THEME I:
Brué, Naber, Semola (Annals of Math, 2025): Milnor conjectured that the fundamental group of a complete Riemannian manifold with Ric ≥ 0 is finitely generated. This paper disproves it with a 7D example having π₁ ≈ ℚ/ℤ.
Kapovitch, Mondino (Geometry & Topology): Studied topological regularity of RCD(K,N) spaces. Defined boundary in this setting and proved stability under Gromov-Hausdorff convergence.
Brué, Naber, Semola (Inventiones Math): Proved that boundaries in noncollapsed RCD(K,N) spaces are (N-1)-rectifiable and homeomorphic to manifolds away from codim-2 singular sets.
Brué, Mondino, Semola (GAFA): Solved a conjecture on metric measure boundaries, proving vanishing measure on RCD(K,N) spaces without boundary. Clarifies structure of infinite geodesics in Alexandrov spaces.
Mondino, Semola (Memoirs AMS): Developed pointwise and viscosity Laplacian bounds on RCD(K,N) spaces. Proved PDE principles for perimeter-minimizing sets and regularity of their boundaries.
Mondino, Semola (AJM): Proved Lipschitz continuity of harmonic maps from RCD(K,N) to CAT(0) spaces. Established synthetic Bochner-Eells-Sampson inequality, generalizing classical smooth results.
THEME II:
Cavalletti, Mondino (Analysis & PDE): Provided precise formulas for the Laplacian of distance functions on RCD(K,N) spaces, capturing both smooth and singular parts.
Ciosmak (JMPA): Extended localization to vector-valued 1-Lipschitz maps. Constructed partitions of ℝⁿ into convex sets and showed conditional log-concavity for log-concave measures. Partial confirmation of Klartag's conjecture.
Ciosmak (Calc. Var. PDE): Developed optimal transport theory for vector measures. Disproved a conjecture by Klartag with a counterexample, but proved it under AC marginals. Generalized Kantorovich duality.
Barilari, Mondino, Rizzi (Memoirs AMS): Proposed a unified synthetic curvature theory covering both Riemannian and sub-Riemannian geometries using gauge metric measure spaces.
Kristaly, Mondino (Proc. LMS): Proved sharp isoperimetric inequality for clamped plate eigenvalues in RCD(0,N) spaces (close to N=2,3). Analyzed rigidity and stability using volume ratios and Bessel functions.
THEME III:
Mondino, Suhr (JEMS): Formulated Einstein equations via optimal transport. Linked Ricci curvature, energy-momentum tensor, and cosmological constant through entropy convexity.
Cavalletti, Mondino (Cambridge J. Math): Developed optimal transport theory in Lorentzian synthetic spaces. Defined timelike Ricci lower bounds and extended Hawking singularity theorem to Lorentzian length spaces.
Cavalletti, Manini, Mondino (Comm. Math. Phys): Studied optimal transport on null hypersurfaces. Gave a transport-based characterization of the Null Energy Condition. Proved sharp and rigid versions of Hawking area theorem.
Cavalletti, Manini, Mondino (preprint): Introduced synthetic null hypersurfaces. Defined entropy-based null energy condition stable under convergence. Extended Penrose singularity theorem to continuous spacetimes.
Mondino, Saemann (preprint): Defined Lorentzian Gromov-Hausdorff convergence via causal diamonds. Proved a compactness theorem and connected it to causal set theory.
These studies collectively advance the synthetic treatment of curvature, topology, and boundary regularity in both Riemannian and Lorentzian, smooth and non-smooth settings. They merge deep insights from geometric analysis, PDEs, and relativity through the lens of optimal transport and metric measure spaces.
