Prof Changfeng Gui joined the University of Macau (UM) in 2023 as the Chair Professor and the Head of the Department of Mathematics in the Faculty of Science and Technology (FST). He received his undergraduate and master’s degrees from Peking University and earned a PhD degree in Mathematics from the University of Minnesota in 1991. Prof Gui is currently a fellow of the American Mathematical Society (AMS) and has received several remarkable awards, including the Research Achievement Award of the Pacific Institute for the Mathematical Sciences (PIMS); the André Aisenstadt Prize in Mathematics from the Centre de Recherches Mathématiques; IEEE Best Paper Award, and the Overseas Cooperation Fund of the National Natural Science Foundation of China (NSFC). His main research interests are nonlinear partial differential equations, applied mathematics, and image analysis and processing. Prof Gui believes that partial differential equations are an essential branch of mathematics and a critical tool in many engineering and scientific fields, such as mechanical engineering and geophysics.

How to study partial differential equations

Partial differential equations are widely used to describe changes and movements in physical phenomena. Prof Gui shared the example of the rate of change of a phenomenon, which is itself a differential, and explained that the study of such phenomena unavoidably involves partial differential equations. Many physical problems remain unsolved, and the solutions to most problems cannot be clearly indicated in formulas. Prof Gui explained that typically there are two methods for studying the corresponding partial differential equations of these problems: one is to study some qualitative characteristics of the problem, such as its symmetry, monotonicity, and asymptotic expansion, even if it is difficult to clearly exhibit the solution of the problem in formulas; the other is to calculate the numerical solution of the equation and study the theoretical problems related to the numerical solution, such as whether there is a reasonable numerical solution and whether the solution is accurate. The study of partial differential equations is therefore very important.

The reasons for choosing this research direction

Prof Gui chose partial differential equations as his research direction due to his passion for mathematical analysis. He shared that he particularly enjoyed the course of mathematical analysis during his undergraduate studies, which included courses requiring a deeper understanding than calculus, such as real analysis and functional analysis. This further sparked his enthusiasm for the “analysis” branch in Mathematics, and the core method of partial differential equations is “analysis”, which integrates ideas from other branches of Mathematics such as geometry and topology and occupies a central position in the field of mathematics. During his doctoral studies in the United States, Prof Gui had more opportunities to interact with renowned scholars in this field. “Because I had interactions with these masters during my studies, I naturally focused on partial differential equations as my research direction.” Currently, Professor Gui’s research focuses mainly on the theory of nonlinear partial differential equations, which includes two special directions: one is the study of a class of best inequalities closely related to physics and geometry, especially on high-dimensional spheres; and the other is a special type of nonlinear equation related to phase transitions in material science, which has important theoretical and practical applications and has been a problem that Prof Gui has been studying for many years. In terms of the application of partial differential equations, Prof Gui focuses on the application of Mathematics in image processing. By designing well-functioning algorithms and utilizing computer processing, he hopes to solve problems in medical imaging and scientific detection while also establishing a solid mathematical theory.

The reasons for choosing the UM

Regarding his decision to join UM, Prof Gui expressed his desire to return to China to further participate in Mathematics teaching and research. “UM provides me with the opportunity to continue my research in a new and dynamic environment with highly motivated students.” He says. As the Head of the Department of Mathematics, Prof Gui hopes that his joining will help to further enhance the research level and atmosphere of the entire mathematics department and lead the department to achieve more and better mathematical results.

Useful advices for students

Regarding young students who interested in pursuing Mathematics research, Prof Gui advises, “It is crucial to establish a solid foundation in Mathematics, while not limit yourself to a narrow field. Student needs to explore and understand various branches of Mathematics.” Regarding Mathematics research, Prof Gui hopes that students should not be too hasty in seeking results after conducting the research projects and should not rush in papers publication. They would not be able to explore the potential of Mathematics or the joy and satisfaction of learning, and they would lose the motivation to study Mathematics if they only pursue the quantity of papers and ignore the pursuit of quality. Prof Gui also suggested that as a student of Mathematics, it is necessary to possess a strong foundation in fundamental mathematical knowledge and techniques, as well as adopting a multidimensional approach to problem-solving and seek out methods for resolving mathematical difficulties. Regardless of focusing on Applied or Theoretical Mathematics, it is only through the cultivation of these skills that student can achieve fruitful results in mathematical research.