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Lee Sang Hyeok, Professor in the Department of Mathematical Sciences
Solving mathematical conundrums appear to be a sudden encounter with their solutions at a “Eureka!” moment while an individual wrestles with these problems. It is a serious misunderstanding. In reality, the journey is an arduous quest, in which an explorer can reach the end after turning over every stone despite perpetual tedium. The advice of Professor Lee Sang-hyeok (Professor at the Department of Mathematical Sciences), who has silently tolerated the process wherein a shift emerges at the end of the road, sounds like a sobering alarm in an atmosphere that emphasizes merely immediate results.
“It was in the third grade of elementary school when I first realized my affection for mathematics. I would spend all summer vacation grappling with a thick puzzle book included in a workbook that I purchased. Around my second year of middle school, I became smitten with mathematics again while learning about triangles and circles. Because geometry was so enchanting, I sometimes studied a dozen pages ahead of class. However, I ultimately chose to pursue my undergraduate degree in electrical engineering rather than mathematics. While majoring in the electrical engineering and computer science program at Pohang University of Science and Technology, my enthusiasm for mathematics was revisited because I was constantly captivated by the beauty of this discipline. Students frequently face numerous unclear theories in their learning. However, mathematics is different. It must be incredibly lucid and perfect for anyone to understand. The pursuit of truth without unnecessary details fascinates me.”
In 2021, when the COVID-19 pandemic was rampant, Professor Lee marked a significant milestone in the history of mathematics by elucidating the “boundedness of maximal functions on space curves”—one of the key challenges to harmonic analysis. Harmonic analysis is a branch of mathematics that investigates complex functions in fundamental wave units, such as sine and cosine. “Since the 1970s, there have been active studies on proving that maximal functions defined by surfaces and curves are bounded* between certain values without diverging to infinity. While the boundedness of maximal functions on plane curves was proven in 1986 by Jean Bourgain, a winner of the Fields Medal—widely regarded as the Nobel Prize of Mathematics—the boundedness of maximal functions on space curves defined in three or more dimensions had remained unresolved. My co-author and I sought to find a solution.” What he proved in his research article is “the maximal space curve theorem, which states that a necessary and sufficient condition for a maximal function for a curve with non-zero torsion and twists in three dimensions to be bounded in a Lebesgue space** is that the integral exponent of that space should be greater than 3.” This research was recognized for its originality and importance, and was published in Inventiones Mathematicae, one of the most prestigious academic journals in mathematics.
Professor Lee likened the process of trial and error in solving mathematical conundrums to his visit to his grandparents’ house in his childhood. “While I was walking across a long, tedious mountain path holding my mother’s hand, I encountered a moment out of weariness, wondering ‘Is this the right path?’ However, the minute I turned around the mountain corner, another path immediately opened up.”
Professor Lee cited Swedish mathematician Anders Karlsson, who won the Abel Prize in 2006, as an example. “Anders Karlsson addressed the problem of whether a given function converges to each point in the Fourier series. In fact, at the age of 17, a brilliant mathematician named Andrey Kolmogorov had already shown that certain functions do not converge. Therefore, most mathematicians had believed, ‘Oh, there would be no converging functions.’”
Karlsson, initially adhering to the common notion in academia, strived to prove the impossibility of convergence. For 10 or more years, he repeated the tedious routine of proceeding and abandoning in the middle of his exploration. Meanwhile, he suddenly realized that “this might work.” He changed the courses of exploration and found a solution two to three years later. He had obtained the answer in the opposite direction.
Professor Lee Sang-hyeok, walking under the curved surface of Pungsan Madang.
His proof on the boundedness of maximal functions on space curves can be summarized with the terms “complexity,” “three-dimensional curves,” and “irregular functions,” as reflected in the façade of Pungsan Madang, with its structures of curves, surfaces, repetitions, and asymmetry.
“I constantly emphasize to my students that they should become creators of new mathematics. Motivation is crucial for creating something new. In this context, mathematics and art share many similarities.”
An audacious confrontation with quandaries in mathematics does not necessarily lead to instantaneous fame or fortune, not to mention an immediate solution. There is a need for enthusiasm about tackling problems. “You should move forward until you reach the dead end of one possibility. If you only dabble in this and that, nothing will ever be resolved. You will end up wasting time and contributing nothing to solving any problem. Shift emerges when you reach the end of the road.”
“Shift cannot be compelled.
It happens naturally, like water overflowing,
as experience accumulates and energy surges.
Shift is an insight and a reward
given to those who never abandon.”
