Summary of "You've Grown Taller"

Key Plot Points

• Lu Zhou leaves his former professor Old Tang's office, reflecting on their conversation.
• He visits Academician Lu's office, but the academician is away at a conference in Daya Bay.
• A master's student, Song Xuewen, is there. Lu Zhou leaves tea as a gift and introduces himself.
• Song Xuewen recognizes Lu Zhou as "God Lu" and excitedly gives him a letter left by Academician Lu.
• Lu Zhou signs an autograph and, when asked how it felt to win the First Prize in Natural Sciences, answers simply: "Pretty nervous."
• Later, on campus, Lu Zhou runs into Han Mengqi, who is carrying a lab sample. They walk to the Computational Materials Research Institute together.
• Han Mengqi hints that she wants Lu Zhou to say something meaningful after their long separation. His response is only: "You've grown taller?" — leaving her speechless.

Character Developments

• Lu Zhou remains modest and slightly socially oblivious despite his academic fame. He is respected by juniors but uncomfortable being idolized.
• Han Mengqi has matured and grown more independent, but still clearly looks up to Lu Zhou, possibly with unspoken feelings.
• Song Xuewen is a devoted, somewhat starstruck junior who admires Lu Zhou.

Significant Themes

• Mentorship and respect across academic generations.
• The awkward gap between public recognition and personal identity.
• Subtle, unspoken romantic tension and the passage of time.

Character Interactions

• Lu Zhou & Song Xuewen: Polite, then fan-like enthusiasm; Lu Zhou responds with calm humility.
• Lu Zhou & Han Mengqi: Warm but emotionally clumsy; her hopeful prompt is deflated by his literal, deadpan observation.
• Academician Lu (off-screen): His forethought in leaving a letter shows his care for Lu Zhou as a former student.

Chapter Summary: “A Seven-Figure Bonus”

Key Plot Points and Events

• Han Mengqi delivers a sample box to Lu Zhou’s laboratory but becomes irritated when he makes a remark that seems to mock her height. Because Lu Zhou is her former mentor, she suppresses her anger, leaves coldly, and refuses to speak further.
• Yang Xu informs Lu Zhou that the graphene superconductivity experiment succeeded. Two graphene layers demonstrated unconventional superconductivity when twisted to approximately 1.1° and cooled to 1.7 K—the “Magic Angle.”
• The researchers discuss the significance of the discovery. Although 1.7 K is far below practical superconducting temperatures, Lu Zhou emphasizes that seemingly limited results are necessary steps toward major breakthroughs.
• The team sets a long-term goal of raising graphene’s superconducting temperature to at least 77 K, but Lu Zhou boldly raises the target to 100 K, relying on information he possesses about the technology’s potential.
• Patent ownership and collaboration become concerns. Professor Herrero may prefer making the technology freely available, so Lu Zhou offers $10 million in research funding to persuade him to cooperate while securing the related patents. If necessary, Lu Zhou is willing to increase the offer to $20 million or proceed without Herrero’s team.
• Lu Zhou announces the laboratory’s year-end bonuses. When Yang Xu asks for a rough estimate, Lu Zhou reveals that the bonuses will be seven figures, shocking the researchers.

Character Developments

• Han Mengqi is shown as easily provoked but restrained by her respect for Lu Zhou as her mentor. Her sulky reaction reveals both her youthful temperament and lingering sensitivity toward him.
• Lu Zhou demonstrates confidence, ambition, and immense financial power. He is willing to invest heavily in research and patents even if the project may not produce immediate profits, suggesting that he has a larger strategic motive.
• Yang Xu acts as the practical and cautious counterpart to Lu Zhou. He recognizes the scientific importance of the work but worries about industrial value, patent complications, and whether the massive investment is worthwhile. His attitude shifts to delighted surprise after hearing about the bonus.
• Professor Herrero is characterized indirectly as an idealistic scholar who prioritizes open scientific progress over personal profit or recognition.

Significant Themes

• Scientific progress through incremental research: Even results with little immediate practical value can form the foundation for transformative discoveries.
• Idealism versus commercialization: The chapter contrasts Herrero’s apparent preference for open access with Lu Zhou’s determination to secure patents and control the technology.
• Ambition and strategic foresight: Lu Zhou sets an unusually high 100 K target and insists on acquiring the technology despite uncertain financial returns, hinting at plans beyond the project’s obvious market value.
• Power and generosity: Lu Zhou’s willingness to offer tens of millions of dollars and distribute enormous bonuses establishes his wealth and reinforces his position as a highly capable, generous leader.
• Mentorship and interpersonal tension: Han Mengqi’s reaction highlights the complicated familiarity between mentor and student, blending irritation, respect, and emotional awkwardness.

Character Interactions

• Han Mengqi and Lu Zhou have a tense, lightly comedic exchange shaped by their mentor-student history.
• Yang Xu and Lu Zhou discuss the successful experiment, the “Magic Angle,” future research targets, and patent negotiations.
• Yang Xu questions Lu Zhou’s extravagant funding plans, while Lu Zhou decisively asserts that the technology must be obtained regardless of cost.
• Lu Zhou reassures the laboratory staff through the promise of substantial year-end bonuses, ending the chapter on a humorous and celebratory note.

Summary: Chapter "A Letter from Germany"

1. Key Plot Points & Events

• Lu Zhou decides to allocate 20 million RMB as a bonus pool for the Jinling Institute of Computational Materials and several million USD for the Sarot Laboratory, distributing bonuses based on contributions to papers and performance—his philosophy being that retaining talent requires compensation.
• He sends an email to White Sheridan (Starry Sky Technology, North American branch manager).
• Receives an email from Professor Lazerson (Germany) announcing that the "helium-3 probe technology" was successfully deployed on the Wendelstein 7-X stellarator. The probes pierced the plasma and collected macroscopic parameters (temperature, density) more accurately than conventional diagnostics. Lazerson proposes submitting a paper to the ITER conference.
• Lu Zhou agrees to the ITER submission, then spends ~30 minutes reviewing the attached experimental data, concluding that extracting a Plasma Turbulence model from it is extremely difficult—even with Level 6 Mathematics and Level 4 Physics.
• He considers using Inspiration Time but decides against it, prioritizing the Navier–Stokes system mission (which takes precedence over Plasma Turbulence research).
• He opens a letter from Academician Lu containing a couplet in semi-cursive calligraphy: "Without indifference to fame and gain, one cannot make one's aspirations clear; without tranquility, one cannot reach far." He resolves to hang it during the New Year.
• Next morning: Year-end bonuses are distributed, causing shock: Qian Zhongming receives 880,000, Liu Bo receives 660,000, Liu Hong receives 100,000 (moved to tears, having previously been underpaid research labor), and Yang Xu receives 1.5 million (the highest), feeling the burden of expectations.

2. Character Developments

• Lu Zhou: Demonstrates his management philosophy (talent retention via fair compensation); exercises scientific judgment (declining to waste Inspiration Time on Plasma Turbulence when the N-S mission has priority); shows appreciation for Academician Lu's gesture despite not understanding calligraphy.
• Yang Xu: Receives the highest bonus (1.5M), feels a heavy sense of responsibility—his plans to save for an apartment have become "unable to keep up with the changes."
• Qian Zhongming & Liu Bo: Stunned by their bonuses, initially suspecting a financial error; their reactions highlight the extraordinary generosity relative to materials science industry norms.
• Liu Hong: Moved to tears by 100K, comparing it to his previous underpaid experience under Wang Haifeng.

