In this talk, we study the empirical likelihood for a partially linear single-index model with a subset of covariates and response missing at random. By using the bias-correction and the imputation method, two empirical log-likelihood ratios are proposed such that any of two ratios is asymptotically chi-squared. Two maximum empirical likelihood estimates of the index coefficients and the estimator of link function are constructed, their asymptotic distributions and optimal convergence rate are obtained. It is proved that our methods yield asymptotically equivalent estimators for the index coefficients.

Numerical differential geometry is a combination of numerical linear algebra and differential geometry, which enables us to do concrete computations on manifolds. In this talk, we will first explain the motivation behind this subject and then we will present two classes of examples: 1) optimization algorithms on special manifolds; 2) geometric method in data analysis.

We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded generation of Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds. This is joint work with Jiming Ma.

In this talk, we will focus on the CR analogue of Yau's uniformization conjecture in a complete noncompact pseudohermitian (2n+1)-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of Kähler geometry. And we obtain the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature. This is a joint work with Professor Shu-Cheng Chang and Dr Chien Lin.

A compact polyhedron $X$ is said to have the /emph{Bounded Index Property (BIP)} if there is an integer $/B>0$ such that for any map $f: X/rightarrow X$ and any fixed point class $/F$ of $f$, the index $|/ind(f,/F)|/leq /B$. $X$ has the /emph{Bounded Index Property for Homeomorphisms (BIPH)} if there is such a bound for all homeomorphisms $f:X/rightarrow X$. In 1998, B. Jiang gave the following question: Does every compact aspherical polyhedron $X$ (i.e. $/pi_i(X)=0$ for all $i>1$) have BIP or BIPH? In this talk, we will survey the progress on this question.

Clustering of extremes usually has a large societal impact. The extremal index, a number in the unit interval, is a key parameter in modelling the clustering of extremes. We build a connection between the extremal index and the stable tail dependence function, which enables us to compute the value of extremal indices for some time series models. We also construct a nonparametric estimator of the extremal index and an estimation procedure to verify D(d)(un) condition, a local dependence condition often assumed when studying extremal index. We prove that the estimators are asymptotically normal.

We study the regularity of the Gromov-Hausdorff limits of Riemannian manifolds with bounded Bakry-Emery Ricci curvature, which include the Ricci soliton and bounded Ricci curvature cases. Our main results are the generalizations of the works of Cheeger-Colding-Tian-Naber when the manifolds are volume noncollapsed. The new ingredients here are a Bishop-Gromov type relative volume comparison theorem on the original manifold without involving weight, and proving that the C/α harmonic radius can be bounded from below, which has relaxed Anderson's result. Our proof of the Codimension 4 Theorem essentially follows the guideline of Cheeger-Naber, but we managed to shorten the proof by using Green's function and a linear algebra argument of R. Bamler. These are joint works with Qi S. Zhang.

In this talk, I will present some recent results on the compactness of the solutions to the Yamabe problem on manifolds with boundary. The compactness of Yamabe problem was introduced by Schoen in 1988. There have been a lot of works on this topic later on. This is a joint work with Sergio Almaraz and Olivaine Queiroz.