The 2016 and 2018 HLMA Gold Award winners discovered the joy of mathematics research during their HLMA journey, and have since aspired to pursue a career in research, with the hope that their research results can inspire the next generation. Ken is currently a PhD candidate in Mathematics at the University of Montréal with a specialisation in analytic number theory. David graduated with First Class Honours in Mathematics from the University of Cambridge, and is now a PhD candidate in Mathematics and conducting research on algebraic geometry at Yale University.

The process of research can be strewn with obstacles and is often likened to navigating in the dark: researchers constantly wonder if they are headed in the right direction, how much further they have to go, or if their paths will lead to anything fruitful at all. In the case of Ken and David, how did their enthusiasm for mathematics drive them to devote themselves to mathematics at such a young age? How did they find the motivation and energy to overcome all kinds of obstacles and keep going?

C:

L: Ken Leung

B: David Bai

L: I am specializing in analytic number theory, and the focus of my current research is the distribution of primes in short intervals and arithmetic progressions. It has been the most inspiring subject that I have worked on to date. Prime numbers, as natural arithmetic objects, are concrete yet mysterious. The study of prime numbers dates back to ancient Greece, but despite its long history, numerous unsolved problems persist in this field, such as the twin prime conjecture, the Goldbach conjecture, and the Riemann hypothesis. It has become my motivation to make an important contribution to this field.

B: I am currently working on algebraic geometry, which is an interesting field that has its origins in the study of systems of polynomial equations and the geometric objects defined by them. Recent developments in algebraic geometry have given rise to new frameworks, contributing to innovative achievements in both geometry and number theory.

I developed an interest in this field when I first learned about the Cayley-Salmon theorem, which states that any smooth cubic surface contains exactly 27 lines. I was fascinated by the fact that a theorem, despite its simplicity, could reveal such an intriguing phenomenon in algebraic geometry.

L: I have always aspired to bring something new and exciting to the mathematics community. The driving force behind my passionate

pursuit of mathematics lies in the profound beauty it encapsulates. Mathematics may seem abstract and inaccessible to most people because its unique beauty and allure can only be truly appreciated through intellectual exploration. As Godfrey Harold Hardy expressed in his book, “A painter makes patterns with shapes and colors, a poet with words, and a mathematician with ideas.”

B: I have always dreamed of creating a consequential theorem of my own. What pushed me most in my mathematical journey is the joy of discovery – the magic moments of unveiling something new and original.

L: At present, we are witnessing major breakthroughs in several mathematical fields. I am optimistic about the developments in analytic number theory, which is my area of specialization. By proving the existence of infinitely many primes with bounded gaps, the well-known research by James Maynard and Zhang Yitang marks, a momentous advancement in addressing the twin prime conjecture. The recent recognition of Maynard’s work with the awarding of the Fields Medal has further underscored the significance of this achievement. Their success demonstrates the importance of believing in oneself and having the courage to tackle a seemingly impossible task. I hope that one day, I can contribute to this field and, more importantly, inspire the next generation of young mathematicians.

B: Currently, the Langlands program is one of the most creative and exciting areas of mathematics research. Widely hailed as the grand unified theory of mathematics, it is a collection of deep conjectures that predict a correspondence between important objects in number theory and representation theory. Although it is not my current focus, the Langlands program has generated ideas that have given me valuable inspiration.

I hope that my work will advance the public understanding of mathematics and pave the way for future generations to engage in mathematics research, in the same way that the mathematicians of the past inspired my pursuit of mathematics.

L: I believe that mathematics has significant applicability in addressing real-world challenges. It plays an indispensable role in artificial intelligence by helping solve complex issues such as cancer and other serious diseases, while supporting efforts to tackle global warming through simulation technologies. However, it is crucial to recognize that the application of mathematics is a double-edged sword that has both positive and negative consequences.

B: The ideas in quantitative reasoning help facilitate and inform key decision-making processes in society, as is particularly evident from the rise of data analytics in recent decades. In the age of data collection, knowledge of how to extract information from the existing data can provide a refreshing perspective for decision-making.

L: Academic research is a prolonged process, during which a significant amount of effort and time does not always translate into immediate results. Hence, it is essential to stay motivated despite the challenges. Mathematicians are not machines. Taking an occasional break to recharge and spend time with family and friends is the key to progress.

B: When I get stuck, I often try to talk to people from other fields of studies, as ideas from different research areas can sometimes lead to unexpected inspiration.

L: HLMA was not my first encounter with mathematics research. I had previously participated in several smaller projects related to elementary number theory, but HLMA has been my most memorable experience. Despite not making much progress for months during the preparation process, I persevered and did not give up. Eventually, I had a stroke of inspiration and managed to complete the proof within days. This experience taught me that while inspiration is crucial, breakthroughs do not happen spontaneously. They are not the products of miracles, but of hard work and repeated trials and errors.

B: HLMA was my first exposure to mathematics research. Many years have since passed, but I still vividly recall the moment I discovered the main theorem of my study. It took me a few attempts to obtain the complete proof, but the moment of realization was a memorable experience that has since been a key driving force in my mathematical endeavors.

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