3. Significant Themes

• Talent retention & fair compensation in research: Lu Zhou's belief that "research is not a one-man job" and that incentives drive team stability and quality.
• Prioritization of scientific goals: Choosing between two research problems—the N-S system mission takes precedence over Plasma Turbulence.
• International scientific collaboration: Successful deployment of helium-3 probe technology on Wendelstein 7-X, with cross-national data sharing and conference submissions (ITER).
• The cost of scientific ambition: Calibrating the difficulty of new fields (Plasma Turbulence is vastly more complex than prior models).
• Recognition and humility: Academician Lu's gift of calligraphy emphasizing detachment from fame and wealth.

4. Character Interactions

• Lu Zhou ↔ Professor Lazerson: Long-distance email exchange—Lazerson excitedly reports success on Wendelstein 7-X, initially doubted by Professor Kleiber; Lu Zhou agrees to the ITER paper submission.
• Lu Zhou ↔ Academician Lu: Off-page, through the gift of a written couplet expressing classical wisdom about ambition and tranquility.
• Office interactions among staff: Liu Bo pokes Qian Zhongming to ask about his bonus; they compare amounts, initially suspecting accounting errors, then confirming that "the finance department probably did not make a mistake." Liu Hong's reaction contrasts his past exploitation with present generosity.

Summary: "Reunited After a Long Separation"

Key Plot Points

• Lu Zhou spends a week at Jinling University, visiting old professors and studying the Navier–Stokes Equations (a Millennium Problem) at the university library and his hotel.
• On January 18, two days before the wedding, Dorm 201 members finally reunite at their old fish restaurant near the school gates.
• The four friends drink beer and catch up over a pot of grilled fish.
• Shi Shang reveals his life has veered off his plans: he got his girlfriend pregnant less than a year after graduation and had to buy a marital home with his parents' money rather than his own savings.
• Liu Rui gets drunk and passes out; Lu Zhou calls Wang Peng to drive everyone to the Purple Mountain Grand Hotel, where Huang Guangming and Liu Rui will stay until the wedding.
• Shi Shang asks Lu Zhou to be his best man.

Character Developments

• Liu Rui: Has visibly matured. He no longer feels depressed comparing himself to Lu Zhou, accepts the widening gap between them with composure, and has become more sociable (started playing basketball, brags to lab mates about his friendship with Lu Zhou).
• Shi Shang: Has changed the most—society has worn down his edges, and he now carries a more mature bearing. However, he is burdened by his unplanned pregnancy and financial struggles, confessing his experience as a "warning" to his roommates.
• Lu Zhou: Remains dedicated to his mathematical research even during personal time; acts as the reliable friend who handles logistics when others are drunk.
• Huang Guangming: Essentially unchanged, still giving off a "punchable vibe" even when not smiling.

Significant Themes

• The widening gap between friends after graduation: Lu Zhou's academic success (national awards, working on a Millennium Problem) starkly contrasts with his friends' more ordinary lives, yet the friendship endures.
• The gap between plans and reality: Shi Shang's carefully imagined post-graduation plan (save money, start a business, marry later) collapses under the reality of an unplanned pregnancy and financial dependence on parents.
• Class and status differences among peers: Despite Lu Zhou's fame, the group maintains their old dynamic of teasing and brotherhood.
• Friendship's resilience: The reunion shows that despite different life paths, the bond between the four roommates remains intact.

Character Interactions

• Liu Rui and Lu Zhou: Liu Rui toasts Lu Zhou, jokingly thanking him for letting him brag to a female lab mate that they had "slept together" (a misunderstanding Lu Zhou quickly corrects with embarrassment).
• Lu Zhou and Liu Rui (on academics): Liu Rui asks about Lu Zhou's current project; Lu Zhou hesitantly reveals he's working on the Navier–Stokes Equations, leaving Liu Rui speechless.
• Shi Shang and the group: Shi Shang drunkenly laments his situation as a "warning" to his friends, but Huang Guangming and Liu Rui call him out for essentially showing off about getting married and having a home; Lu Zhou silently agrees.
• Group dynamic: The friends drink, brag, tease one another, and ultimately take care of each other—old habits preserved despite the passage of time.

Summary: "Brother Fei's Most Glorious Day"

Key Plot Points and Events

• Pre-wedding gathering: Shi Shang organizes a dinner for the groomsmen and bridesmaids, including his former college roommate Lu Zhou and the bride Yajing's three college roommates (Deng Le, Xiao Yunyun, and Qian Hua).
• Casual outing: After dinner, the group goes to KTV, then returns to the Purple Mountain Grand Hotel for the night.
• Wedding day: The next morning, the wedding party completes the traditional bride pickup and ceremony at the hotel. During a lull, the groomsmen pass time chatting and eating sunflower seeds.
• Unexpected speech request: A staff member asks Lu Zhou to give a speech as "representative of the groomsmen" — something Shi Shang never mentioned. After some playful resistance (Huang Guangming refuses, Liu Rui insists), Lu Zhou takes the stage to address the wedding guests.

Character Developments

• Lu Zhou: Shows his willingness to support old friends despite being a distinguished Princeton professor; he actively participates in the wedding and is chosen as the speaker due to his status and eloquence.
• Shi Shang (Brother Fei): Revealed to be deeply nervous on his wedding day, described as experiencing the "most glorious moment" of his life.
• Qian Hua: Still feels awkward around Lu Zhou due to past tension over Wu Yan; the two exchange only polite words.
• Huang Guangming: Playfully teases Shi Shang about the wedding nerves and declines the speaking role, admitting he might ruin the occasion with crude remarks.
• Liu Rui: Acts as the voice of reason, nominating Lu Zhou because of his doctorate and professorship.
• Deng Le and Xiao Yunyun: Show warmth and curiosity toward Lu Zhou, teasing him about his single status.

Significant Themes

• Friendship and loyalty: Longtime college bonds remain strong despite distance and time; Lu Zhou's presence at the wedding demonstrates his commitment to friends.
• Celebration and nostalgia: The reunion of college roommates and dorm-mates evokes shared youth and memories.
• Social transitions: The wedding marks a milestone in the friends' lives, showing their divergent paths (academia vs. other careers) while maintaining their connection.
• Awkward residue of the past: Qian Hua's discomfort illustrates how old conflicts linger even after years.

Character Interactions

• Deng Le and Xiao Yunyun with Lu Zhou: Playful, admiring conversation — the girls express disbelief that Professor Lu actually came; they tease him about not having a girlfriend.
• Lu Zhou and Qian Hua: Tense, polite, and minimal; unresolved history with Wu Yan creates distance.
• Liu Rui and Huang Guangming: Comedic banter about who should give the speech, showcasing their comfortable, teasing friendship dynamic.
• Lu Zhou and Shi Shang: A silent but meaningful bond; Lu Zhou takes the stage to honor his friend, facing the crowd after the host announces him with impressive credentials.

