报告题目：From sum-of-squares to sum-of-modules: theory and algorithms

报告人：邵嗣烘 教授

报告时间：2026年1月20日9:00-11:30

报告地点：数学与统计学院312会议室

邀请人：沈明

邀请单位：福州大学数学与统计学院

报告摘要：As an important branch of combinatorial optimization, graph cut problems are characterized by their wide variety, inherent computational difficulty, and extensive range of applications. To efficiently tackle these problems, researchers have embedded them into continuous frameworks based on sum-of-squares formulations, developing theories and algorithmic tools, such as linear spectral theory and semidefinite programming that approximate the discrete problems via relaxation-rounding approaches. Recently, nonlinear spectral theory and algorithms based on sum-of-modules formulations have shown promising results. For example, the 1-Laplacian spectral theory accurately characterizes the Cheeger cut problem: its second smallest eigenvalue exactly equals the Cheeger constant, and the corresponding eigenvector yields the optimal binary cut. By embedding the maxcut problem into an origin-excluded Euclidean space via the sum-of-modules formulation, a simple continuous iterative algorithm has been developed; this method enjoys explicit analytic solvability of its subproblems and eliminates the need for rounding, achieving the state-of-the-art performance on the G-set benchmark. In this talk, we will show that such equivalence-based spectral theories and algorithms developed from the sum-of-modules formulation can be generalized to a broader range of graph cut problems.

报告人简介：邵嗣烘，北京大学博雅特聘教授，主讲《高维数值方法》，《谱方法》，《数学分析I-III》，《数学模型》和《计算流体力学》等课程。主要开展面向智能、量子和计算的交叉融合研究，落脚点在基础的数学理论和高效的算法设计，强调离散数学结构的设计、分析和应用。具体研究领域包括：高维数值方法、离散建模与组合优化、计算量子力学、图谱理论及算法、微分方程数值解和计算复杂性等，获国家自然科学基金杰青、优青、面上和青年等项目资助。2019年入选北京智源人工智能研究院“智源青年科学家”。2020年获北京大学优秀博士学位论文指导老师。2025年获北京大学优秀本科毕业论文指导教师和北京大学兴证全球基金奖教金杰出青年学者奖。曾获中国计算数学学会优秀青年论文一等奖，北京大学黄廷芳/信和青年杰出学者奖，北京大学学术类创新奖，宝洁教师奖和北京大学优秀班主任等。