Here is a more technical summary, in reverse chronological order. 

In recent projects, I have developed methods to study the shape of high-dimensional orthogonal representations of discrete groups using minimal surfaces and harmonic maps. In particular, I found connections between random matrix theory, group theory and differential geometry here and here. A surprising consequence is that random equivariant harmonic maps from Riemann surfaces to high-dimensional Euclidean spheres typically look like a hyperbolic plane. This result was applied here by M. Ancona, F. Labourie, A. Roig Sanchis and J. Toulisse to answer positively a question of S.T. Yau  (Problem 101 of the 1982 problem list) about negatively curved minimal surfaces in spheres. The uniqueness of the limit of random equivariant harmonic maps into spheres was established here with R. Caniato and X.Z. Li.

I have also worked on several invariants which measure the size of manifolds and maps between manifolds, like the volume entropy, the ADM mass, the minimal volume, the symplectic capacity and the spherical volume. With C.H. Dong, we solved here a conjecture of G. Huisken and T. Ilmanen about the stability of the Euclidean 3-space among spaces with small mass. With D. Kazaras and K. Xu, we found here a counterexample to a conjecture of  I. Agol, P. Storm and W. Thurston about the relation between scalar curvature and volume entropy. With K. Sackel, U. Varolgunes and J. Zhu, we quantified the nonsqueezing theorem of M. Gromov here. I established the  stability of hyperbolic manifolds for the Besson-Courtois-Gallot entropy inequality here.

Since my postdoc, I have often thought about the problem of constructing optimal geometries on closed manifolds. My main proposal is to study "minimal surface geometries", whose existence theory is much more tractable compared to other standard special geometries like Einstein metrics. This is explained here, where I showed that one can naturally associate to any closed negatively curved manifold a certain minimal surface that realizes its spherical volume, an invariant invented by G. Besson, G. Courtois and S. Gallot. 

During my PhD, I studied the construction of minimal surfaces via min-max variational methods developed by Almgren-Pitts and Marques-Neves. Building on this theory, I settled here the conjecture of S.T. Yau (Problem 88 of the 1982 problem list) about the existence of infinitely many minimal surfaces. With F.C. Marques, A. Neves, and with X. Zhou, we found here and here unexpected results on the spatial distribution of minimal surfaces in Riemannian manifolds. With D. Ketover and Y. Liokumovich, we solved here the conjecture of J. Pitts and J.H. Rubinstein about minimal Heegaard splittings in 3-manifolds. I also introduced combinatorial methods to control the topology of minimal surfaces here.