Chapter Summary

Key Plot Points

• Lu Zhou delivers the best man's speech at his good friend Brother Fei's wedding
• He initially freezes onstage, causing concern among the bridesmaids, but then delivers a unique speech framing love as the greatest miracle in the universe from a mathematical/probability perspective
• The speech is well-received, moving the groom to tears
• After the wedding, Qian Hua approaches Lu Zhou to apologize for past harsh words about him
• She offers to buy him a drink, but Lu Zhou declines as he must catch a high-speed train; his remaining holiday plans involve family time before returning to work

Character Developments

• Lu Zhou: Shows a more personal, emotional side through his wedding speech; demonstrates that despite being mathematically focused, he can express warmth in his own way; also shows maturity by letting go of past grievances
• Qian Hua: Apologizes for her past behavior, showing personal growth; appears flustered and possibly developing feelings for Lu Zhou
• Xiao Yunyun: Confident in Lu Zhou's abilities, defends him, shows admiration for his eloquence

Significant Themes

• Love and probability: Lu Zhou frames romantic love as statistically improbable and therefore a miracle, contrasting mathematical thinking with human emotion
• Personal growth: Old conflicts can be resolved with time and maturity
• Different forms of expression: A person can express genuine feeling through their own unique perspective, even a mathematical one

Character Interactions

• Lu Zhou's speech addresses the wedding guests, groom, and bride
• Qian Hua, Xiao Yunyun, and Deng Le discuss Lu Zhou's speech among themselves
• Xiao Yunyun defends Lu Zhou against Qian Hua's skepticism
• After the wedding, Qian Hua seeks out Lu Zhou to apologize; he accepts graciously but declines her drink invitation

Summary: "A New Year"

Key Plot Points and Events

• Lu Zhou arrives at the high-speed rail station for his trip home to Jiangling for the New Year. He dismisses his bodyguard/driver Wang Peng, insisting he go home for the holiday—especially after joking that bringing a man home might make his family misunderstand.
• Lu Zhou takes the high-speed train to Jiangling Ancient City and arrives at his family home bearing gifts.
• He reunites with his mother Fang Mei (who is preparing fish for dinner), his little sister Tong, and his father "Old Lu" who returns from work.
• Lu Zhou presents Spring Festival couplets to his father, written by his graduate advisor, Academician Lu.
• Old Lu proudly mentions he has been reassigned to a logistics position with better benefits, and makes a point of urging Lu Zhou to bring home a daughter-in-law next year instead of gifts.

Character Developments

• Lu Zhou: Demonstrates consideration for others (persuading Wang Peng to go home), familial warmth and longing (his emotional reaction upon seeing his mother), and a playful, teasing dynamic with his family. He also deflects pressure about finding a spouse.
• Old Lu: Revealed to be a somewhat simple, proud working-class father who takes great pleasure in his son's achievements. His promotion indicates the family's circumstances have improved.
• Fang Mei: Shown as a typical loving mother, immediately concerned about preparing a proper meal. Her deft fish-cleaning is depicted with humorous flair.
• Tong: Portrayed as an affectionate, playful younger sister ("bro"), looking for gifts and teasing Lu Zhou.
• Wang Peng: Demonstrates профессиональная dedication to his job; briefly shows a more human side when persuaded to take a break for the New Year.

Significant Themes

• Family reunion and filial piety: The centrality of returning home for the New Year, respecting elders, and familial warmth.
• Honoring parents: Lu Zhou's desire to surprise his parents and his father's pride in his son's achievements.
• Social mobility and recognition: Old Lu's delight at his son having an academician as a mentor; his own improved work situation.
• Tradition vs. modernity: New Year couplets, superstition, and family expectations (marriage) set against Lu Zhou's modern, rational scientific outlook. Old Lu's gentle superstition ("If we invite the Star of Literary Brilliance's talent...") contrasts with Lu Zhou's protest that he is a scientist.
• Status and hierarchy in Chinese society: The "academician" status carries symbolic weight, even for the common father.

Character Interactions

• Lu Zhou ↔ Wang Peng: A lighthearted but firm exchange where Lu Zhou, now anxious to get rid of Wang Peng, uses the pretext that his family might mistake him for a romantic partner. Wry, humorous dynamic.
• Lu Zhou ↔ Tong: Sibling banter; Tong is eager for gifts and teasingly says her brother's neglect "breaks [her] heart"; Lu Zhou raps her forehead.
• Lu Zhou ↔ Fang Mei: Warm, emotional reunion. She scolds him for not calling ahead and quickly goes back to cooking.
• Lu Zhou ↔ Old Lu: A close, jocular father-son conversation covering topics from tea to marriage. Old Lu admires the academician's couplets and treats them with quasi-superstitious reverence. Ending on shared domestic chores.

Chapter Summary: “A Fluid with a High Viscosity Coefficient”

Key Plot Points and Events

• After dinner, Lu Zhou spends time with his family during the holiday. His mother refuses his help with the dishes, so he watches television with his younger sister, Tong.
• Tong asks whether Lu Zhou might win a Nobel Prize. He explains that Nobel recognition may take many years, despite Gerhard Ertl promising to recommend him. Lu Zhou expects the Fields Medal to be a more immediate possibility.
• Lu Zhou encourages Tong, an Economics student, to work toward an ambitious academic goal. He offers to help her mathematically, but emphasizes that she must first develop genuine ability and make meaningful contributions herself.
• The family’s newly displayed Spring Festival couplets attract attention because they were supposedly written by an academician. Old Lu exaggerates Lu Zhou’s reputation and begins accepting requests from neighbors and colleagues for Lu Zhou’s calligraphy.
• Although Lu Zhou has never used a brush, his father pressures him into writing the couplets. Lu Zhou practices and discovers that calligraphy gradually calms him, allowing him to relax from the intense pressure of his scientific work.
• After completing the requested couplets, Lu Zhou notices that the horizontal scroll for Academician Lu’s couplet is still missing. While looking at the thick ink, he suddenly compares it to a highly viscous fluid.
• This observation triggers an abstract mathematical inspiration. He abandons the calligraphy task, returns to his room, and begins working to translate the insight into formal mathematical language.

Character Development

• Lu Zhou remains ambitious but shows a more reflective and patient attitude toward major awards. He understands that recognition, especially the Nobel Prize, often requires years of sustained work.
• He acts as a supportive older brother, encouraging Tong while also stressing that academic success must be earned through real effort rather than borrowed prestige.
• Lu Zhou discovers an unexpected personal interest in calligraphy. The activity becomes a form of meditation, helping him release psychological tension and regain emotional balance.
• Tong is portrayed as playful and somewhat unmotivated, prioritizing her mobile game during vacation, but she is also curious about her brother’s achievements and receptive to his guidance.
• Old Lu is proud of his son and eager to use Lu Zhou’s reputation to impress others. His enthusiasm unintentionally creates extra work for Lu Zhou.

Significant Themes

• Ambition versus patience: Lu Zhou recognizes that the greatest academic honors often come only after long periods of work.
• Authentic achievement: The discussion with Tong emphasizes that recognition cannot be gained merely by attaching one’s name to someone else’s accomplishments.
• Family pride and social status: Old Lu’s pride in his son reflects how Lu Zhou’s achievements affect the family’s standing in the community.
• Rest and balance: Calligraphy offers Lu Zhou a rare moment of tranquility, showing the importance of activities outside intense intellectual labor.
• Unexpected inspiration: A simple everyday object—the viscous ink—becomes the source of a potentially important scientific insight.

