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he/him

Physical & Biological Sciences Division

Mathematics Department

Dr

Faculty

Mathematics

Geometric Analysis

McHenry Library

Office 4192

Mathematics Department

I am primarily interested in the following areas of mathematics: metric geometry, topology, differential geometry, and measure theory.

My research focuses on *metric measure spaces* (m.m.s.) satisfying the *RCD(K,N)* condition (i.e. possibly singular spaces with Ricci curvature bigger than K and dimension smaller than N). The story of these spaces goes back to Gromov's precompactness Theorem:

"Sequences of complete Riemannian manifolds with a lower bound on the Ricci curvature and upper bounds on both the dimension and diameter are precompact for the Gromov–Hausdorff topology."

Since then, substantial attention has been devoted to studying the limits of such sequences, referred to as *Ricci limit spaces*. However, it is essential to note that these limits can be singular and may not support differential calculus. Consequently, extensive research has been dedicated to defining Ricci curvature lower bounds for such singular spaces. A promising notion is the concept of RCD spaces (where RCD stands for *Riemannian Curvature Dimension*)**. **At the moment, the best candidates are m.m.s. satisfying the *RCD(K,N)* condition, defined via *Optimal Transport*.

I am currently studying the following questions:

- Given a topological space X, is there any distance d and measure m (compatible with the topology) such that (X,d,m) satisfies the RCD(K,N) condition?
- If yes, can we describe the space of such structures (in terms of the topological properties of the moduli space of RCD(K,N) structures)?

Oct 4, 2023, Ricci curvature lower bounds for singular spaces, UCSC, Geometry and Analysis seminar

Feb 2, 2023, Moduli spaces of compact RCD(0,N)-structures, Durham University, Geometry and Topology seminar

Dec 13, 2022, Moduli spaces of compact RCD(0,N)-structures, Hausdorff Center for Mathematics, Prof. Sturm’s research group

Aug 9, 2022, Moduli spaces of compact RCD(0,N)-structures, Casa Matemática Oaxaca (CMO), Workshop on Metric measure Spaces with Symmetry and Lower Ricci Curvature Bounds

Nov 16, 2021, Théorème de localisation de Klartag et généralisation aux espaces RCD, Laboratoire de Mathématiques d’Avignon, ANR CCEM sur Avignon

Mondino, A., Navarro, D. Moduli spaces of compact RCD(0,N)-structures. *Mathematische Annalen.* (2022). (Open access).

https://link.springer.com/article/10.1007/s00208-022-02493-7

Navarro, D. Contractibility of moduli spaces of RCD(0,2)-structures. *arXiv preprint: 2202.06659. *(2023). (To appear at Annales de l'Institut Fourier).

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