L4 BK11 The Sieve, the Soil, and the Masters

Why do some education systems produce excellent test-takers, while others nurture people capable of changing an entire field? Through the stories of mathematicians Wang Hong and Deng Yu, this book examines three forces behind exceptional talent: the sieve that identifies potential, the soil that supports long-term growth, and the masters who guide gifted students towards original achievement.

Sieve, Soil, and Master

What the Fields Medals of Hong Wang and Yu Deng Reveal

FANG TIANLIANG

Prologue: Two Medals, Two Paths

Philadelphia, 23 July 2026.

In the opening-ceremony hall of the Pennsylvania Convention Center, the spotlights fell on four faces. The International Congress of Mathematicians, held once every four years, had just announced this year's Fields Medalists — mathematics' most prestigious honor, awarded no more than four times per cycle, and only to mathematicians under the age of forty who have made outstanding contributions.

Two of the four winners were from China. Hong Wang was recognized for breakthrough work in harmonic analysis and geometric measure theory, including a proof, with a collaborator, of the three-dimensional Kakeya conjecture — a problem that had confounded mathematicians for more than a century. Yu Deng was recognized for work at the intersection of partial differential equations and fluid dynamics; his research, with collaborators, achieved a historic advance on a key step of Hilbert's sixth problem, posed in 1900, on the axiomatization of physics.

It was the first time Chinese-born mathematicians had stood on this stage — and, remarkably, the first time two Chinese mathematicians had won in the same year, at the same congress.

That night, back home, social media lit up with two competing emotions at once.

One was pure elation. China had finally achieved a breakthrough in a field often called "mathematics' Nobel Prize" — and with two winners at once. "National pride" and "a mathematical powerhouse" trended across the comment sections.

The other reaction arrived faster, and was more complicated. People dug up Hong Wang's and Yu Deng's academic records and discovered that both had studied at Peking University, and that both had left after finishing their undergraduate or graduate studies there, going on to France and the United States for further training. A question began circulating, sharpening with each repetition:

Who, exactly, deserves the credit for this glory?

Peking University's official account posted congratulations within hours, calling the two "outstanding talents Peking University trained." Some independent commentators offered a sharply different reading: the two were "survivors who escaped Peking University," and their success owed less to the university than to what began only after they left it.

The argument did not stay confined to mathematical circles. It quickly became a question that millions of Chinese families could see themselves in: if two young people now standing at the summit of the world had each, at some point, felt like they did not fit inside the domestic education system, then what did that mean for an ordinary child struggling, confused, or doubting themselves in some classroom right now?

This book does not set out to answer the question of who "deserves" the credit — it will avoid that word altogether. Because once you look closely at how Hong Wang's and Yu Deng's paths actually unfolded, you start to suspect the question itself may be aimed in the wrong direction.

Hong Wang and Yu Deng started from nearly opposite places. One did not come up through competitions; her grades were, for a long stretch, unremarkable; her path was winding, slow, and often overlooked. The other was recognized as a prodigy from secondary school onward — an Olympiad gold medalist, carrying high expectations his whole way up. Two almost-opposite trajectories, arriving at the same podium.

If education only had one correct shape, this should not have been possible.

What is really worth asking, then, may not be "who made them" but something more fundamental: how, exactly, is a person's talent discovered? And under what conditions does it grow from "latent potential" into "actual creation"?

This book takes its title from the plainest metaphor for that question — three of them, in fact: sieve, soil, and master.

Exams, competitions, interviews, credentials, prizes — these are sieves. Their function is to identify certain abilities out of a crowd and pass them through to the next stage. A sieve is not, in itself, a bad thing; any society operating at scale needs some kind of sieve to handle basic sorting and selection.

But being sifted out is only the beginning of the story. A seed, once selected, still needs soil — time, room to fail, and an environment where thinking is allowed to grow slowly. Soil is what decides whether a seed that has been seen will actually take root and bloom.

But soil rarely generates itself. More often, it is cultivated, patiently and deliberately, by a specific person — a teacher, a mentor, an elder — through patience and trust. That person is what this book calls a master. A true master is not necessarily the most accomplished figure in a field, but someone willing to spend the time to truly see a person who has not yet taken shape, and generous enough to let that person surpass them.

The sieve decides who is seen first; the soil decides who keeps growing; and the master decides whether an unfinished person can recognize, sooner rather than later, their true direction.

The stories of Hong Wang and Yu Deng are an excellent way into understanding how these three forces relate to one another. But before telling their stories in full, one thing needs to be cleared up first: more than one version of their lives circulates online. Some come close to the truth. Some selectively emphasize a single part of it. And some do not hold up under scrutiny at all.

Before we step into Hong Wang's and Yu Deng's real stories, we need to work out where these conflicting accounts actually come from.

· · ·

Interlude: Why One Person's Success Comes in Three Different Versions

On the Chinese internet, roughly three versions of Hong Wang's formative years circulate.

The first might be called "relay education." It holds that Hong Wang's success belongs to no single school or country, but is the joint product of early encouragement at home, undergraduate training at Peking University, a master's period in France, doctoral training at MIT, and a series of collaborators afterward — with no single leg of the relay being redundant, and none of them sufficient on its own to explain the outcome.

The second is "the elite-school account." This version emphasizes the foundation Hong Wang built as an undergraduate in Peking University's mathematics program — her systematic training, the caliber of her classmates, the academic resources the university provided — treating these as the true starting point of her later work at the frontier. Peking University's own statement after her win largely follows this framing.

The third version, the most widely shared and the most dramatic, might be called "escape and reversal." It tells the story of a student overlooked, even systematically marginalized, at home, who through sheer personal persistence went abroad, was rediscovered and properly cultivated in foreign academic soil, and rose all the way to the summit of the world. It comes with details vivid enough to belong in a film: grades that languished at the bottom of her class throughout her undergraduate years; not a single mathematics professor willing to write her a recommendation letter; crying alone in a bathroom in one of the teaching buildings.

All three versions are trying to answer the same question — who made her — but the quality of evidence behind them is not the same at all.

The "relay education" account can be reasonably well corroborated across the multiple public interviews Hong Wang and Yu Deng have since given, including their own specific descriptions of what each stage of their education contributed. The "elite-school account" selectively emphasizes one stage of that relay but does not conflict with the known facts. The "escape and reversal" version is true only in the broad sense that both did eventually leave Peking University to study abroad; several of its most dramatic specific details have since been shown not to hold up. China's Joint Internet Rumor-Refutation Platform confirmed that the claims of "grades that languished at the bottom of her class throughout undergraduate study," "failing to meet the bar for exemption-based graduate admission," and "no professor willing to write her a recommendation" are all inaccurate. Chen Dayue, former dean of Peking University's School of Mathematical Sciences, has stated that Hong Wang was one of 49 students named "outstanding graduates" among 269 graduating students — undergraduate, master's, and doctoral combined — from the mathematics school's class of 2011; that she was in fact eligible for exemption-based graduate admission and simply did not apply for it because she had already decided to study abroad; and that she received recommendation letters from faculty members Wang Fuzheng, Tian Qingchun, and Wang Lizhong when applying to study in France, having also worked with Wang Lizhong on undergraduate research.

As for the detail about crying alone in a bathroom — the only sources that can currently be traced for this are anonymous social-media posts by people claiming to be "Hong Wang's classmates at Peking University." This kind of information has a common feature: the chain of transmission cannot be traced back to its origin, and there is no way to verify who first told the story, or whether it was told accurately. It may contain a kernel of real emotion, or it may be something added and dramatized as it spread online. This book will not repeat it as a confirmed episode in her life — but the fact that it exists at all is, in itself, worth examining. Why did the public feel such a need to believe that a young person who had just won the highest honor in mathematics had "nearly been buried" by the domestic education system?

The answer to that question may have less to do with Hong Wang's actual experience than with an emotion nearly everyone who has been through China's education-competition system shares: almost everyone, at some stage, has wondered whether they were the one who went unseen. A story about a genius overlooked and then rediscovered spreads quickly, and keeps accumulating detail, because it channels so much accumulated grievance and hope belonging to ordinary people. In telling Hong Wang's story this way, people are, in a sense, telling their own.

What Happened to an Interview as It Was Reprinted

On the evening of 23 July 2026, Peking University's Beijing International Center for Mathematical Research (BICMR) published its full interview with Yu Deng, titled "A Conversation with Fields Medalist Yu Deng: Traveling Between the Critical and the Long-Time," numbered question by question across twenty-four items. The following day, Peking University's official public account reprinted the piece under the title "An Interview with PKU Alumnus Yu Deng."

Comparing the two versions line by line shows that the reprint cut an entire question from the original — question 21, about whether and when students should study abroad, and how to think about eventually returning to China — and also removed a qualifying remark Deng made while discussing the differences between Chinese and American educational environments. In the original, he noted that he had been away from China's academic environment for a long time, that his first-hand knowledge of current conditions was limited, and that this made it hard for him to offer a sweeping judgment about which environment was better. With that qualifier removed, the remark that immediately follows — about Peking University laying the foundation and MIT offering exposure to the frontier — reads, in the reprint, much more like an unqualified conclusion.

These are the differences that can be directly compared and verified, word for word, between the two public versions. As for why the edits were made, this book does not speculate.

