First, I'd like to explain Littelmann's path models. Let be a symmetrizable Kac-Moody algebra (such as, etc.). In the papers "A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras (1994)" and "Paths and root operators in representation theory (1995)", Littelmann introduced Lakshmibai-Seshadri (LS) paths of shape λ (where λ is an integral weight for), and gave the set of them a crystal structure in terms of his root operators; Kashiwara and Joseph proved independently that if λ is dominant, then the crystal of LS paths of shape λ is isomorphic to the crystal basis of the integrable highest weight module of highest weight λ. Using Littelmann's path model (consisting of LS paths), we can describe

The difference between quantum sl_n and quantum gl_n is just an extra generator in the zero part. However, that in the affine case is huge. We have developed a new approach to study the structure and representations of quantum affine gl_n. In the series of talks, I will discuss the entire theory including a new realisation, the Lusztig form and the representation theory of affine q-Schur algebras

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.