Character Interactions

• Lu Zhou and Tong share a teasing but affectionate sibling relationship. He jokes about carrying her academically, while she humorously challenges his seriousness.
• Lu Zhou and Old Lu clash over the calligraphy requests, with Old Lu casually assigning a difficult task and Lu Zhou protesting his lack of experience.
• Despite the conflict, Lu Zhou ultimately fulfills his father’s requests, demonstrating family loyalty.
• Tong’s humorous remark that Mathematics may have made Lu Zhou “stupid” contrasts with his sudden scientific breakthrough and reinforces the lighthearted family atmosphere.

Summary: The Long-Awaited Princeton Town

1. Key Plot Points and Events

• Lu Zhou fails to produce the horizontal inscription for a couplet but makes unexpected progress on the Navier–Stokes Equations during his break, writing down a reformulation involving a bilinear operator B with a cancellation condition.
• He sets the research aside and enjoys an uninterrupted family holiday through the New Year and Lantern Festival—one of his first truly restful breaks in a long time.
• After the festival, he returns to Princeton. His PhD student Jericho picks him up at Philadelphia Airport and drives him back to Princeton Town.
• The next day, Lu Zhou visits Professor Fefferman at the Institute for Advanced Study.
• Fefferman, who continued working during Lu Zhou's absence, reveals new progress: he has constructed a partial differential equation using an abstract bilinear operator B′ that shares structural similarities with the Euler linear operator B but is distinct from it. If this stronger conclusion can be proved, it would indirectly prove the original Navier–Stokes conclusion.
• Fefferman's two doctoral students, unable to follow the discussion, quietly return to their own work.

2. Character Developments

• Lu Zhou: Shows growing discipline around work-life balance (initially promises not to touch math but instinctively returns to it). He experiences a genuine "reboot" of energy and spirits from the family break. He is eager and ready to resume the demanding research collaboration.
• Professor Fefferman: Demonstrates intellectual generosity and relentless dedication, continuing the research solo while his partner was away. He is philosophical and indirect in his communication, preferring to convey ideas through metaphor (the cigarette) and blackboard mathematics rather than direct exposition.

3. Significant Themes

• Mathematics as obsession: Even when Lu Zhou tries to rest, his mind gravitates toward the research—mathematics is not something he can simply switch off.
• Piety toward family vs. intellectual vocation: The chapter contrasts the warmth of family time with the pull of the academic calling; both are shown as necessary, but the return to Princeton is framed as a homecoming of a different kind.
• The nature of chaos and complexity: Fefferman's cigarette metaphor—orderly swirl becoming chaotic dispersion—mirrors the mathematical content of the Navier–Stokes equations themselves.
• Indirect proof strategies: The chapter highlights an elegant mathematical insight—that proving a stronger, related result can sometimes be easier than proving the original claim directly.

4. Character Interactions

• Lu Zhou and Chen Yushan: A brief text exchange. Chen playfully scolds Lu Zhou for not waiting to travel together (though he had no idea she wanted to), and uses a cutesy emoticon. Lu Zhou notes this out-of-character behavior reminds him of AI.
• Lu Zhou and Jericho: Jericho eagerly picks up his professor at the airport, expresses warmth and respect, and confirms the research group functioned but felt incomplete without him.
• Lu Zhou and Fefferman: The core academic relationship. Fefferman offers Lu Zhou a cigarette (which he declines to smoke but holds), reflects philosophically on fluid dynamics, and walks him through a major new result on the blackboard—treating Lu Zhou as a true intellectual equal and collaborator.

Summary of the Chapter

1. Key Plot Points & Events

• Lu Zhou and Professor Fefferman agree on a general proof strategy for the Navier–Stokes Equations but struggle to construct the required abstract bilinear operator.
• Their calculations increasingly suggest that the 3D Navier–Stokes Equations may experience a finite-time “blow-up” — a singularity where smoothness fails — contradicting physical intuition.
• Both mathematicians are disturbed by this implication. Fefferman, smoking in the conference room, compares the result to the inevitable drift of predictable systems into chaos.
• That evening, Lu Zhou goes running to clear his head but remains preoccupied. He meets Molina, who tosses him a sports drink and teases his declining stamina.
• Lu Zhou asks Molina whether a calm lake could suddenly explode on its own under normal circumstances. She says no, but he replies that his mathematics suggests such a possibility truly exists.

2. Character Developments

• Lu Zhou: Still deeply engaged in the research, but visibly troubled by the abstract conclusion that clashes with common sense. He tries to unwind through physical exertion but cannot escape the problem. His frustration is shown through his intense sprint and vacant stares.
• Fefferman: Experiences a rare loss of composure and confidence. His uncertainty about his own equations and his chain-smoking indicate the psychological weight of the impasse.
• Molina: Appears as a supportive colleague with a slightly teasing, competitive edge. She notices Lu Zhou’s physical condition and jokes about it, but also shares a quiet moment of conversation.

3. Significant Themes

• Nonlinearity and Complexity: The chapter highlights how “nonlinearity” in PDEs leads to extreme difficulty and counterintuitive results.
• Mathematics vs. Physical Intuition: The possibility that the Navier–Stokes equations permit spontaneous blow-up—like a self-exploding lake—illustrates the gap between rigorous math and ordinary experience.
• Chaos and Determinism: Fefferman’s reflection on predictable systems inevitably descending into chaos mirrors the mathematical conclusion.
• Psychological Burden of Research: The emotional strain of pursuing a result that seems absurd is depicted through the characters’ moods and habits.

4. Character Interactions

• Lu Zhou ↔ Fefferman: A tense, candid discussion where both acknowledge the disturbing implications of their work. Fefferman verbalizes his doubts; Lu Zhou shares his disbelief but cannot refute the mathematics.
• Lu Zhou ↔ Molina: A casual evening encounter near Carnegie Lake. Molina offers him a drink, jokes about his fitness, and listens as he confesses that his math suggests even a peaceful lake might “explode.” She laughs it off as bizarre, but Lu Zhou remains serious.

Chapter Summary

Key Plot Points

• Lu Zhou is researching the three-dimensional Navier–Stokes Equations, focusing on why his equations “blow up” (diverge) rather than merely whether global regularity exists.
• Molina briefly interacts with him, suspecting he is overcomplicating the problem, but Lu Zhou insists the complexity is necessary.
• After struggling with existing methods, Lu Zhou has a breakthrough: instead of searching for existing tools, he invents his own—a concept he names the Lu Manifold (English: L-manifold, with L-Flow for short). He constructs an approximate three-dimensional manifold from abstracted fluid molecules and applies topological methods.
• He spends about a month (through April) isolated at home, working intensively on the idea, until he finishes a draft manuscript.
• He then receives an email from the journal Mathematical Annals confirming that a paper proving the Collatz Conjecture—authored by his students—will be published in the latest issue.

Character Developments

• Lu Zhou: Persistent, highly creative, and willing to abandon conventional approaches. He values insight over incremental progress and becomes fully absorbed in research.
• Molina: Sarcastic but genuinely concerned; she questions Lu Zhou’s methods but ultimately acknowledges his seriousness.
• Fefferman: Supportive colleague; temporarily suspends regular discussions to give Lu Zhou space.
• Vera: Assists Lu Zhou by delivering reports, maintaining a background role.