· · ·

Part One: Two Different Paths of Growth

Chapter 1: Hong Wang — A Different Kind of Start

Hong Wang was born in 1991 in Guilin, Guangxi, the daughter of a mathematics teacher. In an interview with China Science Daily, she recalled getting her hands on a primary-school mathematics textbook when she was five or six years old; the "think about it" problems in that book gave her the first taste of the pleasure of solving something — her love of mathematics, from that point on, was a matter of genuine interest, not an assigned task.

But she was not the kind of prodigy popular imagination tends to picture — someone who dominates every stage from childhood on. In secondary school, she never won a major mathematics competition, placing only in an ordinary provincial prize at one high-school math league. In her first year of high school her class rank fell outside the top hundred; by her second year she had risen into the top sixty; only in her third and final year did she break into the top ten. In 2007, having skipped two grades in primary school, Hong Wang was admitted to Peking University at sixteen.

That year, Peking University's mathematics department accepted only one student from Guangxi, and Hong Wang did not make the cutoff. She was placed instead in the School of Earth and Space Sciences. From her first day, she began preparing to transfer — under the rules at the time, a transfer required a strong class rank in both her original major and a transfer examination, a process she herself would later call a "second gaokao." She had to study coursework for two majors simultaneously, and because of credit limits could only formally enroll in two of the mathematics department's three core courses, teaching herself the third, mathematical analysis, on her own. In her second year, she finally transferred successfully into the mathematics department.

The real difficulty began only after that transfer. University-level mathematics bore little resemblance to secondary school, and it took Hong Wang a long time to work out, on her own, how to study mathematics systematically. Most of her classmates had been admitted directly through competition results — including her classmate Yu Deng, later a fellow Fields Medalist, whom classmates at the time called "Deng the God." Hong Wang, by her own account, was simply "struggling — surviving was already an achievement." Tian Gang, a Peking University professor and member of the Chinese Academy of Sciences, offered a similar assessment in a media interview: "[At the time] she wasn't a competition student, and she wasn't the most eye-catching student in her cohort either — but through her love of analysis and years of hard work, she eventually made a breakthrough."

In 2026, a number of online rumors claimed that Hong Wang's undergraduate grades had languished at the bottom of her class, that she had failed to meet the standard for exemption-based graduate admission, and that not a single mathematics professor had been willing to write a recommendation letter for her overseas applications. China's Joint Internet Rumor-Refutation Platform confirmed that none of these claims hold up. According to Chen Dayue, former dean of Peking University's School of Mathematical Sciences, Hong Wang was one of 49 students named "outstanding graduates" among 269 undergraduate, master's, and doctoral students who graduated from the mathematics school's class of 2011. When she applied to study in France, she received recommendation letters from faculty members Wang Fuzheng, Tian Qingchun, and Wang Lizhong, and had worked with Wang Lizhong on undergraduate research. Her grades at the time were, in fact, good enough to qualify her for exemption-based graduate admission — she simply did not apply for it, because she had already decided to study abroad.

In other words, what Hong Wang actually lived through at Peking University was a real, if often oversimplified, situation: she genuinely struggled for long stretches, and genuinely thought of herself as an unremarkable presence among her classmates — but that authentic sense of setback is not the same thing as being "weeded out by the system," "unrecognized by anyone," or "left with nowhere to go." Real self-doubt and real institutional support are not mutually exclusive; they can both be true at once. As an undergraduate, Hong Wang was neither the protagonist of the "triumphant-underdog" story that later circulated online, nor an effortlessly gifted prodigy — she was an ordinary student, surrounded by formidable peers, who found her footing slowly through years of sustained effort.

After graduating, Hong Wang went to France for further study, enrolling at the École Polytechnique. By her own account, her French advisor Yvan Martel supervised a reading course she took and told her that if she ran into any difficulty while working through the material, she could simply save the questions up and ask them at their next meeting. This, she has said, was the first time she realized she did not have to solve every problem alone — that people existed who were willing to help her — and it made her more patient, and less anxious. During her master's, she spent one semester on a detour into architecture, interning at a firm in Paris; that departure ultimately confirmed her choice of mathematics — she later recalled that architecture required persuading others to invest before you could build anything, which limited creative freedom, whereas mathematics could be "constructed" entirely from one's own ideas. The École Polytechnique required students to complete an internship or a research visit before graduating, and Hong Wang chose MIT — Martel helped connect her with colleagues there. In 2013, during that research stint at MIT, she attended a seminar by the mathematician Larry Guth on the Erdős distinct distances problem. Guth, working with Rice University mathematician Nets Katz, had proven the problem, and he explained his reasoning with such clarity that Hong Wang found she could follow most of the proof — and, in doing so, rediscovered the pull of mathematics itself. The following year, she was admitted to MIT's mathematics PhD program and chose Guth as her advisor — a leading authority in geometric measure theory and harmonic analysis.

Hong Wang has described Guth this way: his defining traits are patience and empathy. When she got stuck mid-discussion — what she calls "freezing at the blackboard" — Guth never rushed her, always watching with a smile while she kept thinking, never pressing ahead to the next question. When she brought him half-formed, even flawed ideas, he never interrupted or dismissed them; he listened all the way through, encouraging her to "pour out" everything in her head, however jumbled, before helping her sort through it together afterward. "Those conversations," she has said, "taught me how to clear my own head." Guth also encouraged her to rest — "whenever he heard I was going to go have fun, he was delighted."

Her doctoral research centered on one of the most famous open problems in geometric measure theory: the Kakeya conjecture, proposed in 1917 by a Japanese mathematician — if a needle is allowed to rotate through every possible orientation, how small can the area it sweeps be made? In two dimensions, mathematicians had shown early in the twentieth century that the area could be made arbitrarily close to zero. The three-dimensional case remained unsolved for more than a hundred years afterward.

Proving the Kakeya conjecture was not a target Hong Wang set out for from the start; it emerged gradually, out of a long-running program of research into a family of related problems in harmonic analysis. She earned her PhD from MIT in 2019 and went on to a postdoctoral position at the Institute for Advanced Study in Princeton. In 2020, she found herself stuck on the Falconer distance-set problem. Around that time, she noticed that a colleague had made progress by assuming a certain kind of geometric "stickiness" condition, and she recalled a 2014 blog post by mathematician Terence Tao suggesting that any counterexample to the three-dimensional Kakeya conjecture would likely have to display similarly sticky characteristics. She reached out to Joshua Zahl, who was working on a related question, to propose a collaboration. The two spent two years on the problem — first proving that no such "sticky counterexample" to the Kakeya conjecture could exist, then gradually generalizing the result — and in 2025 completed a full proof of the three-dimensional Kakeya conjecture, in a paper that ran to 127 pages.

She went on to hold positions as an assistant professor at UCLA and a professor at New York University, and in 2025 became a tenured professor at France's Institut des Hautes Études Scientifiques. In 2026, she was awarded the Fields Medal for this work and for a broader series of breakthroughs in harmonic analysis and geometric measure theory — becoming only the third woman ever to win the prize, and the first winner born in the 1990s.

Rather than a single dramatic turning point, Hong Wang's path looks more like a slow, continually adjusted curve of accumulation: from Guangxi to Peking University, from France to MIT, from an unremarkable presence in a competition-driven system to a mathematician who solved a century-old problem. That trajectory itself may be worth understanding more carefully than any single "moment of reversal."

Chapter 2: Yu Deng — From Competition to the Unknown

Yu Deng was born in 1989 and, like Hong Wang, entered Peking University's School of Mathematical Sciences in 2007. He took up the game of Go as a child, and in 2001, as a first-year middle-schooler, competed in China's national Go dan-ranking tournament — an event known informally as the "Go gaokao," with an acceptance rate under ten percent. In an interview with China Science Daily, he recalled learning Go in Shenzhen at a level comparable to Zhou Ruiyang, who would later become a nine-dan professional and world champion: "We played each other a few times, wins and losses on both sides. He didn't make dan rank that year either — he made it the following year. So I think if I had kept playing Go, I might have ended up somewhere similar to him." That year marked a fork in the road he narrowly did not take toward a professional Go career, though the game never really left him — it remained a lifelong hobby he used to balance study and research.

In high school, he won a gold medal at the China Mathematical Olympiad, then represented China at the International Mathematical Olympiad, and was admitted to Peking University's School of Mathematical Sciences on the strength of that gold medal, without needing to sit the national college entrance exam.

According to the China Science Daily interview, Yu Deng was close at Peking University with classmates Liu Yuchen and Xu Disheng among others; the group often worked through problem sets together and once collectively pushed through a particularly demanding topology textbook, spending weekends browsing for math books at Haidian's book market. He has described that period this way: "The foundation I built in those two years was quite solid."

After two years of undergraduate study, Yu Deng transferred to MIT, graduating in 2011, and went on to Princeton for his PhD. Various online accounts attribute his departure from Peking University to dissatisfaction with the domestic emphasis on competition and rote problem-drilling, framing it as a search for "pure mathematics" free of exam pressure. It is worth noting that this account does not entirely match what Deng himself has said in public interviews. In the original BICMR interview, he stated plainly that his first two years at Peking University had given him a very solid foundation, and that the MIT period exposed him to more of the field's frontier — adding that "this difference may mainly come from being at different stages of study, rather than being entirely a difference between the schools or the countries." He did not describe the transfer as a rejection or an escape from his education at Peking University.

