【报告摘要】This talk presents three perspectives on the role of structure in tensor and matrix computations. The first concerns matrix decompositions: using involution models of Grassmannians, classical groups such as SO(n), SU(n), and Sp(2n) can be expressed as products of structured matrix varieties. The second concerns optimization: nonlinear least squares problems over smooth varieties can be analyzed by exploiting the geometry of the parametrization map, leading to generic avoidance of singularities and generic linear convergence for structured tensor approximation algorithms. The third concerns algebraic geometry: multilinear varieties over infinite fields have special properties that provide a geometric framework for comparing invariants of tensors and polynomials. Taken together, these works illustrate a common principle: hidden geometric and algebraic structures can make nonconvex, singular, and high-dimensional problems more tractable.

【报告人简介】叶科，中国科学院数学与系统科学研究院研究员，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。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、2025年数学院科研进展奖。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。