Significant Themes

• Creative problem-solving and the need to build new mathematical frameworks.
• The isolating nature of deep intellectual work.
• The contrast between conventional wisdom and bold, unconventional ideas.
• Mentorship and the success of students reflecting on the mentor’s work.

Character Interactions

• Molina and Lu Zhou: Brief but meaningful exchange; she teases him about having a fever, he defends his approach.
• Vera and Lu Zhou: Practical interaction—she delivers his teaching reports; he barely looks at them, showing his focus.
• Fefferman and Lu Zhou: Indirect support; Fefferman deliberately avoids disturbing Lu Zhou during his intense working phase.

Summary

1. Key Plot Points and Events

• Lu Zhou emerges from seclusion and visits his office at the Institute for Advanced Study.
• He announces that the team’s paper on the Collatz Conjecture has been accepted by Mathematical Annals, with Professor Sarnak as reviewer — a major honor.
• Lu Zhou congratulates his students: Vera, Hardy, Jericho, Wei Wen, and Qin Yue, formally declaring them graduated.
• Each student chooses to continue their studies (doctoral programs at Princeton) rather than leave academia.
• Lu Zhou then meets with Professor Fefferman to discuss his recent work on the Navier–Stokes equation, presenting a new approach using differential manifolds.
• Fefferman shares his own partial progress on a bilinear operator construction. Lu Zhou studies it and suddenly identifies a key flaw: “I think I know where the problem is…”

2. Character Developments

• Lu Zhou: Shows pride and generosity as a mentor, offering support and recommendation letters. His intellectual drive remains central, as he pivots from one Millennium Problem (Collatz) to another (Navier–Stokes) and immediately spots the gap in Fefferman’s equations.
• Qin Yue: Displays humility, crediting Lu Zhou’s tools rather than his own ability, but also shows confidence in continuing academics.
• Vera, Hardy, Jericho, Wei Wen: All express surprise and admiration; Hardy eagerly asks to stay under Lu Zhou’s supervision, while Wei Wen envies Qin Yue’s success.
• Fefferman: Engages as a dedicated collaborator, open to new ideas yet skeptical of the differential manifold approach, and shows persistence in his own attempts.

3. Significant Themes

• Academic achievement and the recognition of hard work (publication in a top journal).
• Mentorship: Lu Zhou’s guidance not only solves problems but also shapes students’ futures.
• Collaborative research: Exchange of ideas across different fields (topology applied to PDEs) and between senior scholars.
• Scientific progress as iterative: breakthroughs come from inspiration, but also from identifying flaws in existing attempts.

4. Character Interactions

• Lu Zhou warmly interacts with his students, praising their work and asking about their future plans.
• Students show collective admiration and gratitude toward Lu Zhou; Qin Yue’s modesty contrasts with Hardy’s easygoing persistence.
• The meeting with Fefferman is an intense, respectful intellectual exchange—Fefferman critiques Lu Zhou’s topological approach, then presents his own equations, which Lu Zhou immediately analyzes and finds a conceptual error in.

Chapter Summary: “Splitting into Two Paths”

Key Plot Points and Events

• Lu Zhou explains to Professor Fefferman that their previous approach using the Abstract Proof Method only proves the existence of a smooth weak solution for a limited interval, ((0,T_1)). The crucial value of (T_1) remains unknown, so the method does not solve the full Navier–Stokes Millennium Problem.
• He also points out that they still need a rigorous way to distinguish Fefferman’s original operator (B) from the newly constructed bilinear operator (B).
• Fefferman accepts that their earlier reasoning was incomplete but insists that the Abstract Proof Method still has potential. Rather than abandoning it, he proposes that they pursue separate research directions.
• Lu Zhou agrees and returns home to revise his paper on the L-manifold. He submits it to Mathematical Annals, where it passes the editor’s initial review and enters peer review.
• Lu decides to present his ongoing Navier–Stokes research at the International Mathematical Union Congress in August, despite admitting that his current understanding of the problem is still incomplete.
• Meanwhile, Lu’s student Qin Yue’s paper on the Collatz Conjecture is published in Mathematical Annals, immediately attracting international attention.
• At Nankai University’s Chen Shengshen Institute of Mathematics, Academician Zhang Wei and Professor Fu Lei discuss Qin Yue’s achievement and Lu Zhou’s influence. They learn that Lu is working on the existence and smoothness problem for the Navier–Stokes Equations, one of the seven Millennium Problems.
• Zhang resolves to recommend Qin Yue for the Thousand Talents Plan, hoping to prepare for his possible return to China or future research there.

Character Developments

• Lu Zhou demonstrates intellectual honesty by recognizing the limitations of the proof method rather than overstating its success. He also shows maturity by accepting Fefferman’s decision to continue independently.
• Lu remains ambitious but realistic: although his Navier–Stokes work is still uncertain, he intends to present it publicly and continues refining his separate L-manifold research.
• Professor Fefferman shows persistence and independence. He acknowledges the flaws in the existing approach but refuses to give it up prematurely, choosing to investigate it on his own.
• Qin Yue emerges as a highly capable student whose work on the Collatz Conjecture is impressive enough to gain international recognition and attract institutional interest.
• Academician Zhang Wei appears supportive and perceptive, recognizing both Qin Yue’s talent and Lu Zhou’s extraordinary potential.
• Professor Fu Lei reacts with disbelief and awe to Lu’s accomplishments, highlighting how unusual Lu’s rapid rise has become.

Significant Themes

• Multiple paths in research: The chapter emphasizes that difficult mathematical problems may require parallel approaches rather than a single agreed-upon method.
• Limits of partial success: Proving local smoothness is valuable but fundamentally different from solving the global Navier–Stokes problem.
• Intellectual honesty and rigor: Lu refuses to treat an incomplete argument as a solution, stressing the importance of identifying unresolved definitions and distinctions.
• Mentorship and academic legacy: Lu’s influence is reflected in Qin Yue’s success, while Zhang and Fu discuss how exceptional scholars shape future generations.
• Extraordinary talent and public expectations: Lu’s achievements have become so remarkable that even major awards and international recognition seem plausible, though his colleagues remain uncertain whether he can solve a Millennium Problem.
• Research value beyond immediate application: The L-manifold may not directly solve Navier–Stokes, but it could become an important independent contribution with future uses in topology, physics, or engineering.

Character Interactions

• Lu Zhou and Fefferman: They engage in a rigorous but respectful debate over the Abstract Proof Method. Their disagreement leads not to conflict but to a practical decision to pursue two independent approaches.
• Zhang Wei and Fu Lei: They discuss Qin Yue’s paper, academic ethics, Lu’s mentorship, and Lu’s work on Navier–Stokes. Fu is skeptical that Lu would give an important result to a student, while Zhang views Lu as a rare, historically significant mathematical talent.
• Zhang Wei and Qin Yue: Though Qin Yue does not appear directly, Zhang’s earlier recommendation and later plan to nominate him for the Thousand Talents Plan demonstrate his continued support.
• Lu Zhou and Qin Yue: Their relationship is shown indirectly through Qin Yue’s successful publication, which reflects Lu’s effectiveness as a mentor and his willingness to support his student’s achievements.