After his PhD, Yu Deng's research did not head directly toward Hilbert's sixth problem. Instead, he spent several years on what was, by comparison, a smaller question: how to rigorously derive the wave kinetic equation — which describes the statistical distribution of wave energy — from the Schrödinger equation, which describes the waves themselves. In 2019, he published the first paper in this line of work with Zaher Hani of the University of Michigan; in 2021, the two completed "Full Derivation of the Wave Kinetic Equation," later published in the top international journal Inventiones Mathematicae. That same year, Yu Deng received a Sloan Research Fellowship, an important early-career honor for young scholars in North America.

The real turning point came only after this earlier body of work had matured. In 2021, while visiting Brown University's Institute for Computational and Experimental Research in Mathematics, Deng had a flash of insight: the technical tools he had built to study wave systems might extend to something far larger — the core of Hilbert's sixth problem, posed in 1900: how to rigorously derive the Boltzmann equation, which governs macroscopic fluid behavior, from Newtonian mechanics describing the collisions of microscopic particles. The question had remained unresolved for a hundred and twenty-five years. In 2023, Deng and Hani met the young mathematician Xiao Ma — a recent Princeton graduate who, like Ma's advisor, had studied under Ioannescu — at an academic conference, and the three formed a collaboration. Between 2024 and 2025, they published two landmark papers, running to hundreds of pages, completing the core derivation.

In 2026, Yu Deng and Hong Wang won the Fields Medal together. He now teaches at the University of Chicago. Asked for advice for young researchers, he said in the interview: do what you genuinely love. "Whether others recognize you, and how your results turn out — these things aren't entirely within your control. But if what you're doing is something you truly enjoy, if you can take pleasure in the process of thinking, exploring, and researching, then that pleasure is itself a form of reward."

Compared with Hong Wang, Yu Deng's path looks, on its surface, much closer to the classic "child prodigy" script — Olympiad gold, direct admission to an elite university, one smooth step after another. But assuming his path was therefore simpler, or easier to replicate, risks missing something important: the transition from a competitor skilled at solving assigned problems within a set time limit, to an original researcher capable of sitting with patience, sitting with confusion, and eventually finding a way through a problem that had defeated everyone for a hundred and twenty-five years, was itself no small transformation. How that transformation actually happens is the question the rest of this book keeps returning to.

· · ·

Chapter 3: Talent Is Not a Finished Work

Before going further, two words need to be defined clearly, because this book will keep returning to them, and they are rarely defined with any precision: talent, and success.

Most of the time, people talk about "talent" as though it were a finished object, simply waiting to be discovered — like a mineral deposit buried in the ground: some people have it, some don't, and all education has to do is find the people who do.

But the stories of Hong Wang and Yu Deng do not fit neatly into that metaphor.

If talent were a finished work, it ought to reveal itself early, in a clear and recognizable form — the way Yu Deng's did, winning an International Mathematical Olympiad gold medal in secondary school, ranking among the best in the world, instantly recognizable to any teacher, any exam. His talent largely does fit the "mineral deposit" metaphor: buried shallow, and easy for standardized selection to find.

Hong Wang was not like that. She never won an Olympiad gold; her undergraduate grades were never the strongest in her cohort; entering the mathematics department required a transfer in the first place. Judged solely by "did she stand out on exams," it is hard to find, anywhere in her undergraduate record, a clear signal pointing toward a future Fields Medalist.

That is not to say Hong Wang lacked talent then — quite the opposite. Her later ability to independently complete the core proof of the Kakeya conjecture could not plausibly have appeared out of nowhere during her doctoral years. The more reasonable explanation is that her talent already existed as an undergraduate; it simply had not surfaced in a standardized form — competition results, exam scores — and so the evaluation systems of the time failed to detect it.

This is where the book's definition of "talent" comes from: not a finished object waiting to be found, but something closer to an as-yet-unformed capacity for learning, a sensitivity to problems, and a sustained curiosity — one that needs the right problem, the right amount of time, and the right way of being observed, before it can move from existing to being seen. The same underlying talent, met by a sieve well-suited to detecting it, may surface early. Met by a sieve poorly suited to detecting it, that same talent may lie dormant for years, or even be mistaken for mediocrity.

"Success" needs the same kind of redefinition. If this book simply equated success with "winning the Fields Medal," it would fall into a contradiction of its own making: criticizing exams and competitions for judging people by a single standard, while itself judging what counts as a successful life by a single world-class prize — which is really just the same sorting logic, with "gaokao score" swapped out for "international award."

So when this book speaks of "success," it does not mean any particular title or honor, but a more basic process: a person's potential converting into sustained capability, and producing original or lasting value in a field suited to them. Winning the Fields Medal is one way that process has manifested in Hong Wang and Yu Deng — but it is certainly not the only way it can manifest. Other figures who appear later in this book never won any major prize at all, yet completed the same transformation from potential to contribution, and their stories deserve to be taken just as seriously as Hong Wang's and Yu Deng's.

Back to Hong Wang and Yu Deng. Placed side by side, what is most interesting about their two stories is not whose talent shone brighter, but that their talent surfaced at completely different times and in completely different ways — while depending, afterward, on remarkably similar conditions. Yu Deng's talent was recognized early, but recognition alone still left him years of work to do, transforming from a competitor skilled at problems that already had known answers into a researcher willing to face open questions no one knew the answer to. Hong Wang's talent waited far longer to find its true stage, but once she reached the right environment, she too needed years of sustained training and accumulation to complete the transformation from a promising student into an independent researcher.

In other words: when a person is seen only determines when they enter the right growth trajectory — not how far that trajectory still has to go. The two of them walked the same essential distance, at different paces. What conditions that distance actually requires — from surfacing potential to genuine creation — is exactly what the book's next two sections, on the sieve and the soil, will examine in detail.

· · ·

Part Two: The Sieve — What We Measure, and What It Costs

Chapter 4: The Inevitable Assembly Line — Why We Need Sieves at All

Before discussing the limits of sieves, it is worth confronting a plain fact directly: in any society with a large population and relatively scarce educational resources, the sieve cannot simply be abolished.

China's gaokao, which tens of millions of families brace for every year; the fiercely competitive selection process for the International Mathematical Olympiad; the elaborate application-and-interview systems at North American universities — all of these are, in essence, sieves of different shapes. The reason for their existence is not hard to understand: in a system with limited information and limited resources, society needs some way to distribute a finite number of places that can scale reasonably well and minimize the influence of personal connections.

Peking University's School of Mathematical Sciences can only admit a limited number of undergraduates each year; MIT's laboratory resources are equally finite. Under resource constraints, "teaching to the individual" and "waiting patiently for each flower to bloom" are expensive ideals, and administrators of large systems need a measurement tool that can produce comparable results quickly.

Seen this way, sieves built around exams and competitions have their own legitimacy — one might even call them a practical invention. They are, in principle, relatively "blind": ideally, a gaokao paper or an Olympiad problem does not look at a candidate's family background, only at the reasoning written down within a set amount of time. For many young people from ordinary backgrounds, this kind of sieve has been an important channel for upward mobility.

Yu Deng's early trajectory is a case of this system working relatively smoothly. Layer after layer of increasingly difficult competition measured, with reasonable accuracy, his computational ability, spatial imagination, and logical steadiness under pressure — and carried him to the front of that particular track.

But acknowledging that sieves are necessary, and relatively reasonable, is not the same as saying they are a perfect tool. Every measurement method carries some side effect. When we try to use a relatively fixed ruler to measure a talent as complex and varied as human ability, that ruler does not just sort people — it also, invisibly, shapes how people judge their own abilities, even whether they believe they are "talented" at all.

Chapter 5: The Illusion of Measurement — What Are We Actually Rewarding?

When a sixteen- or seventeen-year-old solves, in a few hours, an Olympiad problem that would stump many adults, it is easy to fall into an illusion: that we are selecting the world's most promising future researchers.

But there is a distinction, easy to overlook, between "skill at solving problems" and "skill at research." Advanced sieves like the Olympiad are, at bottom, a form of "timed navigation within a closed system": however devious the problem, it comes with one guarantee — a known answer exists, and the problem-setter already has it in hand.

Within that structure, what a sieve is genuinely good at measuring is a specific cluster of abilities: how quickly someone can draw on existing knowledge and methods, how steadily they perform under pressure and against the clock, how efficiently they can solve a problem within a fixed set of rules.

Genuine frontier research looks more like "open-ended exploration within an unbounded system." When proving the three-dimensional Kakeya conjecture, or working on Hilbert's sixth problem, not only does no one tell you the answer — no one can even guarantee the problem has a solution within the current theoretical framework at all.

In that kind of territory, the "short bursts of brilliance" a sieve prizes may not carry nearly the same weight. What frontier exploration actually requires, more often, is the ability to tolerate long stretches of ambiguity, the psychological resilience to keep going for years without positive feedback, and the intuition and courage to propose a question that even one's own advisor is not sure is worth pursuing.

Unfortunately, a few hours of exam time is a poor instrument for measuring patience, and an even poorer one for scoring curiosity. These traits take longer to reveal themselves and resist quick quantification. So the mesh of a sieve naturally tends to catch people who respond quickly and can produce a definite result in a short window — while its ability to detect a different, equally important kind of ability, one that takes longer to surface, is comparatively limited.

