从代数几何、微分几何、计算复杂度和组合四方面研究了Grassmannian上的计算问题，取得五项成果：给出其作为仿射簇次数的组合公式，证明Sturmfels等人的猜想；得到曲率显式矩阵表达式；确定其欧式距离次数；证明其上的无约束二次优化问题为NP-难；证明切片秩函数在域扩张下的稳定性，解决Kazhdan等人的猜想。

Publication:

1. L.-H. Lim and K. Ye, Degree of the Grassmannian as an affine variety, Advances in Mathematics, 2025, Volume 480, Part A, 110459. 

2. Z. H. Lai, L.-H. Lim and K. Ye, Simple matrix expressions for the curvatures of Grassmannian, Foundations of Computational Mathematics, 2025, published online. 

3. Z. H. Lai, L.-H. Lim and K. Ye, Euclidean distance degree in manifold optimization, SIAM Journal on Optimization, 2025, 35(4), pp. 2402-2422. 

4. Z. H. Lai, L.-H. Lim and K. Ye, Grassmannian optimization is NP-hard, SIAM Journal on Optimization, 2025, 35(3), pp. 1939-1962. 

5. Q. Y. Chen and K. Ye, Stability of ranks under field extensions, Discrete Analysis,  2025:27, 29pp.

Author:

Zehua Lai

Department of Mathematics, University of Texas, Austin, 2515 Speedway, Austin 78712, TX, USA

zehua.lai@austin.utexas.edu

Lek-Heng Lim

Computational and Applied Mathematics Initiative, University of Chicago, 5747 South Ellis Avenue, Chicago 60637, IL, USA

lekheng@uchicago.edu

Ke Ye

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55 Zhongguancun East Road, Beijing 100190, China

keyk@amss.ac.cn

Qiyuan Chen

State Key Laboratory of Mathematical Sciences Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190, China

chenqiyuan@amss.ac.cn