As a postdoc and later assistant professor at MIT, Naber and Jeff Cheeger introduced the notion of quantitative stratification to Lower Ricci curvature.

During his time at Northwestern, Naber and Cheeger proved the codimension four conjecture, showing in particular that Einstein manifolds have controlled singular sets.

Near the end of his time at Northwestern, Elia Brue, Naber and Daniele Semola gave a counterexample to the Milnor conjecture, showing the existence of spaces with nonnegative Ricci curvature and infinitely generated fundamental group.

Together they developed a stratification theory for nonlinear harmonic maps, which broadly extended the results of Schoen/Uhlenbeck from Hausdorff dimension estimates to finite measure and rectifiable structure for singular sets.

In 2014 Naber was awarded a two-year Sloan Research Fellowship and was an invited speaker with talk The structure and meaning of Ricci curvature at the International Congress of Mathematicians in Seoul.