This is not to say standardized measurement has no value — only that it measures a portion of the ability spectrum that happens to sit toward the front, and happens to be relatively easy to observe from outside. It is not the whole spectrum.

Shing-Tung Yau's Warning: Competition Is Not a Mathematician's Destination

This concern is not merely an outsider's abstract critique. Shing-Tung Yau, the first ethnic Chinese Fields Medalist and a longtime advocate for cultivating basic-science talent in China, has repeatedly voiced similar worries in recent years. He does not deny the value of exams and competitions, but he objects to turning them into a mere machine for university admission.

In one interview, Yau said he was struck by how much of China's exam-oriented education pushes students to study for the test rather than out of genuine interest. Exams can measure certain foundational abilities, he said, but they cannot select for the creative talent that actually matters. In mathematics specifically, a competition can only demonstrate that a student can solve a given problem within a given time; a truly great mathematician also needs vision, taste, the ability to formulate good questions, and the patience to sit with a problem that may remain unsolved for a very long time.

Tsinghua University's Qiuzhen College offers an instructive case in point. Qiuzhen runs the "Yau Mathematical Sciences Leaders Program," aiming to cultivate, within China, talent with a solid grounding in mathematics and theoretical physics capable of producing original work. The program uses an integrated undergraduate-through-doctoral training model, and some students must pass a preparatory year of coursework and assessment before formal admission.

According to media reports, in the 2026 preparatory cohort, some students did not pass the assessment and were not formally admitted, having fallen short of the required standard in courses such as mathematical analysis and linear algebra. This need not be read as some kind of institutional failure — quite the opposite: it shows that Qiuzhen College is still carefully distinguishing between two different kinds of ability — one already proven through competitions and exams, and another, separate question of whether a student can keep growing within a still higher level of mathematical training.

This is exactly where the sieve is most often misunderstood. Competition prizes, high exam scores, admission to an elite university — these are, of course, meaningful signals, but they are not the finish line. They only show that a person has passed through one particular sieve; they cannot, on their own, prove that the person already possesses the interest, endurance, problem sense, and self-motivation that long-term research demands. Being selected is only the qualification to enter the next stretch of the road. Whether a person can keep walking it depends on the soil that comes after.

Chapter 6: The Cost Beyond Measurement

If a sieve missing certain kinds of seeds is a loss of efficiency in the system, then living for a long time under constant measurement and ranking carries a subtler cost of its own — one that belongs to no single individual, but is something anyone who has spent years inside a high-intensity, standardized competitive environment may have to face.

For those who repeatedly come out ahead in the sieve, the real test often begins only after the selection is over. Someone who has grown used to confirming their own worth through scores and rankings, upon entering a more advanced research environment surrounded by equally or more accomplished peers, and going for long stretches without a comparable result to show, may find that this old way of confirming self-worth suddenly stops working. Education researchers have, in fact, discussed this phenomenon: some young people who consistently ranked at the top of competitive environments experience intense self-doubt, even burnout, once they move into the research stage. This is not any one specific person's story — it is a condition this kind of environment tends, structurally, to produce.

For those who did not stand out in the sieve, the cost can take a different form: living for a long time in an environment that rewards agility and definite output, a slower, more deliberative way of thinking is easily and systematically underrated — sometimes by the very people who think that way.

The limitation of measurement is that it always renders a judgment about a still-developing person based on their performance at one particular point in time. This can burden a temporarily leading person with excessive psychological weight, and it can also cause a temporarily unremarkable person to give up on a path prematurely, doubting whether they belong on it — and neither reaction necessarily reflects how far that person could ultimately go.

Chapter 7: The Limits of a Single Timeline

Different people's abilities surface at different points in time. Some show a given capacity very early; others take much longer for it to gradually emerge.

But many existing selection mechanisms operate only within a relatively narrow age window — the gaokao has a fixed testing age, and major competitions typically carry firm upper age limits for participation. This means that a person who fails to shine within that specific window may miss out on a particular path entirely, even though they might well have continued to develop afterward.

This is not a criticism of any particular selection system as unreasonable — any large-scale selection process needs some kind of time boundary; that is a practical constraint, not a personal failing on anyone's part. But the design itself does raise a real issue worth confronting: a sieve that only operates within a single time window will naturally be better at detecting ability that surfaces early, and comparatively worse at detecting ability that needs more time to appear.

This is also why a healthy talent-development system, ideally, should not rely only on a single, one-time selection window, but should preserve, at later stages, more opportunities for a person to be seen and evaluated again — transfers between departments and majors, interdisciplinary collaboration, second applications. To some degree, these mechanisms exist precisely to compensate for the limits of a single timeline. Hong Wang's undergraduate transfer is exactly this kind of corrective opportunity in action.

Chapter 8: From Solving Problems to Asking Them

Getting through every sieve, earning a degree from an elite university, and standing at the starting line of research does not automatically mean a person already possesses everything original research requires.

The shift from "solving problems that already have answers" to "facing questions that may have no answer at all" is a genuine transition — one that almost every young researcher emerging from a competition system or a high-intensity exam-driven environment has to go through. At the core of that transition is a change in how one judges the value of one's own work: no longer "did I get the result faster than everyone else," but "is this problem, in itself, worth a long-term investment."

This shift rarely completes itself at any single, identifiable moment, and it is hard for outsiders to describe its inner texture with any precision — unless the person involved chooses to talk about it themselves, this book will not speculate, for any specific researcher, about exactly how that transition happened for them. What can be said with confidence is that the transition is nearly universal, and nearly always difficult; only those who genuinely make it through have a chance to grow from someone skilled at solving problems into someone capable of independently posing good ones.

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Part Three: The Soil — How Talent Grows

Chapter 9: Why Freedom Must Coexist with Rigor

The phrase "a relaxed environment" is easy to flatten into "no pressure, no standards." But an environment that can actually sustain original research tends to look quite different: relaxed about direction and pace, yet extremely rigorous about logic and evidence.

A number of accounts circulate online describing Hong Wang's doctoral advisor Larry Guth — for instance, that he "never interrupts a student's train of thought, patiently hears out every idea, and only then helps sort through it together" — a description later corroborated by Hong Wang herself in interviews (see Chapter 1). What can be said with confidence is that geometric measure theory and harmonic analysis have long been recognized as fields demanding enormous patience and an exceptionally high technical bar — the fact that the Kakeya conjecture went unproven for a hundred and eight years is itself evidence that there is no shortcut in this territory that simply "hurrying up" can unlock.

What is really worth emphasizing here is not any single specific remark from any particular advisor, but a more general pattern: in research fields that demand long stretches of open-ended exploration with no guaranteed exit, students and advisors often develop a distinctive kind of working relationship — the advisor does not supply a standard answer, but a form of judgment: when a line of thinking is worth pursuing further, when it is time to abandon it and start again, what actually counts as a proof that holds up. Passing on that kind of judgment takes time, and it requires an advisor willing to keep investing sustained attention even while the student has nothing yet to show for it.

Freedom does not mean the absence of standards — quite the opposite. It is precisely because the final standard is so uncompromising (a mathematical proof either holds completely or it does not; there is no middle ground) that the process leading up to it needs enough freedom for real exploration — room to try, to go down the wrong path, to overturn one's own earlier ideas — before finally arriving at a conclusion that can genuinely withstand scrutiny.

Chapter 10: Failure, Waiting, and the Value of Slowness

A hundred and eight years passed between the Kakeya conjecture being posed and Hong Wang and her collaborator finally proving it. That is not an exaggeration; it is the problem's actual history — generation after generation of mathematicians worked on it, and the overwhelming majority never lived to see the answer.

Problems like this reveal a fact about original research that is often overlooked: going years without a result is the normal condition of this kind of work, not an aberration. If a research system requires participants to produce a presentable result within a fixed cycle — every year, say, or every three years — then the rational choice for most people will be to abandon the genuinely difficult, genuinely open questions and instead pick smaller problems more likely to yield results quickly. This is not a matter of any individual researcher's willpower; it is simply where the incentive structure naturally points.

Yu Deng's work toward Hilbert's sixth problem embodies the value of "slowness" too, though in a different form — not a long stretch of visible stagnation, but a subtler kind of patience that is easy to overlook: polishing a problem until it is fully mature, then waiting patiently for it to naturally extend into a larger one. From 2019 to 2021, he and his collaborator Zaher Hani spent more than two years focused on a comparatively smaller problem: the kinetic derivation of the wave equation. That work was already solid enough to be published in a top international journal and won him a Sloan Research Fellowship. The real pivot toward Hilbert's sixth problem came only after that "smaller" problem had been thoroughly mastered — the tools and intuitions accumulated along the way turned out to open a door onto a far larger question.

In his interview, Yu Deng explained why his publication count is relatively modest: "I hold my papers to a fairly high standard, so there are correspondingly fewer of them... the bar for whether something is worth writing up might be that it has to contain a genuinely new idea that convinces even yourself... the author should be clear about exactly where the key point of the paper lies." This may be the most direct account of the value of slowness one could ask for: real slowness is not procrastination or stagnation, but a refusal to sacrifice quality for quantity — a willingness to let an idea be turned over and interrogated in one's own mind, again and again, until it genuinely holds up, before ever being written down. Being able to work this way requires a support system of its own: some form of funding security, some evaluation method that does not treat short-term paper counts as the sole criterion — something that gives a researcher the standing to work at their own pace and get something truly solid done.

