In this talk, I will introduce our recent work on low-rank methods for Bayesian inverse problems. For linear problems with Gaussian noise and Gaussian prior, the posterior is also Gaussian and characterized by the posterior mean and covariance. We propose a low-rank Arnoldi method to approximate the large dense posterior covariance matrix by making use of tensor computations. For nonlinear systems, the posterior is not Gaussian anymore, however, can often be approximated by a Gaussian distribution using the ensemble Kalman filter (EnKF) or the extended Kalman filter (ExKF).

When estimating the average causal effect in observational studies, researchers have to tackle both self-selection of treatment and outcome modeling. This is difficult since usually there are a large number of covariates that affect people's treatment decision and the true functional form in the model is not known. Propensity score is a popular approach for dimension reduction in causal inference. We propose a new semiparametric estimation strategy using B-spline based on the propensity score, which does not rely on parametric model specification. We further improve the efficiency of the estimator by addressing the error heteroscedasticity. We also establish the asymptotic properties of both estimators.

In this talk, we will first review the existing results and methods on the exact controllability for the 1-D coupled hyperbolic system by means of boundary controls on one side. Then, we will show our recent results on this topic on how the sufficient and necessary boundary coupling effect on the non-control side can be used to shorten the control time as much as possible no matter how the coupling terms appearing in the equations. In particular, this is done under natural conditions on the regularity of the coefficients and arbitrary number of coupled transport PDEs in either direction. Several counterexamples and interesting phenomena will also be shown.

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.