FST MO-HQ Distinguished Talk Series – Extreme Problems of Dirichlet Eigenvalues – A RevisitSpeaker: Prof. Fanghua LIN from New York UniversityHosted by: Prof. Chang Feng GUI

The Faculty of Science and Technology of the University of Macau hosted a distinguished Talk by Prof. Fanghua Lin, a world-renowned mathematician and member of the U.S. National Academy of Sciences. In his lecture, Prof. Lin revisited the classical yet deeply challenging topic of extreme problems associated with Dirichlet eigenvalues of the Laplacian.

The talk began with an introduction to Weyl’s asymptotic formula, which describes the qualitative behavior of large eigenvalues on bounded domains in Euclidean space. Prof. Lin skillfully bridged classical results with contemporary research, highlighting recent advances in obtaining more quantitative estimates. He discussed the well-known Polya’s conjecture and explored several natural variational and extreme problems related to eigenvalues, offering insight into both theoretical and applied implications. Prof. Lin’s presentation was both rigorous and accessible, striking a balance between deep mathematical detail and intuitive explanation. He engaged the audience with clear examples and historical context, making complex topics such as spectral geometry and PDE theory approachable for graduate students and faculty alike. The talk was particularly enriched by Prof. Lin’s distinguished background, which includes seminal contributions to Ginzburg-Landau theory, partial differential equations, and geometric measure theory.

This Distinguished Talk was not only a testament to Prof. Lin’s scholarly excellence but also a valuable academic event that inspired further inquiry into one of mathematics’ enduring and evolving domains.

澳門大學科技學院舉辦了一場具啟發性的傑出學者講座，主講人為國際著名數學家、美國國家科學院院士林芳華教授。在講座中，林教授重新探討了拉普拉斯算子狄利克雷特徵值極值問題這一經典且極具挑戰性的課題。

講座從介紹外爾漸近公式 (Weyl’s asymptotic formula) 開始，該公式描述了歐幾里得空間中有界域上大特徵值的定性行為。林教授巧妙地將經典成果與當代研究相結合，重點介紹了近年來在獲取更精確的定量估計方面所取得的新進展。他討論了著名的波利亞猜想，並深入探討了若干與特徵值相關的自然變分與極值問題，為相關理論及應用方向提供了深刻見解。林教授的演講既嚴謹又易懂，在深邃的數學細節與直觀的解釋之間取得了良好的平衡。他透過清晰的實例與歷史背景引導聽眾，使得譜幾何、偏微分方程理論等複雜議題對研究生和教師而言都變得易於理解。

林教授卓越的學術背景——包括在金茨堡–朗道理論、偏微分方程和幾何測度論等方面的開創性貢獻——進一步豐富了講座的内涵。這場傑出學者講座不僅是林教授卓越學術成就的體現，也是一次極具價值的學術活動，激發了在數學這一持續發展與演進的領域中進一步探索的熱情。