On Branching Indices of Affine A-D-E Diagrams : A Geometrical Characterization by Kleinian Singularities

Stability conditions and Kleinian singularities

We describe the spaces of stability conditions on certain triangulated categories associated to Dynkin diagrams. These categories can be defined either algebraically via module categories of preprojective algebras, or geometrically via coherent sheaves on resolutions of Kleinian singularities. The resulting spaces are related to regular subsets of the corresponding complexified Cartan algebras.

McKay correspondence for the Poincaré series of Kleinian and Fuchsian singularities

We give a conceptual proof that the Poincaré series of the coordinate algebra of a Kleinian singularity and of a Fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two Coxeter elements. These Coxeter elements are interpreted geometrically, using triangulated categories and spherical twist functors.

通过锥猜想建立自同构群的几何有限性

Geometrical finiteness for automorphism groups via cone conjecture

本文通过锥猜想研究K3、恩里ques、Coble表面及不可约辛流形的自同构群在超球面上的等距作用的几何有限性，证明此类群非正曲率特性，并揭示K3表面的Kleinian晶格与(-2)曲线关系。

The Geometry of Noncommutative Singularity Resolutions

We introduce a geometric realization of noncommutative singularity resolutions. To do this, we first present a new conjectural method of obtaining conventional resolutions using coordinate rings of matrix-valued functions. We verify this conjecture for all cyclic quotient surface singularities, the Kleinian D_n and E_6 surface singularities, the conifold singularity, and a non-isolated singularity, using appropriate quiver algebras. This conjecture provides a possible new generalization of the classical McKay correspondence. Then, using symplectic reduction within these rings, we obtain new, n…

非单连通李代数中次正则胞腔上的仿射Kazhdan-Lusztig多项式：及其在特征标公式中的应用（附Roman Bezrukavnikov、Vasily Krylov和Kenta Suzuki的附录）

Affine Kazhdan-Lusztig polynomials on the subregular cell in non simply-laced Lie algebras: with an application to character formulae (with an appendix by Roman Bezrukavnikov, Vasily Krylov, and Kenta Suzuki)

本文将仿射Kazhdan-Lusztig多项式技术推广到非单连通情形，计算次正则幂零轨道上的显式特殊值，从而得到仿射李代数不可约表示及顶点代数的特征标公式，并应用于味对称性为$G_2$和$B_3$的Argyres-Douglas理论的Schur指数计算。