For a fibration f: X → Z over the filed of complex numbers, Iitaka conjectures κ(X) ≥ κ(Z) + κ(F), where F is the geometric generic fiber of f and κ denotes the Kodaira dimension. The conjecture is usually denoted by Cn,m, n=dim X, m=dim Z. We will introduce the progress of the conjecture on a positive characteristic field, including the recent results of Cn,n-1 and C3,1 on a positive characteristic field. These are joint work with Lei Zhang and Caucher Birkar.

Solutions of an algebraic differential equation have a rich geometric structure. In his landmarking speech in 1900, Hilbert described a problem on the existence of Fuchsian type equation having prescribed monodromy group, which is now named the Hilbert’s 21stproblem. In this talk, I will introduce Grothendieck school’s formulation and solution of this problem.

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.