“人大数学时间I”第四十三期：Enrique Zuazua、韩青
为促进学科交流融合、拓宽师生学术视野、释放科研创新活力，助力人大数学学科走向一流，JAVxxx 设立“人大数学时间”，以专题研讨、高端学术论坛为载体，搭建数学思想充分碰撞、优秀人才不断涌流、创造活力竞相迸发的舞台。“人大数学时间”将持之以恒，久久为功，立志通过交流与创新、提出重大问题，引领数学学科及相关领域的创新与发展，成为对我国数学发展有贡献意义的平台。以下为“人大数学时间I”第四十三期信息：
议程：
3月19日（星期四）
10:00 Enrique Zuazua教授报告及前沿问题探讨
地点：立德楼603
16:00 韩青教授报告及前沿问题探讨
地点：立德楼610
报告题目：Machine Learning from an Applied Mathematician's Perspective
主讲专家：Enrique Zuazua，欧洲科学院院士，ICM报告人
摘要：
Machine Learning has emerged as one of the most transformative forces in contemporary science and technology. In this lecture, I discuss Machine Learning through the lens of applied mathematics, highlighting its connections with control theory, partial differential equations, and numerical analysis. The presentation is organized around three goals: representation, generalization, and generation.
We revisit the links between Machine Learning and systems control (cybernetics), interpreting representation and expressivity in deep neural networks in terms of ensemble controllability of neural differential equations. Within this framework, generalization appears as a stability property with respect to perturbations in the data and the model.Next, we discuss neural-network architectures as tools for numerical approximation, using the Dirichlet problem for the Laplace equation formulated as an energy minimization problem under neural-network constraints.Finally, we present a PDE-based perspective on generative diffusion models, interpreting their convergence through the asymptotic behavior of Fokker–Planck equations driven by the score field, and highlighting how classical tools shed light on their regularization and convergence properties.
报告人简介：
Enrique Zuazua教授为欧洲科学院院士，巴斯克科学文学和人文学院荣誉院士，埃尔朗根-纽伦堡大学首位国际校使，同时也是动力学、控制、机器学习与数值计算领域的亚历山大•冯•洪堡学者。Enrique Zuazua教授连续担任2006年，2026年国际数学家大会特邀报告人，系西班牙国家研究计划数学小组首位科学经理(1999-2002)，巴斯克应用数学中心创始科学主任(2008-2012)；任法国INSMI-CNRS,CERFACS等多家国际研究机构科学委员会成员，并担任众多应用数学与控制理论领域顶级期刊的主编及编委。
报告题目：Global and Exterior Solutions to the Minimal Surface Equation
主讲专家：韩青，美国圣母大学数学系终身教授，美国纽约大学库朗数学研究所博士
摘要：
A characterization of global solutions to the minimal surface equation has been known by the efforts of Bernstein (1914), De Giorgi (1965), Almgren (1966), Simons (1968), and Bombieri, De Giorgi, and Giusti (1969). In this talk, we first review relevant results. Then, we switch to exterior solutions and aim to present a complete characterization of solutions to the minimal surface equation near infinity. It is well-known that Dirichlet boundary value problems in exterior domains do not always admit solutions. We demonstrate that prescribing asymptotic behaviors forms a new type of problems leading to all solutions near infinity. The harmonic functions determining the asymptotic behaviors play the role of “free data” as the boundary values in the boundary value problems.
报告人简介：
韩青，美国圣母大学数学系终身教授，美国纽约大学库朗数学研究所博士，美国芝加哥大学博士后，获美国Sloan Research Fellowship；韩青教授长期致力于非线性偏微分方程和几何分析的研究，在等距嵌入、Monge-Ampere方程、调和函数的零点集和奇异集、退化方程等方面做出了一系列原创性的重要研究成果。
