## Selected papers

*Topological regularity and stability for noncollapsed spaces with Ricci bounded below*(with E. Bruè and A. Pigati), preprint (2024), [arXiv].*Six dimensional counterexample to the Milnor conjecture*(with. E. Bruè and A. Naber), preprint (2023), [arXiv].*Stability of Tori under Lower Sectional Curvature*, (with E. Bruè and A. Naber), accepted for publication by(2023), [arXiv].**Geom. Topol.***Fundamental Groups and the Milnor Conjecture*, (with E. Bruè and A. Naber), accepted for publication by(2024), [arXiv].**Ann. Math.***Lipschitz continuity and Bochner-Eells-Sampson inequality for harmonic maps from RCD(K,N) to CAT(0) spaces*, (with A. Mondino), accepted for publication by(2023), [arXiv].__Amer. J. Math.__***Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds*, (with G. Antonelli, E. Pasqualetto and M. Pozzetta), accepted for publication by(2023), [arXiv].__Ann. Sci. Éc. Norm. Supér.__*Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds,*(with A. Mondino), accepted for publication by(2023) [arXiv].__Mem. Amer. Math. Soc.,__*The metric measure boundary of spaces with Ricci curvature bounded below,*(with E. Bruè and A. Mondino),**Geom. Funct. Anal.****33**(2023), no. 3, 593-636, [arXiv].*Rectifiability of the reduced boundary for sets of finite perimeter over RCD(K,N) spaces*, (with E. Bruè and E. Pasqualetto),**J. Eur**(JEMS)**. Math. Soc.****25**(2023), no. 2, 413-465, [arXiv]. *Boundary regularity and stability for spaces with Ricci bounded below,*(with E. Bruè and A. Naber),**Invent. math.***228**Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows*, (with E. Bruè) Comm. Pure Appl. Math., 73 (2019), 1141-1204, [arXiv]__.__*Rigidity of the 1-Bakry-Émery inequality and sets of finite perimeter in RCD spaces*, (with L. Ambrosio and E. Bruè), Geom. Funct. Anal. 29 (2019), no. 4, 949-1001, [arXiv].

In [BNS23a] we construct a family of seven-dimensional, complete Riemannian manifolds with nonnegative Ricci curvature and infinitely generated fundamental groups. The examples disprove a long-standing conjecture in Riemannian Geometry raised by Milnor in 1968. They illustrate an unexpected flexibility of Ricci curvature lower bounds. The construction is based on two main tools of independent interest. The first one is a topological method to build manifolds with infinitely generated fundamental groups, based on an inductive gluing construction. The second main point is a careful analysis of the relationship between the connected components of the group of diffeomorphisms Diff(S^3xS^3) and the space of positively Ricci curved Riemannian metrics on S^3xS^3. This tool is fundamental for the gluing steps of the topological construction. The result opens the way to the search for new examples of Riemannian manifolds with constraints on their curvature and for stronger rigidities in low dimensions. More recently in [BNS23c] we managed to lower the dimension of the counterexamples to six. The Milnor Conjecture remains open in dimensions four and five.

In [APPS22] we prove a sharp second-order differential inequality for the isoperimetric profile of complete non-compact Riemannian manifolds with a lower bound on the Ricci curvature. The main result generalizes to non-compact manifolds a classical statement for compact ones originally due to Bavard-Pansu and later improved by Bayle. The main challenge in the non-compact setting is the possibility that isoperimetric regions, i.e., domains minimizing the boundary area for a prescribed value of the enclosed volume, do not exist for all volumes. The key idea of the paper is to combine a generalized existence result for these isoperimetric regions, whose proof is based on the concentration-compactness method, with a series of tools of geometric measure theory in ambient spaces of low regularity, partly developed in the earlier [MS21]. These tools are fundamental to estimating the first and second variation of the area for isoperimetric regions living in the possibly singular ``pointed limits at infinity'' of the complete Riemannian manifold. This seems to be the first application of the RCD theory to the proof of a new sharp geometric inequality on smooth Riemannian manifolds.

In [BMS22] we settle an open question about the existence of infinite geodesics on Alexandrov spaces (i.e., metric spaces where a triangle comparison theorem holds) with empty boundary first raised by Perelman and Petrunin in 1996. The result is obtained by establishing a connection between the geometric notion of boundary and an analytical notion, that was conjectured by Kapovitch-Lytchak-Petrunin in 2017. The main new idea illustrates the strong potential of Analysis techniques in Alexandrov Geometry.

In [BNS20] we develop a boundary regularity and stability theory for manifolds with lower Ricci curvature bounds and lower volume bounds. The results extend the more classical theory, originally due to Cheeger and Colding, that dealt with manifolds with empty boundary. Moreover, they answer several open questions raised in earlier works by De Philippis-Gigli and Kapovitch-Mondino. Broadly speaking, the statements and the methods that we develop fit into the (boundary) regularity theory for solutions of geometric PDEs. From a technical standpoint, the main challenge is to prove that the existence of a single ``almost flat'' boundary point forces the presence of a definite amount of them.

In [BS18] we prove the ``constancy of the dimension'' conjecture for Riemannian metric measure spaces with Ricci curvature bounded from below, the so-called RCD spaces. The conjecture was motivated by an earlier breakthrough by Colding and Naber in 2011 in the case of Gromov-Hausdorff limits of smooth manifolds. The proof requires introducing a series of new tools to study the regularity of flows of Sobolev vector fields.

## Books

*Lectures on Optimal Transport,*(with L. Ambrosio and E. Bruè)

*La Matematica per il 3+2, (130), Springer (2021), 250 pp.*

## Academic works

**PhD thesis**,

*Recent developments about Geometric Analysis on RCD(K,N) spaces.*Advisors: Prof. L. Ambrosio and Prof. A. Mondino.