Slowness is not a synonym for laziness or inefficiency. It is a rhythm intrinsic to the work of original research itself — a genuinely unknown problem does not reveal its answer any faster just because someone is in a hurry.

Chapter 11: A Comparative Case in Long-Term Basic Research

Looking only at Hong Wang's and Yu Deng's experiences risks creating a false impression — that "being allowed to take your time" is something that exists only in Western academic environments. The story of the Japanese scientist Yoshinori Ohsumi offers a useful counterpoint.

Ohsumi was born in Fukuoka, Japan, in 1945, and earned his doctorate from the University of Tokyo in 1974. Japan's university entrance exams are every bit as famously competitive as China's, and elite institutions like the University of Tokyo are known for selection processes no less rigorous than any exam-heavy education system in the world. Ohsumi came up through exactly that kind of highly competitive system.

What makes his story worth remembering is what came after — decades of research that followed. According to multiple public accounts, Ohsumi began studying yeast cells almost as a "side project" during his postdoctoral years, after his main research had stalled. By 1988, when he began focusing specifically on the phenomenon of cellular autophagy, it was, at the time, an almost completely neglected corner of biology — many researchers assumed it was little more than an ordinary garbage-disposal mechanism inside the cell, nothing especially interesting. Over roughly five years, through a systematic screen of more than a thousand yeast mutants, Ohsumi identified fifteen key genes involved in autophagy, publishing the results in 1993. It would take nearly another decade before the field truly grasped the significance of that work — including its later-discovered connections to diseases like cancer and Parkinson's. He finally won the Nobel Prize in 2016, nearly thirty years after he first began the research.

Ohsumi's story shows that a society can maintain a highly competitive selection mechanism at the earliest stage of education without that necessarily meaning its research system is hostile to "slow research." What really determines whether an original direction can be sustained is usually not how fierce the entrance exam was, but whether, once inside the research world, a person can retain the standing and the resources to keep going through years, even decades, in which almost no visible progress is apparent to anyone on the outside. This is a reminder that "the sieve" and "the soil" are, in fact, two relatively independent systems — a society can remain strict at the point of selection while remaining patient at the point of cultivation; the two are not naturally at odds.

Chapter 12: Peers, Collaborators, and Opportunity

Breakthroughs are rarely made alone.

The key collaborators in Yu Deng's work on Hilbert's sixth problem were Zaher Hani of the University of Michigan and the mathematician Xiao Ma. Hani had worked with Deng for years — the two had been collaborating on wave kinetic equations continuously since 2019. Ma was someone Deng and Hani met at an academic conference in 2023 — by Deng's own account, the two "realized he was very skilled in this area, so we invited him to join the project." It is worth noting that Ma, like Deng, studied under Ioannescu, making him Deng's academic "younger brother" from their shared time at Princeton. In any field, finding a collaborator who both understands your research direction and complements it is never easy — it typically depends on a larger academic community that spans institutions, even national borders: regularly held conferences, a mentorship lineage descended from a shared advisor, and colleagues willing to spend the time listening to where, exactly, someone else is stuck.

It is worth stressing that collaborations and opportunities like this rarely appear out of thin air, to someone who is entirely unprepared. That Deng and his collaborators were able to form an effective partnership so quickly after meeting depended on each of them having already, independently, invested enough deep thought in this direction beforehand — opportunity, in cases like this, functions less like a light switching on out of nowhere, and more like two already well-prepared but separately stalled lines of thinking finally being reconnected.

What peers and collaborators offer is often not just technical complementarity, but a harder-to-quantify kind of support: when someone has been stuck on a problem for years and starts to wonder whether the path is even passable, is there anyone nearby who understands that this kind of stagnation is normal, anyone willing to keep walking that road together with them? That kind of support is very difficult for a person to provide for themselves alone — it requires the existence of a community.

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Part Four: The Master — How a Person Is Truly Seen

Chapter 13: Guth — How One Seminar Called a Mathematician Back

I. A Brief Detour into Architecture: Not a Failure, but a Search for Direction

After finishing her undergraduate degree, Hong Wang went to France for further study. During her master's, she spent one semester on a detour into architecture, interning at a firm in Paris. This episode should not be reduced to a near-abandonment of mathematics; it looks more like a genuine experiment: if the path of mathematical research did not work out, was there another shape her life could take? Architecture offered her a comparatively relaxed pace, but it never gave her a real sense of direction — she later recalled that architectural work required persuading others to invest before you could create anything, limiting creative freedom, whereas mathematics could be "built" entirely from one's own ideas. That brief detour ultimately confirmed something for her: more than architecture, what she really wanted was work she could build, step by step, using nothing but her own logic and judgment.

Having settled that question, Hong Wang returned to the École Polytechnique to finish her engineering degree. The school required students to complete an internship or research visit before graduating, and she chose MIT — her French advisor, Yvan Martel, helped connect her with colleagues there.

II. 2013: A Seminar About Distance

In 2013, during her research stint at MIT, Hong Wang attended an academic seminar given by a thirty-six-year-old mathematician named Larry Guth. Guth's talk described the reasoning behind a proof he had completed with Rice University mathematician Nets Katz of the Erdős distinct distances problem — a question asking: given any finite set of points in the plane, what is the smallest possible number of distinct distances that can occur between pairs of those points?

Guth explained the problem, and the solution's connection to incidence geometry, with remarkable clarity. Hong Wang was startled to find that she could follow most of the argument, and could even picture the corresponding geometric image in her mind as she listened. The talk reignited something in her — not the thrill of solving a competition problem, but the pleasure of understanding a structure for its own sake. The following year, she applied to and was admitted into MIT's mathematics PhD program, choosing Guth as her advisor.

It is worth being precise here about what the seminar actually changed. It did not change Hong Wang's ability — it changed her understanding of what mathematical research could be. For the first time, she felt, directly and intuitively, that frontier mathematics could be clear and comprehensible, rather than an impossibly distant kind of difficulty.

III. Guth as Advisor: Not Choosing the Student's Path, but Helping Her See Her Own

Guth's real significance lies not in having made a choice for Hong Wang, and not in having "discovered" a genius, but in three specific things he provided: a frontier problem, clear judgment, and unconditional trust.

Hong Wang has described Guth's mentoring style this way: when a discussion stalled, he never rushed her, always watching patiently, with a smile, while she kept thinking. When she brought him ideas that were still unfinished, even flawed, he never dismissed them outright — he listened all the way through, then helped her sort out which parts actually held up. What this style of guidance taught her was not any specific technique for solving a problem, but how to turn a vague intuition into an argument that could actually be tested — arguably the hardest, and most easily overlooked, skill in frontier research.

Guth's trust also showed in how he treated Hong Wang's results once they existed. After she and Zahl published their proof of the three-dimensional Kakeya conjecture, Guth personally presented the work at the Bourbaki Seminar in France, introducing it to mathematicians around the world, and repeatedly emphasized, in multiple settings, that the breakthrough belonged to Hong Wang's own creativity and intuition. He did not claim his student's achievement as his own, and he did not lock her research direction into a path he had already proven effective — the territory she went on to open up in harmonic analysis and geometric measure theory has since extended well beyond the scope of Guth's own original research.

This may be the truest meaning of the word "master": not making a choice for someone who has not yet taken shape, but offering a clear enough problem and a patient enough ear that the person can rediscover a direction they already possessed, but had not yet recognized in themselves.

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Chapter 14: Hu Shi and Gu Jiegang — Peking University's Own Tradition of Masters

The mentorship between Hong Wang and Guth can easily leave the impression that "a master who recognizes, supports, and makes a student" is a tradition unique to Western academia. But looking back into Peking University's own history turns up a remarkably similar, and often overlooked, precedent: Hu Shi and Gu Jiegang.

I. 1917: A Class Begins

In 1917, Hu Shi, invited home by Cai Yuanpei, returned to China to take up a professorship at Peking University, teaching the history of Chinese philosophy. In his classroom sat a student named Gu Jiegang. Compared with the fame Gu would later earn for his theory of "the layered construction of ancient Chinese history," he was, at the time, an unremarkable auditor — but in the course of answering questions and grading assignments, Hu Shi noticed that this particular student had an unusually sharp judgment when it came to ancient Chinese texts.

II. A Letter of Forty-Eight Characters

The two men's real, sustained exchange began around 1920. On or around 10 November 1920, Hu Shi wrote to Gu Jiegang asking about the whereabouts of a text by the Qing-dynasty scholar Yao Jiheng, the Jiujing Tonglun — a letter only forty-eight characters long. That short note is now generally recognized as the true starting point of the "Doubting Antiquity" movement in Chinese historiography — Gu Jiegang himself included it in the first volume of his landmark collection Debates on Ancient History, treating it as the beginning of that lifelong scholarly pursuit. Hu Shi later reflected: "That my forty-eight characters could give rise to a three-hundred-thousand-character book, that they could push Mr. Gu onto the road that would define his life's work in ancient history — this was something I never dreamed of at the time."

