Abstract: 

We will talk about recent joint work with Aaron Naber and Wenshuai Jiang. Let $(M^n_i,p_i)\stackrel{d_GH}{\longrightarrow} (X^n,x)$ denote a pointed Gromov-Hausdorff limit space with ${\rm Vol}(B_1(p_i))>{\rm v}$ and ${\rm Ric}_{M^n_i}\geq -(n-1)$. It is known that the singular set $S\subset X^n$ can be dense. However, from our work with T. Colding, it is known that for all $\epsilon>0$, there is a decomposition, $X^n=\mathcal R_\epsilon\cup S_\epsilon$, where $S_\epsilon$ is closed, $S=\bigcup_\epsilon\, S_\epsilon$, the Hausdorff dimension of $S_\epsilon$ is $\leq n-2$ and $\mathcal R_\epsilon$ is $\theta(\epsilon)$-bi-H\"older equivalent to a smooth riemannian manifold, where $\theta(\epsilon)\to 1$ as $\epsilon\to 0$. Our main theorem states that $S_\epsilon$ (and hence, $S$) is rectifiable and there is a Hausdorff measure bound $\mathcal H^{n-2}(S_\epsilon\cap B_1(p_i))\leq c(n,{\rm v},\epsilon)$. These statements are consequences of stronger effective bounds on the so called "quantitative stratification" and still more general statements on "neck regions".

Introduction of the speaker: