Core Viewpoint - A research team from South China University of Technology, Xi'an Jiaotong University, and the University of Macau has solved a key problem related to the "hotspot conjecture" in mathematics, with results published in the journal "Advances in Mathematics" [1] Group 1 - The research focuses on the behavior of heat distribution in an insulated room, where heat diffuses from high to low temperatures, leading to the existence of "hot" and "cold" points before equilibrium is reached [1] - The "hotspot conjecture," proposed by American mathematician Rauch in 1974, suggests that extreme temperature points are more likely to occur at the edges of a convex region rather than within its interior [1] - The research team conducted a systematic analysis over 13 years, addressing an open problem posed by Fields Medalist Terence Tao in 2012 regarding the precise location of maximum values [1]

Core Viewpoint - The research team led by Associate Professor Yao Ruofei from South China University of Technology, in collaboration with professors from Xi'an Jiaotong University and the University of Macau, has successfully solved the "hotspot conjecture," a significant mathematical problem that has persisted for over 50 years [1][2]. Group 1: Research Background - The "hotspot conjecture" was proposed by American mathematician Rauch in 1974, suggesting that in an insulated room, the hottest and coldest points are more likely to be found at the boundaries rather than inside the room during the transient state before thermal equilibrium is reached [2][3]. - The conjecture is mathematically equivalent to stating that for convex regions in a plane, the maximum and minimum values of the second Neumann eigenfunction of the Laplace operator can only occur at the boundary of the region [2][3]. Group 2: Research Achievements - The research focused on the case of triangles, which, despite their simple structure, present significant challenges in analyzing the behavior of eigenfunctions. This work not only addressed a public problem posed by Fields Medalist Terence Tao but also advanced critical conclusions related to the critical points discussed in the Annals of Mathematics [3][4]. - The team provided systematic and rigorous conclusions regarding several key structural issues of the second Neumann eigenfunction, which can serve as important references for subsequent research in spectral geometry and partial differential equations [4]. Group 3: Research Process - The research journey spanned approximately 13 years, with 5 years of concentrated effort leading to the submission of the paper. The team faced challenges and periods of stagnation but made significant progress through continuous refinement and collaboration [5][6]. - A pivotal breakthrough occurred when the team shifted their approach to "directly proving symmetry," inspired by previous work published in 2018. This change in strategy led to a clearer path for the proof [5][6]. Group 4: Publication Challenges - The publication process involved three rounds of revisions, with rigorous peer review standards that demanded near-perfect arguments. This process significantly enhanced the rigor and completeness of the work [6]. - The support for young faculty members at South China University of Technology played a crucial role in the success of this research, highlighting the institution's commitment to fostering academic growth and collaboration [6].