From then on, the two men kept up a steady correspondence, discussing textual criticism, evidentiary method, and how to research ancient Chinese history and literature. In 1923, Gu Jiegang published "A Letter to Mr. Qian Xuantong on Ancient History" in the journal Reading, formally proposing his theory that Chinese ancient history had been "constructed in layers" — the idea that the further removed in time a given account of remote antiquity is, the more layers of embellishment tend to have been added to it, meaning scholars must work to identify each legend's earliest form and then strip away, layer by layer, everything later generations had added. The idea caused a sensation in scholarly circles at the time.

III. Hu Shi's Support

Faced with this genuinely startling thesis, Hu Shi chose to stand publicly and firmly behind Gu Jiegang. In 1926, when the first volume of Debates on Ancient History — edited by Gu — was published, Hu Shi declared, in a public lecture at Peking University, that this was "the greatest joy of my life," and praised the theory of "layered construction" as having "opened a new era for the field of Chinese historiography." Beyond that public endorsement, Hu Shi went on to provide Gu Jiegang with sustained, concrete support over the years — in publishing his work, securing funding, and recommending him for positions.

It should be noted that certain personal details sometimes attributed to Gu Jiegang's early life — such as whether he developed a stammer as a result of his early private-school education — have not, as of this writing, been traced to a source this book can directly verify, and are not treated here as established fact. What is documented, however, is the sustained, multi-year correspondence between Hu Shi and Gu Jiegang, Hu Shi's public support for the "layered construction" theory, and the concrete assistance he provided at the early stage of Gu's academic career — all of which can be traced through clear documentary records.

IV. Peking University's Own Tradition

The story of Hu Shi and Gu Jiegang is a reminder: a master who recognizes, supports, and makes a student is not a culture exclusive to overseas academic systems. China's own tradition of modern universities once nurtured this same kind of relationship. Hu Shi was to Gu Jiegang what Guth would later be to Hong Wang — what each provided was never a path already paved for the student, but earnest listening, public endorsement, and sustained, concrete support.

This is why setting Hong Wang's and Guth's story alongside Peking University's own history with Hu Shi and Gu Jiegang comes closer to the truth, and carries more force, than simply placing "Peking University" and "Guth" on opposite sides of a divide. Peking University gave Hong Wang her first door into mathematics, and gave her rigorous training. What later turned mathematics into her life's work came from a different kind of experience, offered in France, at MIT, and by Guth — a frontier problem, unhurried time, an advisor's trust, and the feeling of being truly seen. Peking University has not always lacked a tradition of masters; this time, the master who made Hong Wang simply happened not to be there.

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Chapter 15: A Master Is Not Someone Better at Choosing People, but Someone Better at Making Them

Yu Deng did his doctoral work under Ioannescu at Princeton. Various online accounts describe this advisor as someone who "holds the technical details of a proof to an exacting standard, but places almost no restrictions on what a student chooses to research or how" — supposedly even allowing Deng, at one point during his PhD, to drift between several different problems in a period of uncertainty, trying and retrying. These accounts currently rest mainly on secondhand retellings; no first-hand interview from Deng himself or his advisor has yet been found to confirm them directly, so this book does not treat them as established fact — only as a claim worth further verification, recorded here for that reason.

Setting the specific details aside, however, a more general kind of advisory role genuinely deserves serious discussion: in fields that demand long-term exploration, good mentorship often combines two seemingly contradictory qualities at once — uncompromising rigor about the logic and precision of a proof, paired with real autonomy over the direction and pace of the research itself. Neither quality can stand alone: rigor without autonomy tends to turn a student into an instrument for executing an advisor's predetermined ideas, leaving little room to develop independent judgment; autonomy without rigor lets exploration drift without direction, and can produce conclusions that do not hold up to scrutiny.

What may matter even more is how an advisor responds when a student has, for the time being, nothing to show. A doctoral student spending a long stretch of time on a direction with no visible progress is an exceedingly common condition in original research. Whether the advisor pushes the student to switch quickly to an easier, more productive topic, or instead allows them to keep searching along the original path — or steps in to help redirect them toward a better-suited question — is a choice that, to a considerable degree, determines whether the student ultimately reaches the point of independently posing and solving problems of their own.

What a genuinely excellent mentoring relationship aims for, in the end, is not turning the student into a copy of the advisor, perpetuating a research path the advisor has already proven works, but helping the student gradually develop their own judgment — until, eventually, they can walk independently into territory their advisor never reached either. The directions Hong Wang and Yu Deng each went on to open up have both extended beyond their respective advisors' original work. That may be the best possible test of this kind of relationship: a truly successful mentorship is one in which the student, in the end, no longer needs the mentor.

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Interlude Two: Two Fates of Math Fever — When Myth Replaces Method

How a society treats mathematics often reveals, more than almost anything else, how it understands knowledge itself.

Some eras treat mathematics as training, embedding it in a long, rigorous, disciplined process of learning. Others prefer to treat it as symbol, attaching it to genius, solitude, austerity, and miracle. The first emphasizes method; the second, myth. Method shapes scholarship; myth shapes emotion. The two are not entirely opposed — but once the balance tips too far, they can rewrite what belongs to the order of knowledge into the narrative of fate.

Modern Chinese intellectual history has seen its own complicated wave of "math fever." It arose partly from the rising prestige of modern science, and partly from how mass media concentrated its portrayal of what a mathematician looked like. Mathematics stopped being merely a discipline and became a symbol of intellect, will, purity, and nobility. Against that backdrop, some people were inspired toward genuine scholarly creation — and others were drawn into something closer to a tragic obsession.

Understanding the inner structure of this "math fever" requires three figures who cannot be avoided: Chen Jingrun, Liu Hanqing, and Lu Jiaxi. Chen Jingrun turned the myth of the mathematical hero into reality, and in doing so became the center of that myth; Liu Hanqing was someone struck deeply by that myth, who mistook the legend for a method, and paid a price of thirty years for the misunderstanding; Lu Jiaxi offers a different, heavier kind of proof — that even at the margins of the system, under extremely impoverished conditions, a person can genuinely reach the summit of mathematical creation, which only makes the debt owed by his circumstances more glaring.

Placing these three figures side by side turns the discussion from individual fates into a deeper question: when a society loves mathematics, what exactly is it loving — the labor of mathematics, or its myth?

I. Chen Jingrun: The Making of a Myth

In contemporary Chinese public memory, Chen Jingrun is without question one of the most iconic mathematicians. Without him, mathematics might never have entered the center of popular culture the way it did; without the publicity surrounding him, the so-called "Goldbach fever" could never have achieved such broad social resonance.

His life did not follow an easy road to glory. In 1950, he was admitted to the mathematics and physics department at Xiamen University; he graduated early, in 1953, and was assigned to teach at Beijing's No. 4 Middle School, but his heavy accent and poor speaking manner meant the school could only assign him to grade homework. He was hospitalized six times within a single year and was ultimately dismissed under the pretext of "suspended duty for health recovery," returning to his hometown of Fuzhou, where he made a living selling books from a street stall. According to Xiamen University's archives, in 1954, Wang Yanan, then president of Xiamen University, happened to pass by on a business trip, recognized Chen Jingrun at his book stall, and decided on the spot to bring him back to work in the reference room of the university's mathematics department. This was hardly an illustrious start — if anything, it carried a note of hardship and awkwardness. And yet it was precisely this unglamorous, real starting point that later became essential to the myth that grew around him: how did someone once nearly unemployed, scraping by on a street stall, reach the summit of mathematics through sheer will and extraordinary diligence?

What is most often retold about Chen Jingrun is his "obsession." He devoured books, nearly forgetting himself; immersed in mathematics to the point that his awareness of ordinary daily life dulled. He is said to have read Hua Luogeng's Additive Prime Number Theory more than thirty times, until it had become second nature to him. Those who knew him well have pointed out that his later breakthrough was anything but a stroke of accidental inspiration — it rested on an extraordinarily deep foundation.

Back at Xiamen University, Chen Jingrun completed a paper, "On Tarry's Problem," improving on a result of Hua Luogeng's in additive prime number theory, which caught Hua's attention. In 1957, on Hua's recommendation, he transferred into the Institute of Mathematics at the Chinese Academy of Sciences as a research assistant, specializing in number theory. In 1966, he proved the "1+2" case of the Goldbach conjecture — that every sufficiently large even number can be written as the sum of a prime and a number that is either prime or the product of at most two primes — publishing a summary in Kexue Tongbao; after years of further refinement, a fuller, more mature version of the proof was formally published in 1973. "Chen's Theorem" has stood ever since as a clear, solid landmark in the history of modern number theory. In 1982, he received China's first-class National Natural Science Award for this work.

Wang Yanan recognizing Chen Jingrun on the street, and Hua Luogeng arranging his transfer to the Chinese Academy of Sciences — these two moments were themselves an early rehearsal of the "master" pattern this book takes up fully in Part Four: someone willing, at a moment when everyone else saw only a down-on-his-luck young man selling books from a stall, to recognize the real ability underneath, and willing to use their own position and standing to bring that person back onto a path where creation could continue. This thread will resurface later in this book, alongside the stories of Guth and Hu Shi.