Chapter Summary: “Two Invitations”

Key Plot Points and Events

• The Collatz Conjecture is announced as proven in a paper published in Mathematical Annals. The proof confirms that every natural number eventually reaches the cycle (4 \to 2 \to 1).
• The achievement attracts worldwide attention, especially because the paper’s main contributors are Lu Zhou’s doctoral students.
• Since the International Congress of Mathematicians is approaching, the International Mathematical Union (IMU) arranges a special forty-five-minute presentation on the proof.
• Professor Sanak delivers the invitation to Lu Zhou, but Lu declines because he already has a one-hour presentation scheduled and is focused on the Navier–Stokes problem.
• Lu reveals that he believes his newly developed “L-manifold” is the key tool needed to solve the Navier–Stokes Equations before the August congress.
• Lu then meets with Vera, Qin Yue, and Hardy. He proposes that Vera, who completed most of the Collatz proof, give the IMU presentation in his place.
• Qin Yue and Hardy immediately support the idea, effectively deciding the matter despite Vera’s nervous objections.
• Vera worries because she has never attended or presented at the IMU Congress. Lu reassures her by citing her successful Berkeley presentation and reminding her that she has four months to prepare.
• When Vera asks Lu for help, he explains that he is about to enter another period of intense seclusion and that she must prepare independently.

Character Developments

• Lu Zhou demonstrates confidence in his research and prioritizes solving the Navier–Stokes Equations over receiving additional public recognition for the Collatz proof. He also shows strong trust in his students by giving Vera the prestigious presentation opportunity.
• Vera is revealed to be the principal contributor to the Collatz proof, but she lacks confidence in public speaking and feels overwhelmed by the scale of the opportunity. Her acceptance of the responsibility marks the beginning of a new professional challenge.
• Qin Yue and Hardy show loyalty to the team and confidence in Vera by immediately endorsing her as presenter.
• Professor Sanak shifts from skepticism about Lu’s L-manifold to admiration after realizing it may be instrumental in solving a major Millennium Problem.

Significant Themes

• Recognition versus responsibility: Lu gives up the chance to present the Collatz proof, believing the person who did the core work should receive the recognition.
• Mentorship and independence: Lu encourages Vera but refuses to guide her directly, forcing her to grow beyond dependence on her mentor.
• Confidence in unfinished work: Lu’s certainty that the L-manifold will help solve Navier–Stokes contrasts with the academic community’s uncertainty about its meaning and usefulness.
• Collective achievement: Although the Collatz proof brings fame, the chapter emphasizes the contributions of the entire research team and the importance of collaboration.
• Pressure of greatness: Vera must confront the intimidating expectations attached to presenting a historic mathematical result before the world’s leading mathematicians.

Character Interactions

• Sanak and Lu: Sanak offers Lu the IMU invitation and questions whether he can solve Navier–Stokes so quickly. Lu confidently explains his faith in the L-manifold, impressing Sanak.
• Lu and his students: Lu organizes a team meeting and proposes Vera as speaker, treating the students as capable mathematicians rather than merely assistants.
• Vera, Qin Yue, and Hardy: The two male students immediately support Vera, while she reacts with nervous resistance. Their exchange highlights both their confidence in her and her own insecurity.
• Lu and Vera: Lu encourages Vera by recalling her previous success, but he ultimately tells her she must rely on herself, reinforcing her transition toward independence.

Chapter Summary: “An Uneasy June”

Key Plot Points and Events

• Lu Zhou commits to presenting his work on the three-dimensional incompressible Navier–Stokes Equations at the International Congress of Mathematicians.
• With the congress scheduled for early August, he gives himself only until early July to complete the paper, leaving roughly three months for the research and writing.
• By early June, after working intensely and often pulling all-nighters, Lu Zhou finally updates his official conference information and registers his report topic.
• His announcement causes a major stir throughout the mathematical community because the Navier–Stokes problem is one of the Clay Mathematics Institute’s Millennium Prize Problems.
• Online discussions become polarized. Some mathematicians dismiss Lu Zhou’s claim, arguing that his expertise in Number Theory should not translate to solving a major problem in Partial Differential Equations. Others remain cautiously hopeful because he previously solved the Goldbach Conjecture.
• Terence Tao comments publicly that Lu Zhou is ambitious but unlikely to pursue such a difficult problem without substantial confidence. Tao connects Lu Zhou’s recently developed L-manifold to a possible bridge between Topology and Partial Differential Equations, suggesting it may be relevant to Navier–Stokes.
• Professor Lions travels from Paris to New York to meet his old friend, Professor Fefferman, and learn whether Lu Zhou has genuinely made progress.
• Fefferman explains that he and Lu Zhou once worked together on Navier–Stokes but eventually separated into different approaches. He admits he does not know the details of Lu Zhou’s current progress.
• Lions asks to meet Lu Zhou, but Fefferman says Lu Zhou has secluded himself completely for his research. Fefferman humorously describes this as “closed-door cultivation.”

Character Developments

• Lu Zhou is shown as intensely focused, disciplined, and willing to isolate himself from ordinary life to pursue a difficult mathematical goal. His confidence is implied by his decision to announce the topic publicly, despite the immense skepticism it generates.
• Terence Tao appears as a careful but perceptive observer. He refuses to make a definite judgment before seeing the proof, yet recognizes the potential importance of Lu Zhou’s L-manifold and believes Lu Zhou likely has a serious strategy.
• Fefferman reveals both respect and bemusement toward Lu Zhou. Though he cannot explain Lu Zhou’s current work, he has become accustomed to his extreme habits and defends the idea that Lu Zhou is socially pleasant despite being highly reclusive while researching.
• Lions is portrayed as curious, serious, and eager to investigate the claim personally. His willingness to cross the Atlantic shows how significant Lu Zhou’s announcement is to experts in the field.

Significant Themes

• Isolation and sacrifice in pursuit of genius: Lu Zhou’s “closed-door cultivation” emphasizes the extreme dedication required to attempt a Millennium Prize problem.
• Skepticism versus possibility: The mathematical community doubts that one person can excel in unrelated fields, yet Lu Zhou’s past achievements and new methods prevent complete dismissal.
• Interdisciplinary mathematics: The L-manifold represents a possible connection between Differential Geometry, Topology, and Partial Differential Equations.
• Reputation and proof: Despite Lu Zhou’s fame, the community insists that his work must be judged by the actual paper rather than by his previous accomplishments.
• Anticipation and uncertainty: The chapter builds tension around whether Lu Zhou has truly solved Navier–Stokes, while keeping the result unknown.

Character Interactions

• Lu Zhou and the mathematical community: His conference registration triggers widespread speculation, criticism, and excitement.
• Tao and Lu Zhou: Although they do not meet directly, Tao analyzes Lu Zhou’s recent work and offers cautious support.
• Lions and Fefferman: Their reunion becomes an inquiry into Lu Zhou’s progress. Lions seeks concrete information, while Fefferman can only provide limited knowledge because of their split research paths.
• Fefferman and Lu Zhou: Their former collaboration has evolved into independent approaches. Fefferman’s comments suggest admiration for Lu Zhou’s abilities, even as he finds his research habits extreme.

Chapter Summary: “Answering the Millennium Problem!”

