Alberto Roncoroni is a researcher at the Department of Mathematics of the Politecnico di Milano.
Who are you and what do you do?
My name is Alberto Roncoroni and I am a tenure-track researcher (RTT) at the Department of Mathematics of the Politecnico di Milano.
I graduated in Mathematics from the University of Insubria and obtained my PhD in 2019 at the University of Pavia. After my PhD, I carried out research as a research fellow and postdoc at the University of Florence and then at the University of Granada, in Spain.
Since 2022, I have been at the Politecnico.
I work in geometric analysis, a field that brings together differential geometry and the study of partial differential equations. In particular, I am interested in understanding how the geometric properties of a space (such as curvature and symmetries) influence the solutions of certain equations, and vice versa.
Geometric analysis… it’s a fascinating but somewhat mysterious name. What exactly is it?
It is an area of mathematics in which analytical tools are used to answer geometric questions, and vice versa.
For example: which surfaces minimize area in a given space? What shape must the solutions of a certain equation have if the space has positive curvature? What can we say about the global structure of a space knowing only its local geometry?
A now classical achievement, which originated in this field, is the proof of one of the Millennium Prize Problems: the Poincaré conjecture by Grigori Perelman in the early 2000s. It was a topological question (“when is a three-dimensional manifold a sphere?”), but the solution came through a refined analysis of geometric flows, in particular the Ricci flow, a parabolic equation that “deforms” the geometry of a manifold over time.
And how does your work connect to this type of problem?
Of course, I am not working on the Poincaré conjecture (someone already got there first!), but I work on problems that are very similar in spirit and approach.
I study the behavior of geometric equations on manifolds, especially in contexts where the curvature or topology of the space strongly influences the solutions. We often ask when a solution is unique, or when it must have symmetry properties.
In some cases, we prove that the only possible solutions are the “simplest” or “canonical” ones, somewhat like in the Poincaré conjecture, where geometry imposes a very rigid classification of shapes.
How do you see geometric analysis within modern mathematics?
It is a rapidly evolving and very active field, also because of its natural interdisciplinarity.
On the one hand, it is fundamental: many ideas in general relativity, field theory, and optimization originate here. On the other hand, it is creative: it pushes you to think “with the mind of a geometer” but also “with the hands of an analyst.”
For me, it is one of the most fertile areas of contemporary mathematics.
A piece of advice for those who want to approach research in this field?
Always start from problems, not from tools: a good question is worth more than a thousand techniques.
Of course, one must build a solid “toolbox” of analytical and geometric methods.
And above all: do not rush. Mathematics needs time to mature, but when it reveals itself, it is surprisingly beautiful.
Do you teach? Do you enjoy it?
Of course! I mainly teach Geometry and Linear Algebra to Engineering students. It is one of the first subjects they encounter, so I like to convey from the very beginning the beauty (and usefulness!) of mathematics, strictly at the blackboard.
It is very rewarding when I manage to “unlock” an intuition in students, or when what they learn in my courses proves useful in other contexts.
Useful contacts if I want to know more about you?
My webpage is https://sites.google.com/view/albertoroncoroni.