It must be emphasized that Chen Jingrun's achievement was a genuine scholarly accomplishment, not an illusion manufactured by publicity. That he could become the mathematical hero of the public imagination was possible only because there was solid scholarly substance behind the heroic narrative. But this is exactly where the trouble begins: once facts like these enter mass circulation, they tend to be rearranged, their emphasis shifted. What the public tends to remember is rarely the refinement of a sieve method, the complexity of a proof, or decades of accumulated mathematical grounding — it is the more easily transmitted silhouette: a cramped room, a solitary battle fought late into the night, stacks of draft paper, a direct confrontation with one of the world's hardest problems. In this way, an achievement built on years of demanding, high-threshold professional training gradually transforms, in the retelling, into a more universally stirring heroic template.

In 1978, Xu Chi's reportage "Goldbach's Conjecture," published in People's Literature, played a decisive role in this process. With tremendous literary force, it thrust Chen Jingrun into the public eye and brought a modern mathematical problem into popular culture with a clarity it had never had before. It stirred society's respect for scientists and inspired a great many young people to aspire toward mathematics and science — a positive effect that should not be underestimated. But the logic of literary narrative and the logic by which science actually forms are not the same thing. Literature naturally emphasizes character, fate, spirit, and symbol, and gives comparatively little space to the scholarly community, methodological training, or technical detail. It can ignite enthusiasm, but it cannot supply method; it can manufacture the desire to climb the summit, without teaching anyone how to climb.

Chen Jingrun's place in modern Chinese culture is therefore something of a double one. He is both the genuine fruit of serious scholarly labor, and a spiritual symbol shaped by mass communication. Precisely because of this, his existence proves not only that a mathematical summit can be reached, but also opens up a dangerous possibility of its own: when people see only the summit, without understanding the path that leads there, myth can end up replacing method — and become the source of a great many later misunderstandings.

II. Liu Hanqing: A Man Struck by the Myth

Liu Hanqing's story is one of the most representative consequences of exactly that misunderstanding.

By ordinary social standards, Liu Hanqing began from a rather bright starting point. In 1980, at sixteen, he scored 398.5 and was admitted into the materials science program at Harbin Institute of Technology — the first student from his small hometown ever to be admitted to a major national university. Left on its original course, his life could have proceeded smoothly through his studies into a career as an engineer. But in his third year of university, after reading Xu Chi's "Goldbach's Conjecture," the direction of his life shifted.

From that point on, Liu Hanqing threw himself entirely into number theory, borrowing every related book the library held, determined to do even better than Chen Jingrun. Outside of number theory, he came to feel that everything else was a waste of time. It may have been this mindset that drove his subsequent choices — neglecting his coursework, repeatedly failing exams, being held back a year, and eventually leaving university without a degree. For more than thirty years after that, he held no job and had little contact with the outside world, spending his years alone poring over this ancient conjecture, submitting his work repeatedly to academic journals without ever being accepted, all while dealing with insomnia, tinnitus, and anxiety, and living for a long time on a monthly subsistence allowance of 400 yuan. In 2017, with help from former university classmates and the media, his circumstances improved somewhat; he began relearning other subjects in preparation for reentering the workforce. He has said that mathematics remains his dream, and that he will keep working on it.

What happened to Liu Hanqing was not simply "failing to succeed" — the deeper issue was that his choices rested on a misunderstanding of what mathematics actually is. The Goldbach conjecture is often simplified in popular retellings to "1+1," a shorthand that easily creates a dangerous illusion: that the problem is somehow simple in form, intuitively obvious, and therefore something anyone might reasonably attempt. But the actual Goldbach conjecture states that every even number greater than 2 can be written as the sum of two primes. Its statement may be concise, but behind it lies some of the most forbidding technical terrain in modern analytic number theory. It has remained unsolved for centuries not because the problem is superficial and was simply neglected by chance, but precisely because it sits at one of the hardest points method itself has yet to reach.

This means that a person lacking systematic training, fluency in modern mathematical language, and basic entry into the scholarly community, however great their passion and willpower, is almost certain to be locked out by attempting to charge straight at a problem like this. Mathematics, unlike some other fields, is especially unforgiving of the impulse to skip the foundations. A summit is a summit not because it is majestic, but because it can only be reached through a long ascent.

Liu Hanqing's story is a reminder that when science communication tips too far into myth-making, it can produce an effect that actually works against education. It leads people to believe that the most worthwhile thing to do is charge directly at the hardest problem, rather than first entering the language, structure, and community that problem demands. It encourages people to gaze up at the summit without ever showing them where the trail begins.

III. Lu Jiaxi: A Proof Grown in Barren Soil

But it would be too simple to conclude, from Liu Hanqing's story, that everyone working on hard problems outside the mainstream academic system is merely a casualty of myth. Lu Jiaxi stands as a very different, and far heavier, kind of counter-example.

Lu Jiaxi was born in 1935 into a poor family in Shanghai. He left school after middle school, worked as an apprentice, and later as a factory worker at an electrical-machinery plant in Harbin. In 1957, he was admitted to the physics department at Northeast Normal University. He was not the kind of "young prodigy" groomed step by step along a conventional academic track, but someone who fought his way back into the world of knowledge through sheer force of self-study. During university, he became deeply drawn to Kirkman's schoolgirl problem in combinatorics, a fascination that set him on the path toward combinatorial design theory.

The crucial difference between Lu Jiaxi and Liu Hanqing is that Lu never stopped at the symbolic allure of a hard problem — he genuinely entered its interior. After graduating in 1961, he moved between several posts — Inner Mongolia's Baotou Iron and Steel Institute, the Baotou Education Bureau's research office, teaching positions at Baotou's No. 8, No. 5, and No. 24 middle schools — before finally settling at Baotou's No. 9 Middle School as a physics teacher. By any practical measure, this was about as far from the international frontier of mathematics as one could get: a middle-school teacher, in a remote industrial city, with a heavy teaching load, no research materials, no funding, and almost no scholarly contact. He taught during the day and researched at night; to consult the literature he needed, he traveled and paid for materials out of his own pocket, sometimes taking hard-seat trains to Beijing and sleeping in the train-station square when he could not afford a hotel room.

And yet it was in exactly this environment that Lu Jiaxi produced mathematics that would genuinely withstand the scrutiny of the international community. By 1965, he had already completed a full proof of Kirkman's schoolgirl problem and submitted it to a leading domestic mathematics journal. According to some accounts, the paper was not taken seriously because "the author's affiliation was a middle school" — this particular detail currently traces to only a single source, and readers should weigh its credibility for themselves — but what can be confirmed is that the result was not made known to the world until 1981, when it appeared in the American Journal of Combinatorial Theory, Series A — a full sixteen years after he had completed the proof. He went on to concentrate on the existence problem for large sets of disjoint Steiner triple systems, a question that had confounded the international combinatorics community for more than a hundred and thirty years. Starting in September 1981, he submitted a series of six papers reporting his results to the Journal of Combinatorial Theory. The Canadian mathematician Ronald Mullin called it "one of the major achievements in combinatorial design in the past twenty years," and wrote to the principal of Baotou No. 9 Middle School urging that Lu Jiaxi be transferred to a university post.

This was not a folk legend in the loose sense of the phrase — it was, in the strictest sense, a modern scholarly achievement. It was published, read, tested, accepted, and absorbed into the community, becoming part of the discipline's history. Which is exactly why Lu Jiaxi's circumstances are so painful to consider. Scholars abroad referred to him respectfully as "Dr. Lu"; in his actual daily life at home, he remained, without a degree, without a title, a middle-school teacher living on a modest income. A person with world-class original ability had to sustain his research through something close to self-depletion — and that is not an inspirational story. It is evidence of an environment failing to do its job.

In October 1983, at the National Conference on Combinatorial Mathematics held in Wuhan, Lu Jiaxi — the only invited speaker who was a middle-school teacher — announced that he had resolved the remaining six exceptional cases of the large-set problem. But years of poverty and exhaustion had already worn down his health; on the 31st of that same month, he died of a sudden heart attack at home in Baotou, at the age of forty-eight. In 1987, his research on the large sets of disjoint Steiner triple systems was awarded China's first-class National Natural Science Award; the award ceremony was held at the Great Hall of the People in Beijing in March 1989 — an honor that arrived six years too late, one of only six ever given in the mathematical sciences since the award was established in China in 1956, alongside earlier recipients including Hua Luogeng and Chen Jingrun.

What Lu Jiaxi's story reveals is not the shallow lesson that "you can succeed no matter how harsh the environment." Quite the opposite: it reveals that even when real creation manages to occur in extremely barren soil, that does not make the barrenness justified. The more outstanding the individual, the more glaring the environment's debt becomes; the more genuine the achievement, the more unnecessary — and painful — the cost.

IV. Three Shapes Within the Same Fever

Placed side by side, Chen Jingrun, Liu Hanqing, and Lu Jiaxi make the inner structure of China's "math fever" far easier to see.

Chen Jingrun was one of the sources of that fever. He proved that modern China really could produce someone who, through sheer extraordinary tenacity, reached the summit of world mathematics. His success was not fiction, and so it carried an unusually powerful demonstrative force. And precisely because that force was real and immense, he was quickly transformed from a scholarly figure into a cultural symbol.

Liu Hanqing represents the collateral damage that symbol caused as it spread. His experience shows that once myth is separated from method, and only the summons remains, it can pull certain people toward a long, largely unproductive investment of the self. The myth itself need not be false — but if what survives its spread is only its glorious outward appearance, stripped of the methodological groundwork underneath, it can turn from inspiration into misdirection.

