Research

My main research interests lie at the intersection of Optimal Transport, Non-Smooth analysis, and Calculus of Variations.

Unbalanced Optimal Transport

I am interested in the generalization of the classical (i.e. balanced) Optimal Transport problem to the broader setting of pairs of measures which may have different total non-negative masses. Together with Giuseppe Savaré we have studied in [SS24] a very general approach to the class of Unbalanced Optimal Transport problems, providing a very natural framework where the familiar primal-dual formulation of the classical case is available in suitably generalized forms, among many other relevant properties. The approach we have used is novel also in the classical balanced setting, and has proved to be fruitful also in that case [SS22].

The most relevant cost in this framework is the celebrated Hellinger-Kantorovich (HK) distance. Currently I am working in collaboration with Nicolò De Ponti and Luca Tamanini to better understand the interplay between the HK distance and the classical Wasserstein and Hellinger distances [DST25].

Calculus on metric spaces and spaces of measures

Part of my research deals with the study of Sobolev and BV spaces on metric-measure spaces. I have worked with Massimo Fornasier and Giuseppe Savaré (and I am currently working with Enrico Pasqualetto, see [PS25]) to provide general criteria for the density of smooth subalgebras of continuous functions in these spaces, both in general and when the base space is the Wasserstein space of probability measures or the Hellinger-Kantorovich space of non-negative measures, see [FSS23], [S23] and [DS25]. In these latter cases, the algebra of smooth cylinder functions plays a key role and turns out to be strongly dense in the metric Sobolev space.

This, besides being interesting from the theoretical point of view, also has natural applied consequences: since cylinder functions are easily computable, one can use them to approximate numerically functions defined on measures, as we have shown with Massimo Fornasier and Pascal Heid in [FHS25].

Geometry and evolutions in spaces of measures

The metric-Sobolev approach to the differentiability of functions defined on measures is not the only way to define notions of Dirichlet energy and gradients for functions on (probability) measures: there is an (a priori completely unrelated) approach based on Dirichlet forms and more geometric-related notions of gradients for smooth functions. In collaboration with Lorenzo Dello Schiavo we have proved in [DS25] that, for suitable choices of the reference measure, these approaches are the same, providing a unified setting to the study of the geometric/differential structure of spaces of measures. Currently, we are working to better understand the related diffusion process that could rightfully be considered the Brownian motion on the space of measures.

I am also collaborating with François Delarue and Mattia Martini in the study of PDEs on the space of probability measures, and the generalization of Feynman–Kac formulas therein.

Together with Giulia Cavagnari and Giuseppe Savaré we have studied the generalization of maximal monotone evolutions to the Wasserstein space of probability measures, see [CSS23] and [CSS23b]. We have studied a large class of operators acting on probability measures, far beyond the well-known framework of Wasserstein gradient flows, providing a robust theory which includes the latter as a particular case. This has also led to the discovery of interesting properties of geodesically convex functionals which are approximable by discrete measures, and also to the study of monotone operators in Hilbert and Banach spaces which are invariant by groups of isometries [CSS25].

In a separate collaboration with Ulisse Stefanelli and Goro Akagi, we are investigating doubly non-linear fractional gradient flows in Hilbert spaces. These are evolution equations of gradient flow type, where the time derivative is replaced by its fractional counterpart and nonlinearities appear both inside and outside the time operator.

Together with Ulisse Stefanelli and Pierre-Cyril Aubin-Frankowski, I am also working on gradient flow evolutions in metric spaces—or more generally, in spaces where the distance function fails to satisfy the triangle inequality. The idea is to adapt the metric structure to the functional being minimized, allowing flexibility in the choice of pseudo-distances while still ensuring convergence of the minimizing movement scheme to actual minimizers.