Marco, what do you work on in your research?
I work in Geometric Analysis, a field of Mathematics in which ideas and tools typical of Mathematical Analysis, such as the limit, the derivative, or the integral, are used to solve problems concerning objects encountered in Geometry, such as curves or surfaces.
From a historical point of view, what we now call Geometric Analysis has its roots in the theory founded by Gauss and Riemann in the second half of the 1800s. During the 20th century, Gauss’s and Riemann’s geometry underwent many developments in various directions, also because it found several applications, for example in the formulation of the Theory of General Relativity.
Problems in Geometric Analysis very often involve quantities such as the length of a curve, or the area of a surface, or various ways of computing the curvature of a geometric object, all quantities that are defined and treated using the tools of Analysis.
Can you give us an example of a problem studied in Geometric Analysis?
Of course! I can give an example of a problem to which I have devoted quite a lot of time in recent years: the so-called isoperimetric problem. In its most traditional formulation, the problem poses the following question: among all geometric figures in the plane having a given perimeter, which one encloses the greatest area? In practice, if we have a rope of a certain fixed length lying on the floor, we are asking what shape we can give it so that it encloses as much area as possible.
Spoiler… the answer is: a circle!
The isoperimetric problem and its solution are classical topics in the so-called Calculus of Variations, a part of Analysis in which problems concern precisely the maximization or minimization of certain quantities. There are nevertheless many open questions, in particular if one wants to study this problem on a surface that is no longer flat, as the plane is, but is instead curved. The interaction between the isoperimetric problem and the geometry of the space in which it is studied is extremely rich, and not yet completely understood.
Can you tell us something about your educational path? Has Mathematics always been a passion of yours?
Definitely not! In fact, I obtained my bachelor’s degree right here at the Politecnico in Physical Engineering (as a teenager I had rather confused ideas…). Fortunately, during the first years I had some excellent mathematicians as professors, who brought me closer to the subject, in addition to my bachelor’s thesis advisor, who encouraged me greatly to follow my passions. Thus, realizing that Mathematics interested me much more, I moved to Pisa for my master’s degree and PhD, and there I began to study Geometric Analysis and met many of my current collaborators.
The figure of the professor seems important in this story…
It absolutely is! Beyond competence, ideas and the approach that a teacher conveys, even at the university level, can change a student. I believe that teaching is a great responsibility and that we should take very seriously the ideas we pass on to our students.
Do you enjoy teaching? What advice would you give to a student interested in Mathematics?
Yes, I really enjoy teaching! The only advice I would give to a student interested in Mathematics, as well as to any student, is to have fun. That is, to spend one’s time on the subject that truly attracts them, intrigues them, satisfies them, and in the process makes them enjoy themselves.
What are your passions, besides mathematics?
Besides mathematics, I really like going to the mountains. I enjoy card magic tricks, and I play chess, even if I must admit that I am still rather bad at it!