and in the hundred and first it gives a different answer. If instead a mathematical model is found which justifies a certain outcome, the result will always and only be that, even if the experiment is repeated a thousand times. In a nutshell, a mathematical theorem gives certainty. The issue is one of time: it is obvious that a computer simulation is a thousand times faster. But a mathematical proof offers greater stability and certainty, because it guarantees that research heads in the right direction.
Is it difficult to get funding to do mathematical research?
It depends on the type of research. Currently, many mathematicians and engineers are studying cloaking, that is, ways of making objects invisible. From a mathematical standpoint, the question is a very interesting one. It’s a matter of figuring out how to deflect certain light frequencies so that when they head towards a particular object they pass around it, to then resume their original trajectory – that is, the one they would have followed had that object not been there. As is obvious, this research has very interesting implications, even in military terms. This is why it gets huge funding. The situation is different for non-applied research. My funding, for example, comes entirely from the National Science Foundation, the public body that finances research in all sciences, particularly pure sciences. Unfortunately, even in the US there have been funding cuts. The response, however, has been to select better projects through increased competitiveness, but at the same time giving more worthwhile research the opportunity to continue. This is because in America there is a cultural approach that values science and research, and which, alongside public funding, attaches great importance to and encourages private donors. This is a model that has still to take off in Europe.
How can the research system in Italy be made more competitive?
The Italian system, even with its humanities-based educational structure, is one that I believe still works. I think that the most important thing is to give people a solid educational grounding, and to instill a capacity to reflect and think, and to have a little initiative. At school level, a liberal arts education is great, but kids need to have their curiosity aroused more, making science more accessible to them and explaining to them that disciplines such as mathematics are not boring at all.
I, for example, came to the decision to be a mathematician when I participated in the Mathematical Olympiad as a student of a high school specializing in classical studies. Luckily for me, I enjoyed it.
In fact, I’m convinced of the need for an education system that knows how to bring together and value both the humanities and the sciences. Let’s not forget that having an interdisciplinary grounding is increasingly valued, and that, in universities and research organizations, there is a growing need for people with cross-disciplinary fluency to establish channels of communication between different fields and disciplines. The real problem in Italy is the lack of research funding and merit-based criteria for the allocation of resources. In addition, the lack of rational recruitment planning is helping to fuel a steady brain drain, thereby “robbing” the country of talent. I think these are the important areas to invest in, before the future of Italian universities and research is permanently jeopardized.
Born in Rome on April 2, 1984, Alessio Figalli graduated in mathematics at Pisa’s Scuola Normale Superiore in 2006. In 2007, he received a double PhD from the Scuola Normale Superiore of Pisa and the École Normale Supérieure of Lyon. After working as a researcher at the CNRS (Centre National de la Recherche Scientifique), he became a professor in 2008 at the École Polytechnique in Paris. In September 2009, he transferred as an associate professor to the University of Texas at Austin, where in September 2011 he was promoted to full professor.