1. Key Plot Points and Events

• Lu Zhou isolates himself for the final week of research on the Navier–Stokes existence and smoothness problem, convinced that only one final obstacle remains.
• He begins a 168-hour countdown, entering an intense state of heightened mathematical intuition in which numbers and abstract forms seem almost vividly interconnected.
• On the submission deadline for the International Congress of Mathematicians, Vera teaches Lu Zhou’s class at Princeton while students anxiously ask about the progress of his research.
• Vera admits she does not know the details of Lu Zhou’s work but confidently declares that he will succeed, based on her faith in him and the miracles he has already helped make possible in her life.
• After the rain ends, Lu Zhou sees a rainbow and finishes his research. The timer reaches zero, marking the end of his weeklong seclusion.
• Professor Fefferman urgently calls, reminding Lu Zhou that the deadline for submitting his conference paper has arrived and suggesting that he change his presentation topic if the Navier–Stokes work is unfinished.
• Lu Zhou reveals that he has not obtained a general solution to the equations, but he has proved that their solutions exist and remain smooth—effectively solving the Millennium Prize problem.

2. Character Developments

• Lu Zhou: His confidence and determination deepen as he pushes through the final stage of the proof. His extreme concentration temporarily suppresses his emotions, but completing the work restores his humanity and brings back his smile. He remains calm and even somewhat absent-minded about the conference deadline, showing how completely absorbed he has been in mathematics.
• Vera: She has grown from an insecure student into a capable and admired teaching assistant, especially after proving the Collatz Conjecture. Her faith in Lu Zhou reflects both her gratitude toward him and her personal transformation.
• Professor Fefferman: Usually composed and easygoing, he becomes visibly anxious and irritable because of Lu Zhou’s silence and the approaching deadline. His reaction emphasizes the enormous significance of Lu Zhou’s research.

3. Significant Themes

• Faith and belief: Vera’s confidence in Lu Zhou is based not on direct knowledge of his proof but on trust built through his past accomplishments and influence on her life.
• Human intellect and transcendence: Lu Zhou’s heightened mathematical intuition is portrayed almost spiritually, comparing mathematics to the language of God and presenting him as temporarily godlike within his field.
• Perseverance and isolation: The chapter emphasizes the sacrifices and intense solitude required to solve an extraordinarily difficult problem.
• Reality versus public expectation: While the academic world speculates about Lu Zhou’s progress, he remains detached from publicity, focused only on completing the mathematics.
• Miracles through effort: Vera’s life and Lu Zhou’s breakthrough both suggest that seemingly impossible achievements can become real through talent, persistence, and guidance.

4. Character Interactions

• Vera interacts with her students, who seek information about Lu Zhou’s progress. Her confident defense of him strengthens her role as both a respected teacher and a devoted supporter.
• Lu Zhou and Fefferman’s phone conversation provides comic relief through Lu Zhou’s mistaken “Walmart fruit sale” answer, but quickly turns serious when Fefferman worries that Lu Zhou has missed the submission deadline.
• Their exchange culminates in Lu Zhou’s understated announcement that he has proved the existence and smoothness of Navier–Stokes solutions, leaving Fefferman stunned.

Chapter Summary: "The World Was Unexpectedly Small"

Key Plot Points

• Professor Fefferman rushes to Lu Zhou's home after a phone call, breathless and demanding to know if Lu Zhou truly solved the Navier–Stokes Equations.
• Lu Zhou leads him upstairs, cleans up his chaotic study, and hands over his roughly hundred-page manuscript.
• Fefferman reads for about an hour, skimming while checking for errors, and finds none. He declares the work "incredible."
• Fefferman explains the broader significance: Lu Zhou's introduction of "L-Manifolds" brings differential geometry techniques into partial differential equations, and incorporates topological principles—an unprecedented fusion.
• Fefferman refuses to let Lu Zhou rest. He insists the manuscript must be organized into electronic form and uploaded within three days (at most, a week), and offers to contact Shigefumi Mori, chairman of the International Mathematical Union, on Lu Zhou's behalf.
• In a surprising reveal, Fefferman recounts the 2000 Clay Mathematics Institute Millennium Conference and its Navier–Stokes declaration—then reveals he was the speaker who presented that Millennium Problem.
• Fefferman congratulates Lu Zhou, calls him and Lu Zhou witnesses to a new page in history, and advises him to preserve the manuscript as a potential historical artifact.

Character Developments

• Lu Zhou: Displays a mix of exhaustion, quiet pride, and modesty. He hesitates to claim credit, needs prompting to understand the full scope of what he has done, and is caught off guard by Fefferman's revelation. He is initially resistant to the pressure to immediately upload the paper.
• Professor Fefferman: Revealed to have a deeper personal stake in the Navier–Stokes problem than previously known—he was the one who formally presented it as a Millennium Problem. He is rigorous, enthusiastic, and generous, treating Lu Zhou's achievement with historic reverence and offering practical administrative help.

Significant Themes

• The weight of solving a Millennium Problem: The chapter emphasizes the gravity and historical significance of what Lu Zhou has accomplished, with Fefferman framing it as a turning point in mathematical history.
• The importance of timely publication: Fefferman's urgency underscores the norms of the mathematical community—discoveries must be submitted promptly to be verified and credited.
• Pioneering mathematical innovation: Fefferman argues that the L-Manifold method itself is as significant as the solution, comparing it to Gromov's pseudoholomorphic curves and Yau's geometric analysis—possibly founding a new field.
• The connectedness of the academic world: Lu Zhou's surprise at Fefferman's role highlights how small the world of elite mathematics truly is.

Character Interactions

• Fefferman's urgency vs. Lu Zhou's calm: Fefferman is breathless and insistent; Lu Zhou is composed, even joking about his own cleverness.
• Mentorship and recognition: Fefferman moves from examiner to advocate, offering praise, practical help (contacting the IMU), and framing the achievement in historical terms.
• The closing exchange: When Fefferman recites the original Millennium Problem declaration, Lu Zhou initially finds it oddly familiar, then realizes the significance only after Fefferman reveals he delivered it—creating a memorable dramatic moment.

Summary of the Chapter

Key Plot Points and Events

• Lu Zhou calls his student Wei Wen to help input a manuscript into the computer, inviting Jericho as well.
• For three days and nights, Lu Zhou, Wei Wen, and Jericho work to complete the digital version of Lu Zhou's paper on the Navier–Stokes Equations.
• Professor Fefferman had already contacted the International Mathematical Union on Lu Zhou's behalf, securing a one-week deadline extension.
• The IMU reschedules Lu Zhou's one-hour lecture to a special session before the opening ceremony, dedicated to the Navier–Stokes Equations.
• The paper is uploaded three days later than expected, causing a massive traffic spike on the ICM website that nearly crashes the server.
• The mathematics community initially falls silent as experts work through the dense hundred-page paper.
• Terence Tao publishes a blog post praising the paper's novel approach, saying he can find no obvious flaws and eagerly anticipates the conference.
• Within two weeks, discussion of the Navier–Stokes Equations explodes across mathematics forums, including the Chinese Shuimu Forum, where users debate the paper's validity and celebrate Lu Zhou's achievement.
• The chapter ends with the arrival of July 31, the day before the ICM opening ceremony—though for many, the event has already effectively begun.

