During my PhD, I specialized in the study of partial differential equations, which remain the primary focus of my research today. Although these equations are often derived from physics, my approach is mainly theoretical: the goal is often to prove the existence of solutions and analyze specific properties, such as uniqueness or symmetry. A problem that has fascinated me for several years is the search for solutions of elliptic equations in which the nonlinear term has super-critical growth in the Sobolev sense. The main difficulty in approaching these equations is the lack of Sobolev embeddings: this is an abstraction to the application of classical techniques for the search of solutions. Over the years, I have applied various methods to overcome this obstacle: variational, topological, shooting, and combinations of these techniques. Carrying out numerical simulations has sometimes been useful in gaining insights into the expected results.
Entering the world of research was initially a tough challenge for me, in many respects. The aspect that struck me the most is the similarity between the work of a researcher and that of an artist: both face a "blank canvas," sometimes without knowing exactly what the object of study is, or where to start to tackle it. Research mathematics in many ways does not resemble the mathematics studied in school: it is not a straight and well-defined path but rather a trail full of forks and branches, in which it is easy to get lost.
Fortunately, in the early years of my PhD, I benefited from the support of my supervisor and a close-knit group of colleagues. The community that deals with partial differential equations is very cohesive, and conferences are valuable opportunities to exchange ideas and establish new collaborations. Teamwork has been fundamental to my professional growth, and I still consider collaboration very important: in the work of a researcher, even in mathematics, one plus one makes more than two!
The path to obtaining a permanent position in Italy was not easy. After my PhD, I obtained several scholarships in Italy, France, and Belgium, and later secured a permanent position at the University of Amiens, in France. I spent about seven years abroad, an enriching experience in many ways, but also demanding, especially due to the difficulty of reconciling constant travel with family needs.
Since 2020, I have been working permanently in the Department of Mathematics at Politecnico di Milano, an extremely active and stimulating environment. In addition to research, I am passionately dedicated to teaching, investing a lot of energy in this area. Recently, I have started to take an interest in active learning techniques, and I am introducing the use of software for interaction in large classrooms into my courses.