Dimitri Navarro

I am primarily interested in the following area of mathematics:

• metric geometry,
• measure theory,
• differential geometry,
• topology.

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, there has been much work to understand the properties of (possibly singular) limits of such sequences called Ricci limit spaces. The most recent idea to study such limits is to define what it means for a m.m.s. to have Ricci curvature bigger than K and dimension smaller than N. 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:

1. 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?
2. 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)?
3. Given a m.m.s. (X,d,m) satisfying the RCD(K,N) condition, what can we say about the topology of (X,d)?