UM scholar publishes paper in world-leading mathematics journal

MACAU, January 21 - A paper co-authored by Gui Changfeng, chair professor and head of the Department of Mathematics in the Faculty of Science and Technology at the University of Macau (UM), has been published in Inventiones Mathematicae, one of the world’s leading journals in mathematics. The paper, titled Uniqueness of critical points of the second Neumann eigenfunctions on triangles, represents a major research achievement and marks the first time a paper by a Macao scholar has been published in one of the top four mathematics journals. The publication reflects the core competitiveness and growing international influence of UM’s Department of Mathematics on the global academic stage.

The study focuses on the ‘hot spots conjecture’, proposed by American mathematician Jeffrey Rauch in 1974, which uses physical phenomena to illuminate a core mathematical problem. Consider an insulated room where heat from a localised source diffuses, leading to temperature extrema during the process. Intuitively, these temperature extremes should occur only at the boundaries of the room, not in its interior. This physical intuition is formalised in the framework of the heat equation with Neumann (insulating) boundary conditions. Mathematically, the conjecture asserts that, in a convex planar domain, the maximum and minimum of the second Neumann eigenfunction of the Laplacian—the mode that governs the slowest decay to uniformity—occur solely on the boundary. Over the past half-century, the problem has attracted the attention of many leading mathematicians, including Fields Medalists Terence Tao and Wendelin Werner, as well as International Congress of Mathematicians speakers Richard F. Bass, David Jerison, and Nikolai Nadirashvili. Although significant progress has been made across various geometries and special cases, the planar triangle—the simplest convex polygon—has long remained a formidable challenge.

The research team focused on the triangular case, conducting a systematic and in-depth analysis that yielded several major breakthroughs. First, the work resolves an open question posed by Terence Tao in 2012 in Polymath Project 7 concerning the precise location of the maximum of the second Neumann eigenfunction. Second, it refines and fully proves the boundary critical point conjecture proposed by Judge and Mondal in their 2020 paper in Annals of Mathematics. Third, it provides affirmative answers to questions raised by David Jerison regarding the monotonicity of eigenfunctions. In addition, the study offers definitive resolutions to several open problems concerning nodal set distributions and mixed boundary eigenvalue inequalities. These results are rigorously established using innovative symmetry arguments and eigenvalue bounds, not only deepening the understanding of the second Neumann eigenfunction on triangles but also providing foundational tools and insights for future research in spectral geometry, partial differential equations, and related fields.

This publication not only affirms the strength of UM’s research capabilities, but also lays a solid foundation for the Department of Mathematics to further deepen its engagement in cutting-edge international research and expand global academic collaborations. The paper is co-authored by Gui, Yao Ruofei, associate professor in the School of Mathematics at South China University of Technology, and Chen Hongbin, professor in the School of Mathematics and Statistics at Xi’an Jiaotong University. The full version of the study is available at: https://link.springer.com/article/10.1007/s00222-025-01398-x.

Inventiones Mathematicae is widely regarded as one of the four premier journals in mathematics, alongside Acta Mathematica, Annals of Mathematics, and Journal of the American Mathematical Society, and serves as a key benchmark for the highest level of international mathematical research.