Xiaochun Rong was born on June 17th, 1954. He graduated from Beijing Normal College (now Capital Normal University) in 1982, and received his Ph.D. from Stony Brook University in 1990. Rong was a Sloan Research Fellow in 1996-1998, and an NSF research grant awardee in 1994-2013. He has been a distinguished professor at Rutgers University in New Brunswick since 2008. Rong was a 2002 ICM invited speaker, and a 2017 Fellow of the American Mathematical Society.

For a Gromov-Hausdorff convergent sequence of closed manifolds $M^n_i\xrightarrow{GH} X$ with $ {\rm Ric}\ge -(n-1)$, $ {\rm diam}(M_i) ≤ D$, and $ {\rm vol}(M_i) ≥ v > 0$, we study the relation between $π_1(M_i)$ and $X$. It was known before that there is a surjective homomorphism $ ϕ_i : π_1(M_i) → π_1(X)$ by the work of Pan-Wei. In this paper, we construct a surjective homomorphism from the interior of the effective regular set in $X$ back to $M_i$, that is, $ψ_i :π_1(\mathcal{R}^◦_{ϵ,δ} )→π_1(M_i)$. These surjective homomorphisms $ϕ_i$ and $ψ_i$ are natural in the sense that their composition $ϕ_i◦ψ_i$ is exactly the homomorphism induced by the inclusion map $R^◦_{ ϵ,δ}\hookrightarrow X$.

Let $F$ be a closed subset in a finite dimensional Alexandrov space $X$ with lower curvature bound. This paper shows that $F$ is quasi-convex if and only if, for any two distinct points $p,r∈F$, if there is a direction at $p$ which is more than $\frac{π}{2}$ away from $⇑^r_p$ (the set of all directions from $p$ to $r$), then the farthest direction to $⇑^r_p$ at $p$ is tangent to $F$. This implies that $F$ is quasi-convex if and only if the gradient curve starting from $r$ of the distance function to $p$ lies in $F$. As an application, we obtain that the fixed point set of an isometry on $X$ is quasi-convex.

In this paper, we show that the uniform $L^4$-bound of the transverse Ricci curvature along the Sasaki-Ricci flow on a compact quasi-regular transverse Fano Sasakian $(2n+1)$-manifold $M$. Then we are able to study the structure of the limit space. As consequences, when $M$ is of dimension five and the space of leaves of the characteristic foliation is of type I, any solution of the Sasaki-Ricci flow converges in the Cheeger-Gromov sense to the unique singular orbifold Sasaki-Ricci soliton and is trivial one if $M$ is transverse $K$-stable. Note that when the characteristic foliation is of type II, the same estimates hold along the conic Sasaki-Ricci flow.

In this paper, the notion of hyperbolic ellipsoids in hyperbolic space is introduced. Using a natural orthogonal projection from hyperbolic space to Euclidean space, we establish affine isoperimetric type inequalities for static convex domains in hyperbolic space. Moreover, equality of such inequalities is characterized by these hyperbolic ellipsoids.

In this survey paper, we discuss various examples of Ricci solitons and their constructions. Some open questions related to the rigidity and classification of Ricci solitons will be also discussed through those examples.

In this paper, we consider a fixed metric space (possibly an oriented Riemannian manifold with boundary) with an increasing sequence of distance functions and a uniform upper bound on diameter. When the fixed space endowed with the point-wise limit of these distances is compact, then there is uniform and Gromov-Hausdorff (GH) convergence to this space. When the fixed metric space also has an integral current structure of uniformly bounded total mass (as is true for an oriented Riemannian manifold with boundary that has a uniform bound on total volume), we prove volume preserving intrinsic flat convergence to a subset of the GH limit whose closure is the whole GH limit. We provide a review of all notions and have a list of open questions at the end.