Character Developments

• Lu Zhou remains remarkably calm and unexcited even after solving a Millennium Problem, showing his maturity and detachment from fame. He is more interested in the work itself than the glory.
• Wei Wen is depicted as intensely passionate and obsessive—despite exhaustion, he refuses to sleep, repeatedly checking the paper for errors, embodying the devoted student.
• Jericho is shown as completely drained after the marathon work session, collapsing into sleep the moment they finish.

Significant Themes

• Creativity and Methodology: The chapter emphasizes Lu Zhou's innovative use of differential manifolds and topological methods to solve a problem in partial differential equations—elevating a "research style" into a genuine "method."
• Academic Recognition and Pressure: The global mathematics community's intense scrutiny and anticipation, with experts racing to verify or challenge the proof.
• The Human Cost of Intellectual Work: The exhaustive toll of producing groundbreaking work, reflected in the students' exhaustion.
• Legacy and Institutional Response: How major figures like Fefferman and Tao validate and amplify Lu Zhou's achievement, and how institutions like the IMU accommodate it.

Character Interactions

• Lu Zhou calls Wei Wen, establishing a working relationship built on mutual trust and shared purpose.
• Lu Zhou expresses genuine concern for his students' well-being, offering to treat them to a meal and telling them to rest.
• Wei Wen shows deep respect and admiration for Lu Zhou, but also frustration at Lu Zhou's nonchalant attitude toward the historic nature of the work.
• Fefferman acts offstage as a mentor-like sponsor, using his influence to secure accommodations from the IMU.
• Tao's blog post serves as an indirect bridge between the author and the broader mathematical community, affirming the paper's significance and creating anticipation.

Chapter Summary: “A World-Renowned Event”

Key Plot Points and Events

• On July 31, 2018, mathematicians from around the world gather in Rio de Janeiro for the International Congress of Mathematicians.
• The main attraction before the official opening ceremony is Professor Lu Zhou’s presentation on the Navier–Stokes Equations, one of the most important unresolved problems in mathematics. His potential success could represent not only a million-dollar prize but also the beginning of a new era in mathematical research.
• Crowds arrive early at the Barra da Tijuca Hotel, and the lecture hall becomes completely packed well before the presentation begins. Media organizations and film museums prepare to record the event in case Lu Zhou achieves a historic breakthrough.
• Professor Xu Chenyang arrives with top mathematics students from Yanshan University. He discusses Lu’s work and its controversial, innovative methods with the students.
• Xu reunites with Zhang Wei, another celebrated Chinese mathematician and one of the conference’s invited speakers. They discuss the intense competition for the upcoming Fields Medals.
• In the lecture hall, Vera anxiously waits for Lu’s presentation. She is joined by Pierre Deligne, who expresses complete confidence in Lu as his most accomplished student.
• Faltings arrives and openly criticizes Lu’s work, suggesting that the presentation may be nonsense. Deligne calmly challenges his skepticism, but the two do not make a bet. As the presentation is about to begin, anticipation reaches its peak.

Character Developments

• Lu Zhou is established as the central figure of the chapter, although he does not yet appear directly. His work has attracted worldwide attention and may revolutionize mathematics.
• Xu Chenyang is portrayed as an accomplished, respected scholar and a supportive mentor. Though unfamiliar with Lu’s specific field, he recognizes the importance of Lu’s research and encourages his students’ interest.
• Zhang Wei is revealed to be a major mathematical prodigy, admired as “God Wei.” Despite his impressive achievements, he shows humility and insecurity about his chances of winning a Fields Medal.
• Vera demonstrates deep emotional investment in Lu’s success. Her nervousness shows both admiration and personal concern for him.
• Pierre Deligne displays strong confidence and pride in Lu, calling him his most accomplished student.
• Faltings serves as a skeptical and antagonistic intellectual presence, openly challenging Lu’s credibility and Deligne’s praise.

Significant Themes

• The pursuit of groundbreaking knowledge: Lu’s work represents humanity’s effort to solve one of mathematics’ greatest challenges.
• Innovation versus established opinion: The novel methods in Lu’s paper have divided the academic community, highlighting how revolutionary ideas often provoke controversy.
• Academic competition and prestige: The chapter emphasizes the intense competition surrounding the Fields Medal and the elite status of the mathematicians attending the congress.
• Humility among geniuses: Despite their extraordinary accomplishments, Xu and Zhang acknowledge the difficulty of standing out among the world’s strongest mathematicians.
• The weight of history: The cameras and international attention suggest that Lu’s presentation could become a defining moment remembered for generations.

Character Interactions

• Xu discusses Lu’s research with his students, encouraging their academic curiosity while admitting the limits of his own expertise.
• Xu warmly reunites with Zhang Wei, and the two discuss the upcoming Fields Medal selection, with Xu offering Zhang reassurance.
• Vera quietly prays for Lu’s success and is comforted by Deligne’s confidence.
• Deligne and Faltings exchange sharp remarks over Lu’s abilities and the validity of his work, reflecting their long-standing rivalry and contrasting attitudes.
• The chapter ends with Deligne confident, Faltings skeptical, and the entire audience waiting for Lu Zhou’s historic presentation to begin.

1. Key Plot Points and Events

• Lu Zhou delivers a high-stakes lecture at the Institute for Advanced Study, presenting his proof of the existence and smoothness of solutions to the 3D incompressible Navier–Stokes equations.
• He opens with philosophical questions about why speeding cars don't tear apart or lakes don't explode, framing the mathematical problem as one about singularities and finite-time blowup.
• He introduces his proof strategy: combining partial differential equations with topological methods, specifically using an "L-manifold" to break through the main difficulty.
• The proof concludes successfully; the final line shows the solutions exist and are smooth. The audience erupts in applause.

2. Character Developments

• Lu Zhou: Demonstrates calmness under immense pressure; his clarity and pace accelerate as the lecture progresses, showing complete mastery. He positions himself as a mathematician who bridges many fields.
• Faltings: Leaves early after fully understanding the proof, dismissively calling it "average" — a display of his characteristic pride and inability to openly praise others.
• Deligne: Teases Faltings about his grudging attitude, showing their long-standing friendship and playful rivalry.
• Vera: Feels proud of her advisor, emotionally affected by witnessing his success.
• Terence Tao & Fefferman: Silently acknowledge the brilliance of the proof; Tao's excitement and Fefferman's knowing smile signal expert recognition.

3. Significant Themes

• Mathematical rigor vs. intuition: The chapter contrasts everyday "obvious" phenomena with the need for rigorous proof, emphasizing that math demands certainty, not vague words like "perhaps."
• Interdisciplinary unity: The proof merges analysis (PDEs) with topology/geometry (manifolds), illustrating that breakthroughs often come from unifying seemingly separate branches.
• Recognition across generations: The reactions of senior mathematicians (Deligne, Faltings, Fefferman) and younger ones (Tao, Vera, Xu, Zhang) highlight the communal nature of mathematical validation.

4. Character Interactions

• Xu Chenyang and Zhang Wei: Discuss the difficulty of following the PDE proof; Zhang notes Lu Zhou's expertise spans number theory, algebraic geometry, and more, speculating even algebraic geometry is within his grasp.
• Deligne and Faltings: After Faltings deems the work "average," Deligne mocks him, "Aren't you embarrassed to say that?" — and returns Faltings' earlier gift, a playful jab at his grudging compliment.
• Audience reaction: The lecture hall falls silent at the start, roars with applause at the end, showing collective awe and approval.