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For a smooth projective curve with genus g(X)>1 and a degree 1 line bundle L on C, let M:=SU_C(r,L) be the moduli space of stable vector bundles of rank r over C with the fixed determinant L. In this paper, we study the small rational curves on M and estimate the codimension of the locus of the small rational curves. In particular, we determine all small rational curves when r=3

Let X be a smooth projective surface over an algebraic closed field k with positive characteristic p, H an ample divisor on X. Suppose that the cotangent bundle $/Omega_X^1$ is semistable of positive slope with respect to H. We will give a restriction on p such that for any stable bundle W, the direct image F_*(W) under Frobenius morphism is stable, where F:X->X is the absolute Frobenius morphism on X. This is a joint work with Ming-shuo Zhou.

Let X be a smooth projective curve over C. Denote by U^L_X (resp. P^L) the moduli space of semistable parabolic vector bundles (resp. generalized parabolic sheaves) of rank r and fixed determinant L on X. In this talk, we prove the Frobenius-split type of the moduli space U^L_X and P^L. This is a joint work with Prof. Xiaotao Sun.

Let X be a smooth projective curve of genus g>1 over an algebraic closed field k of characteristic p>0. Let F: X->X be the absolute Frobenius morphism, and E a semistable vector bundles on X. It is natural to ask whether the length of the Harder-Narasimhan filtration of F^*(E) is at most p. In this talk, we construct a counterexample to above question.

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.