Campus Directory

Pronouns he/him

Division Physical & Biological Sciences Division

Department/College/Unit Mathematics Department

Title Dr

Affiliation Faculty

Faculty Type Visiting Scholar

Areas of Expertise Mathematics
Geometric Analysis

Primary Office Location McHenry Library
Office 4192

Mail Stop Mathematics Department

Research Interests

My primary research fields are topology and differential geometry. In particular, I specialize in Riemannian geometry, where I study spaces with a lower bound on their Ricci curvature. 

In the eighties, Gromov showed that sequences of manifolds with Ricci curvature bounded below always have subsequential limits called Ricci limit spaces. In the late nineties, Cheeger and Colding studied extensively such limits. Since then, it became clear that studying Ricci limit spaces is essential to understanding the topology of manifolds with Ricci curvature bounded below. However, a significant difficulty is that the limits of manifolds (in the Gromov-Hausdorff topology) are generally singular. In particular, such objects do not have a well-defined smooth structure or Ricci tensor.

Thanks to extensive research (most notably Lott-Villani, Sturm, and Ambrosio-Gigli-Savaré), a synthetic definition of Ricci curvature lower bound has been created over the last twenty years. This new definition gave birth to RCD spaces (for Riemannian Curvature Dimension), which are (possibly) singular spaces with Ricci curvature bounded below.

The following questions may summarize my current work:

1. What are the topological properties of RCD spaces?
2. What are the topological properties of their moduli spaces?

Teaching Interests

• Differential geometry
• Complex Analysis
• Differential equations
• Topology

Selected Presentations

1. Oct 4, 2023, Ricci curvature lower bounds for singular spaces, UCSC, Geometry and Analysis seminar
2. Feb 2, 2023, Moduli spaces of compact RCD(0,N)-structures, Durham University, Geometry and Topology seminar
3. Dec 13, 2022, Moduli spaces of compact RCD(0,N)-structures, Hausdorff Center for Mathematics
4. Aug 9, 2022, Moduli spaces of compact RCD(0,N)-structures, Casa Matemática Oaxaca (CMO), Workshop on mms with symmetry and Lower Ricci Curvature Bounds
5. Nov 16, 2021, Théorème de localisation de Klartag et généralisation aux espaces RCD, Laboratoire de Mathématiques d’Avignon, ANR CCEM sur Avignon

Selected Publications

1. Mondino & Navarro "Moduli spaces of compact RCD(0,N)-structures." Mathematische Annalen (2023) https://link.springer.com/article/10.1007/s00208-022-02493-7
2. Navarro "Contractibility of moduli spaces of RCD(0,2)-structures." arXiv preprint: 2202.06659 (2023) (To appear at Annales de l'Institut Fourier) https://arxiv.org/abs/2202.06659
3. Navarro, Pan, & Zhu "On the topology of manifolds with nonnegative Ricci curvature and linear volume growth." arXiv preprint: 2410.15488 (2024) https://arxiv.org/abs/2410.15488