After the Eilenberg–Steenrod’s axiomatic cohomology theory. The Grothendieck school makes an evolution to this field by their theory of derived category and Grothendieck six operators. This new approach is more flexible so that it provides a ‘Poincare duality’ for singular spaces. In this talk, I will explain how the Grothendieck school rewrite the classical cohomology theory. In the end, we will make a quick travel to the l-adic generalization which provides a perfect background to attack the Weil Conjecture.

It is quite challenge to develop model-free feature screening approaches directly for missing response problems since the existing standard missing data analysis methods cannot be applied directly to high dimensional case. This paper develops a novel technique by borrowing information of missingness indicators such that any feature screening procedures for ultrahigh-dimensional covariates with full data can be applied to missing response case. This technique is developed by proving that the joint set of the active predictors on the response and missingness indicator equals to the set of the active predictors on the product of the response and missingness indicator.

A variety X in characteristic p is called Frobenius split if there is a "p-th root" map σ: X→X, that is, an additive map satisfying σ(f^pg)=fσ(g) and σ(1)=1 (in particular, σ(f^p)=f, so that σ is an O_X-linear splitting of the Frobenius map F: O_X→F_*(O_X). Such varieties enjoy very nice properties. In this talk, We will give some example of Frobenius splitting variety. In addition, I continue to introduce some questions about Frobenius splitting variety.

A natural method for approximating out-of-sample predictive evaluation is leave-one-out cross-validation (LOOCV) --- we alternately hold out each case from a full data set and then train a Bayesian model using Markov chain Monte Carlo (MCMC) without the held-out; at last we evaluate the posterior predictive distribution of all cases with their actual observations. However, actual LOOCV is time-consuming. This talk introduces two methods, namely iIS and iWAIC, for approximating LOOCV with only Markov chain samples simulated from a posterior based on a full data set. iIS and iWAIC aim at improving the approximations given by importance sampling (IS) and WAIC in Bayesian models with possibly correlated latent variables. In iIS and iWAIC, we first integrate the predictive density over the distribution of the latent variables associated with the held-out without reference to its observation, then apply IS and WAIC approximations to the integrated predictive density.

学 术 报 告报告人：戴民教授（新加坡国立大学）题目：Asymptotic for Merton problem with capital gain tax and small interest rate时间：2014年12月11日，星期四下午4：00—5：00地点：数学系致远楼107欢迎各位参加

We develop a constructive approach to estimating a sparse linear regression model in high-dimensions. The proposed approach is a computational algorithm that generates a sequence of solutions iteratively, based on support detection using primal and dual information and root finding according to a modified KKT condition for the L0-penalized least squares criterion. We refer to the proposed algorithm as SDAR for brevity. Under certain regularity conditions on the design matrix and sparsity assumption on the regression coefficients, we show that with high probability, the errors of the solution sequence decay exponentially to the minimax error bounds in the Gaussian noise case. Moreover, with high probability, it takes no more than O(log(R)) steps to recover the oracle estimator, where R is the relative magnitude of the nonzero coefficients.

Treatment effect is an important index in comparing two-sample data in survival analysis, industry manufacture, clinical medicine and many other applications. In this paper, we propose a unified semiparametric approach to estimate different types of treatment effects under a case-control sampling plan with the logistic regression model assumption, which is equivalent to a two-sample density ratio model. For different treatment effects, we construct different estimating functions and the nuisance parameters in estimating functions are estimated firstly by the empirical likelihood method. Here, we allow that the functions are nonsmooth with respect to parameters. The confidence interval for the treatment effect based on the empirical likelihood ratio method is also presented. We prove that the estimator based on the estimating equation is consistent and asymptotically normal and the empirical log-likelihood ratio statistic has a limiting scaled chi-square distribution.

In this talk, we consider the modules for the Heisenberg-Virasoro algebra and the W-algebra W(2, 2). We determine the modules whose restriction to the Cartan subalgebra or maximal toral subalgebra (modulo center) are free of rank 1 for these two algebras. We also determine the simplicity of these modules. These modules provide new simple modules for the W-algebra W(2, 2).