Mathematically, such laws are expressed through differential equations or, more generally, dynamical systems.
Dr. Shouchuan Hu, professor of mathematics at Missouri State University, has devoted his career to the study of differential equations, dynamical systems and nonlinear analysis. This area of mathematics largely asks theoretical questions. More often than not, it is impossible to write down the explicit solution of the differential equations that describe physical processes.
“Often in mathematics we’ve been taught to say that graphics will show you or give you some intuition and ideas, but they will never serve as a proof.” — Dr. Shouchuan Hu
Therefore, Hu and others in this field investigate whether or not solutions exist. If a solution exists, the mathematician determines if the solution is unique and under what conditions and studies the qualitative behavior of the solution — a topic that often relates back to behavior of physical systems.
As an example, think about planetary motion.
“Certainly if you look at the planet orbits, they are periodic,” said Hu. “Such motions can be governed by a common differential equation, and sometimes a solution of such an equation is exactly the orbit of this motion.”
The ideas and methods used to study the differential equations of planetary motion can be applied to biological questions, like whether an animal species population could be periodic or environmental conditions such as seasonal weather changes.
One of Hu’s long-term collaborators is Dr. Nikolaos Papageorgiou, professor of mathematics at National Technical University in Athens, Greece. Together they produced two textbook volumes on multivalued analysis – a subfield of nonlinear analysis – that has been praised for its exhaustive depth in the field.
Papageorgiou is excited by the number of non-mathematicians who benefit from their work.
“Almost all interesting problems in nature and society are highly nonlinear and the traditional tools of classical linear functional analysis do not suffice to tackle them,” said Papageorgiou. “Today, many nonmathematical disciplines use the notions, techniques and theories of nonlinear analysis to come up with more realistic models of the phenomena they study and to analyze them.”
When Hu came to Missouri State in 1987, he was the first professor from mainland China to join the faculty, and he quickly made a name for himself in his field. In 1995, he helped launch the international journal Discrete and Continuous Dynamical Systems and has served as the editor-in-chief ever since. He also serves as the co-editor-in-chief for the international journal Communications on Pure and Applied Analysis.
Though it can be a time-consuming and intimidating task to produce a peer-reviewed academic journal, he is proud of the quality of contributors it attracts. Seven Fields Medal recipients have contributed papers over the years.
“The Fields Medal – it’s basically like a Nobel Prize for mathematicians. In a way it’s even more prestigious than the Nobel Prize because it’s only distributed every four years,” he said.
He’s also been able to bring that level of esteem to one of his other notable achievements: In 1996, Hu was the principal organizer of an international conference on Dynamical Systems, Differential Equations and Applications and has served as the chairman for this conference since its inception.
Many Field medalists have presented their research at this biennial conference, which has grown to be the largest conference in the field. From 250 attendees in the first year, it catapulted to regularly include 1,500-2,500 attendees. This international group had its first meeting at Missouri State, and Hu said, it put the university mathematically on the map.
“It’s the largest, most attended conference in the field,” Hu said. “This conference series has become so influential, and it has fulfilled a need in the scientific and mathematic community. Everywhere you go, at least in every mathematical department, some of the faculty will know us.”
In 2000, Hu helped form, and currently directs, the American Institute in the Mathematical Sciences. AIMS is now responsible for the organization of the conference, and it publishes 14 journals in a variety of areas of the mathematical sciences, and publishes a book series related to dynamical systems and applications. To a large extent, AIMS represents Hu’s vision toward furthering mathematical research.
He has also published 76 refereed research articles – many in collaboration with Papageorgiou – in the area of differential equations and dynamical systems since becoming a faculty member at Missouri State.