Lu Jiaxi provides a different kind of correction. He reminds us that just because myth can hurt people does not mean every act of solitary exploration at the margins of the system should be dismissed as illusion. Some people standing outside the mainstream academic structure really are chasing fantasies — but others have genuinely grasped the method, and some have even reached the heart of a problem before the system did. This is exactly why the responsibility borne by the environment is so heavy: it must not only guard against people being misled by myth, but also be capable of recognizing and supporting real talent early — rather than only ever acknowledging it after the fact.

What this chapter is ultimately concerned with, then, is not simply the personal fates of three mathematicians, but a more fundamental question: when a society reveres mathematics, what exactly is it revering? If what it reveres is miracle, solitude, the ultimate unsolved problem, and the posture of the hero, then what it produces will mostly be emotion and worship. If what it reveres is training, proof, method, patience, and community, then it may actually be capable of nurturing creation.

In that sense, the sieve certainly matters — but a sieve can only detect ability that has already surfaced. What ultimately determines what a piece of land can grow is still the soil. A society that worships only the myth of genius will keep generating misunderstanding; a society that genuinely respects the labor behind knowledge will be willing to build training, resources, mentorship, community, and time. This is as true of mathematics as it is of the growth of any form of world-class creativity.

· · ·

Part Five: From Being Selected to Becoming Oneself

Chapter 16: Conclusion — Personal Effort and External Conditions

Looking back over the full arc of Hong Wang's and Yu Deng's growth, the easiest mistake to make is attributing the outcome to a single cause — either "this was entirely the result of their own effort, and the environment didn't matter," or "this was entirely a matter of encountering the right mentor and the right environment; someone else in their place might not have made it." Both claims can point to real supporting fragments — but no single fragment, taken alone, explains the whole picture.

Personal talent and persistence were clearly irreplaceable: no one could have worked through that 127-page proof in Hong Wang's place, and no one could have polished Yu Deng's "smaller problem" to the necessary degree of rigor and then patiently waited for it to extend into something larger, on his behalf. But it is equally clear that this persistence depended, at some point, on encountering a teacher willing to write a recommendation, an advisor willing to wait, colleagues willing to collaborate, and a stretch of time in which not seeing results yet did not mean being weeded out.

To put it in terms of the metaphor this book has built: the sieve determines when, and in what form, a seed is first seen; the soil determines whether, once seen, that seed can actually take root, grow, and bloom. Talent, interest, mentorship, autonomy, a network of peers, secure time, tolerance for failure — these conditions relate to one another less by simple addition than by mutual amplification: talent left unnourished by environment may quietly go dormant; an environment lacking an advisor who knows how to listen may lack direction; and an advisor who does not respect a student's independent judgment may end up producing a copy of themselves, rather than a genuine originator. Real success tends to happen at the exact point where individual gift and external condition happen to intersect.

Chapter 17: Helping a Person Become Themselves

By this point, what this book ultimately wants to leave with its readers is not a verdict on which education system is better, but a handful of questions worth carrying back into everyday life, to keep asking of oneself.

For parents: when a child performs unremarkably in some exam or competition at a given stage, does that really indicate a lack of talent — or simply that this particular sieve happens not to be measuring the kind of ability the child actually has? Before drawing a conclusion, has the child been given enough time for different kinds of ability to have a chance to surface?

For teachers and mentors: faced with a student who has shown no visible progress for a long time, even one who looks, on the surface, like they are "wasting time" — can you tell whether this reflects a lack of effort, or the stagnation that original exploration inevitably passes through? Is what you are offering that student a path you yourself once walked and hope they will simply retrace — or genuine help in finding a path of their own?

For team managers, and anyone responsible for evaluating others: what abilities can your current evaluation methods actually see in the people on your team, and which ones do they systematically miss? Is there enough patience reserved for people who, for now, have nothing to show, but are genuinely working on something difficult?

For anyone who has ever doubted themselves because of a grade, a ranking, or a single evaluation: does failing to be selected by one particular sieve really reveal the ceiling of your ability — or simply that this sieve happens not to be well suited to measuring the particular kind of talent you have?

None of these questions has a standard answer — just as, before the Kakeya conjecture and Hilbert's sixth problem were actually solved, no one knew in advance what the answer would look like. But just as Hong Wang and Yu Deng, by different means and on different timelines, both eventually arrived at the same podium, whether a person can find their own path often depends less on which sieve first selected them, and more on whether, afterward, they ever have the chance to find the soil that belongs to them.

Chapter 18: Outliers in the Wilderness — Three Samples from Extreme Circumstances

The stories of Hong Wang and Yu Deng still unfolded within the relatively conventional soil of the modern academic system — elite universities, top-tier advisors, an international community of peers. Turning to the broader history of science reveals some far more extreme cases: instances where a person's talent went unrecognized by the system for a long time, even met with the system's sustained rejection, and what, in the end, allowed them to complete their work regardless.

For each of the three figures below, this book presents only the facts supported by public reporting; regarding their psychological states, motivations, or feelings at the time, this book does not speculate or project onto them, and offers no diagnosis, unless they have spoken about it publicly themselves — not because this material is uninteresting, but because that kind of judgment lies outside what this book is able to verify.

Yitang Zhang

While pursuing his PhD at Purdue University, Yitang Zhang came into disagreement with his advisor over the reliability of a proof, and after graduating went for years without the broad recognition from academia — including the recommendation letters so often essential to finding a job — that might otherwise have followed. For years afterward, he held no stable academic position, working at various points as an accountant and at a Subway sandwich shop to make ends meet, all while continuing to work on questions in number theory on his own. In 2013, at the age of 58, Zhang submitted "Bounded Gaps Between Primes" to the Annals of Mathematics, proving an important weakened version of the twin prime conjecture — a result that stunned the mathematical world. He went on to join the faculty at UC Santa Barbara.

What is most worth recording about Zhang's story is that a man no longer young, and long positioned at the margins of the academic system, kept working on this research anyway, and eventually completed it. As for the specific psychological mechanisms that sustained him through that period, he has offered his own accounts in various interviews over the years; readers interested in more can consult those first-hand sources directly. This book does not summarize or dramatize them on his behalf.

Katalin Karikó

Karikó spent a long career researching messenger RNA (mRNA) technology, at a time when the field was, for a considerable stretch, not taken seriously by the academic mainstream — mRNA molecules were seen as highly unstable outside the body and prone to triggering immune reactions, and many colleagues doubted the technology had much practical future. While working at the University of Pennsylvania, she was demoted at one point after struggling to secure research funding, and for a long time had no laboratory of her own. Her sustained work with collaborator Drew Weissman on mRNA modification later became a foundational basis for the mRNA COVID-19 vaccines; the two were awarded the Nobel Prize in Physiology or Medicine in 2023.

What is worth recording here is that a line of research the mainstream evaluation system had, for a very long time, failed to recognize, ultimately proved to hold critical value. As for how Karikó herself understands that period, readers are likewise encouraged to consult her own accounts and interviews directly; this book makes no psychological speculation on her behalf.

Grigori Perelman

Between 2002 and 2003, Perelman posted three papers on a preprint server, proving the Poincaré conjecture — one of the seven Millennium Prize Problems. In 2006, he was awarded the Fields Medal, but declined it; in 2010, the Clay Mathematics Institute moved to award him the accompanying million-dollar Millennium Prize, which he also declined. He subsequently resigned his research position and has since lived a comparatively reclusive life.

Perelman has said relatively little in public about his reasons for declining these honors, and various claims circulating in media reports have not been individually verified. This book likewise makes no judgment about the specific psychological reasons behind his decisions, and presents only the sequence of publicly documented facts.

A Brief Reflection

What these three figures share is that their work, for a long stretch of time, went unrecognized — or undetected — by the evaluation systems of their era. This is a reminder that a "sieve" can fail not only to catch talent that has not yet surfaced, but also, over long periods, to recognize work that already exists but does not yet fit prevailing standards of judgment. As for what inner resources each of these three drew on to get through that period, it is very difficult for outsiders to genuinely stand in their place, short of their own willingness to describe it. Rather than imagining a psychological journey on their behalf, this book leaves the question, as it stands, for the reader: if you are working on something that has gone unrecognized for a long time, what kind of soil would you want for yourself?

· · ·

Epilogue

After winning the Fields Medal, Hong Wang gave a forty-five-minute lecture at the International Congress of Mathematicians. The hall was full to capacity — even the moderator gave up their seat to an audience member who could not find one, and stood through the entire talk.

Her doctoral advisor, Larry Guth, should by rights have had one of the best seats in the room. Instead, arriving right on time only to find every seat already taken, he was left outside. According to students who were there, he simply stood outside the window, an old backpack worn through at one corner slung over his shoulder, and listened to the full forty-five minutes through the glass, smiling the whole time.

A single photograph can never capture the entirety of a mentoring relationship. But it leaves behind an image worth sitting with: the student now standing in the world's spotlight; the mentor standing quietly just outside it.

Perhaps this is what successful mentorship actually looks like, in the end — a day arrives when the student no longer needs to live in the mentor's shadow.

The highest achievement in cultivating talent is not manufacturing success in some predetermined shape, but preserving, for every distinct life, the possibility of finding its own way.

END