【报告摘要】In this talk, we present the quadratic models of Grassmannians, which realize a Grassmannian as a real affine variety. We will discuss computational properties of Grassmannians from three aspects:

1) We obtain the degree formula of Grassmannians as affine varieties. In particular, our formula resolves the Devriendt--Friedman--Sturmfels conjecture.

2) We consider the Euclidean distance degree (ED degree) of Grassmannians as affine varieties.

3) We prove that quadratic optimization over a Grassmannian is NP-hard.

Since degree, ED degree, and computational complexity measure different aspects of computational difficulty, an important implication of our results is that computations over Grassmannians are challenging in multiple ways.

【报告人简介】叶科，中国科学院数学与系统科学研究院副研究员，研究兴趣主要集中在代数与几何方法在计算复杂度理论、（多重）线性代数、数值计算以及优化问题中的应用。研究工作分别解决了T. Y. Lam、B. Sturmfels和D. Kazhdan提出的猜想，并回答了Y. Saad的公开问题。相关学术成果发表于Adv. Math., Found. Comut. Math., Math. Program.等国际知名期刊。目前担任《SIAM Journal on Applied Algebra and Geometry》编委。曾入选海外高层次人才引进计划（青年项目）。获得吴文俊计算机数学青年学者奖、华为技术合作成果转化二等奖、优秀合作项目奖。指导的学生工作获得国际符号与代数计算会议（ISSAC）最佳学生论文奖。