Finding elegance in the everyday: Making mathematics matter for everyone

The goal of the IDM is to highlight the value of mathematics and mathematics education in scientific research and in everyday life. This year, the IDM seeks to do so through its theme, Mathematics for Everyone, which is meant to prove that mathematics isn’t an arcane body of knowledge accessible only to geniuses and people adept at arithmetic.

For IDM this year, the National University of Singapore (NUS) is spotlighting three of its eminent academics. From information theory to group testing and fluid dynamics, these scholars are proving that mathematics is not merely an abstract field of study but is a diverse area with a wide variety of potential applications.

As a kid, Assoc Prof Vincent Tan gravitated toward puzzles that, though challenging, had particularly elegant solutions. He couldn’t have known it at the time, but this eye for elegance would be a major driving force behind his career.

In school, Assoc Prof Tan’s excellent track record in mathematics and science won him a scholarship to study engineering at Cambridge University. In his third year, taken by the “elegance of the algorithms and theorems in signal processing”, he decided to specialise in electrical engineering as it was “the most mathematical of the engineering disciplines”. He went on to pursue a PhD degree in electrical engineering and computer science from the Massachusetts Institute of Technology.

He returned to Singapore to do part of his post-doctoral training at the Agency for Science, Technology and Research (A*STAR) and later took on a tenure-track position as an Assistant and then Associate Professor at the Department of Mathematics and the Department of Electrical and Computer Engineering (ECE) at NUS. Along the way, however, his passion for elegance would bring him to one of his biggest career moments.

All forms of electronic and wireless communication occur over channels that are inherently noisy and, as a result, have fundamental limits on the amount of information that they can transmit. In 1948, American mathematician Claude Shannon defined this maximum amount of information—known as the capacity—which has since served as a theoretical target for research groups to strive for.

 “While elegant, Shannon’s channel capacity is a so-called asymptotic notion,” Assoc Prof Tan explained, adding that very long codes are needed to achieve it, which can be highly impractical in real-world settings.

With fellow ECE Associate Professor Marco Tomamichel, Assoc Prof Tan significantly extended Shannon’s theory to “provide a more refined benchmark for communication engineers and code designers”. Their work has already served as a benchmark for new families of codes and could lead to better communications technologies in the future.

As with elegance, this drive for applicability also underpins much of Assoc Prof Tan’s career. “Most, if not all, of my work is theoretical and mathematical in nature,” he said. However, his research questions almost always stem from real-world needs and potential applications.

 Mathematics, after all, is the foundation on which the applied sciences are built on. Citing ChatGPT as an example, he elaborated, “The inner workings of ChatGPT is machine learning, and specifically generative models. The amount of mathematics that goes into designing such a complex model that can give us such high-quality text is simply mind-boggling!”

In fact, mathematics is also responsible for safeguarding our transactions when we use our credit cards. They rely on secure data transmission enabled by “the practical difficulty of factoring a large number into prime numbers,” Assoc Prof Tan explained.  

“Thus, an area as esoteric as number theory has immense practical applications,” he said. “Mathematics is all around us.”

The inclination towards practical applications is also an important driving force behind the career of Assoc Prof Yao Yao from the NUS Department of Mathematics.

Since young, she had always enjoyed mathematics puzzles, which made mathematics an easy choice for her undergraduate programme.

Getting through university, however, was not as simple. “I actually struggled with it during my undergraduate years,” Assoc Prof Yao said. “Abstract algebra was difficult for me, but analysis is more natural to me since one can visualise things to gain intuitions.”

This factored in her decision to pursue partial differential equations (PDE) research, a branch of mathematics that lends itself to applications in physics and other sciences.

At first, Assoc Prof Yao found the area a bit too technical. After she had spent time familiarising herself with the field’s central equations, she started seeing the central ideas behind the technical details.

“Whenever I finally connect the dots to see the big picture – these are the moments that bring me great joy and sustain my drive,” she said, and in chasing these professional highs, Assoc Prof Yao arrived at one of her most pivotal career moments to date.

While fluids are very common in nature, their movement and dynamics remain as some of mathematics’ most complex and enigmatic phenomena. Pinning down fluid mechanics equations are so difficult that some questions have remained unanswered for centuries.

Together with collaborators, Assoc Prof Yao was able to devise various mathematical techniques to study fluid equations. Though still hypothetical, her work could deepen the understanding of how fluids move at small scales, and help other research groups advance their own analyses.

“These findings may have applications in developing more accurate numerical simulations and understanding the behaviour of real-world fluids,” she said.   

Aside from her research work, Assoc Prof Yao’s drive to anchor mathematics in concrete, practical scenarios also shape much of her teaching. In her classrooms, she grabs her students’ interest by relating the course material to real-life examples and problems. She uses playground swings, for instance, to make forced vibration in differential equations more relatable.

“I personally find these examples a lot of fun, and I hope my students enjoy them as much as I do,” she said.

Like Assoc Prof Tan and Assoc Prof Yao, Asst Prof Jonathan Scarlett, who has appointments in the Department of Computer Science, Department of Mathematics and Institute of Data Science at NUS, was captivated by the elegance of mathematical theory and the breadth of its applicability.

Mathematics was also an early academic favourite for Asst Prof Scarlett, as he had always been more comfortable with “logical and rigorous” areas of study, as opposed to those that were more subjective. Later, in university, he was drawn to the field of information theory “because it gave a unique combination of being very mathematically elegant and also very practically useful.”

Eventually, however, Asst Prof Scarlett found himself in the field of data science. One of the reasons that the field appealed to him is that its inspirations and rationales for research are often grounded in real and practical problems. “There can of course still be quite sizeable gaps between theory and practice, but at least in terms of motivation, things have usually been concrete right from the start,” he said.

One of his topics of primary interest is group testing, which has proven valuable for maximising the efficiency of public health testing strategies in earlier decades and also in recent years, including during the COVID-19 pandemic.

“The idea is to pool the samples of individuals together and test the entire combined sample using just a single test,” Asst Prof Scarlett explained. “If the test is negative, then we can classify all of the pooled individuals as being healthy. If the test is positive, then additional testing can be performed to resolve the remaining uncertainty.”

During the early days of the pandemic, countries like the US, China and India employed group testing because this approach could help get the most out of the limited resources available.

For his work in group testing, Asst Prof Scarlett was recognised in the 2021 MIT Technology Review Innovators Under 35 Asia Pacific List.

Moving forward, he hopes to continue bridging the most important gaps in group testing theory and practice by focusing on its more practical aspects, such as testing constraints and application-specific prior knowledge. On a grander scale, however, Asst Prof Scarlett wants his work to shape the field of data science at large.

“I hope to strengthen ties between the mathematical foundations and practical developments of problems in data science. I believe that information theory has a lot of potential to benefit these areas in the same way that it originally impacted communication,” he said.