Recaps

Xiao Yi completed the afternoon exam in thirty minutes, drawing the attention of the academicians monitoring the fifteen universities. As he wandered around USTC, he realized that his thoughts on the sieve method were beginning to form a unified idea that might bring him closer to solving a problem.

• Students at the university examination halls struggled with the unfamiliar questions, while some gradually applied the Principle of Inclusion-Exclusion to the first one.
• Xiao Yi handed in his paper at USTC after thirty minutes, prompting the remote proctors to pull up the surveillance feed.
• Hu Guangde told Liao Huan that Xiao Yi had written a mathematical paper and had finished the morning exam in twenty minutes.
• Liao Huan and the other academicians grew curious about Xiao Yi’s paper, but Liu Min explained that it could not be viewed before the exam ended.
• Xiao Yi wandered around campus, thinking about the second question and the sieve methods in Zhang Yitang’s paper as his scattered ideas began to cohere.

Peking University academicians discovered that Xiao Yi’s exam paper went beyond Zhang Yitang’s method on the twin-prime problem, while IHES editor-in-chief Bojielun read Peter Scholze’s enthusiastic review of a paper whose ideas could help refute Shinichi Mochizuki. The chapter left both groups stunned by the paper’s mathematical significance.

• Liao Huan found a completed exam paper whose solutions to the first excerpt and problems were exceptionally strong, and suspected it belonged to Xiao Yi.
• Liao Huan found that the paper’s treatment of Zhang Yitang’s twin-prime method went beyond what he could understand, so he called Wang Jianxin over.
• Wang Jianxin studied the work and declared that Xiao Yi had improved Zhang Yitang’s method by approaching it from a different angle.
• Bojielun opened an email from Peter Scholze, who said the paper was unquestionably correct, highly valuable in Far Abelian Geometry, and had helped him identify a key point for proving Shinichi Mochizuki wrong.
• Peter Scholze told Bojielun he planned to discuss the matter with Jacob Stix and hoped the journal would publish the paper soon.

Bojielun and Evans learned that Peter Scholze had praised Xiao Yi’s paper on Far Abelian Geometry, which the journal accepted for publication. Xiao Yi received the news at home and told Liu Bin, who was stunned by the message.

• Bojielun learned that Peter Scholze had recognized the value of Xiao Yi’s paper and could use it in work on the abc conjecture.
• Evans confirmed Scholze’s assessment, and Bojielun instructed him to notify Xiao Yi and arrange online publication after the fee and contract were completed.
• Evans emailed Xiao Yi to congratulate him on the paper’s acceptance, explain the publication steps, and pass along Scholze’s request that Xiao Yi post future work on preprint websites.
• Xiao Yi read the email at home, realized Scholze had reviewed his paper, and sent a screenshot and message to Liu Bin.
• Liu Bin saw Xiao Yi’s message after returning from grading the joint mathematics examination and exclaimed in shock.

Hu Guangde celebrated Xiao Yi’s brilliance, then learned that Xiao Yi’s paper had been accepted by IHES. Xiao Yi paid the publication fee with funding from USTC High School and discovered that his joint-examination score was 101 out of 100.

• Hu Guangde learned that academicians had praised Xiao Yi’s paper and that Liao Huan had admitted Hu’s student was impressive.
• Hu Guangde received Liu Bin’s message that IHES had accepted Xiao Yi’s paper and reacted with shock.
• Liu Songlin and other USTC High School leaders visited Xiao Yi, thanked him for his achievement, and transferred 100,000 yuan to him.
• Xiao Yi paid IHES’s publication fee and signed the copyright transfer agreement.
• Xiao Yi checked the joint-examination results and saw that he had ranked first with a score of 101.

Xiao Yi’s perfect written-exam score and an extra point stunned the joint-exam candidates, who joked that he should act like a human being. He arrived at USTC’s interview, where Hu Guangde began by asking how he had slept.

• The candidates in the group chat struggled to explain how Xiao Yi earned 41 points on a 40-point question, and Luo Xiao said aloud that the graders must have awarded an extra point for his answer.
• Xiao Yi copied the group’s repeated message, prompting the candidates to condemn him for acting inhuman.
• Xiao Yi arrived early for USTC’s interview and heard other candidates describe the difficult questions they expected from Academician Hu.
• Hu Guangde and the other interviewers welcomed Xiao Yi, and Hu Guangde asked how he had slept the previous night.

Xiao Yi finished his brief interview and learned that his paper had been accepted by IHES and published online. News of the paper spread among the joint examination candidates, who discovered Xiao Yi was the Chinese mathematician whose work had inspired Peter Scholze.

• Hu Guangde ended Xiao Yi's interview after asking only whether he had slept well, and Xiao Yi left the examination room in under two minutes.
• Xiao Yi received an email confirming that IHES had accepted his paper and made it available online.
• Candidates in the joint examination group discussed Peter Scholze's upcoming conference and the paper that had reportedly inspired him.
• A blogger named Math and Physics Events identified Xiao Yi as the author of a newly published IHES paper and cited Peter Scholze's praise for it.
• A commenter shared a USTC High School post congratulating alumnus Xiao Yi, prompting the blogger to reply, "Huh?"

News that Xiao Yi, a seventeen-year-old USTC High School graduate and Hui Province’s top science student, had published a paper in IHES stunned students, professors, and the wider public. The story left the mathematics community buzzing as the news spread across social media and reached the trending searches.

• Netizens reacted with disbelief to USTC High School’s post about Xiao Yi’s IHES paper, then confirmed the announcement on the school’s official account.
• Luo Xiao shared the post in the Joint Examination Group, and Chen Lue and Lu Ping connected the paper’s author to USTC’s Xiao Yi.
• Professor Liu Min and other faculty members learned about the paper, and Tsinghua’s mathematical sciences faculty group erupted with questions.
• The news spread through China’s mathematics community and reached Weibo’s trending searches.
• Math and Physics Events confirmed with IHES that Xiao Yi had written the paper, noting that he was seventeen and had scored 741 on the Gaokao.

News of seventeen-year-old Yi Xiao’s paper spread from China to the international mathematics community, where Peter Scholze and Jacob Stix discussed how it had helped their work on Shinichi Mochizuki’s IUTT papers and the abc conjecture. After Gerd Faltings told them the paper was online and revealed that Yi Xiao had just finished the college entrance exam, Scholze and Stix were stunned.

• Yi Xiao’s paper became a top-ten Weibo trending topic, drawing astonished comments from mathematics students and ordinary netizens.
• At the Max Planck Institute for Mathematics, Peter Scholze explained ideas from Shinichi Mochizuki’s IUTT papers to Jacob Stix, and Stix agreed they had identified a key point.
• Scholze said they were close to proving the abc conjecture remained unproven and hoped to speak with the paper’s author, Yi Xiao, though he could not find that name in the University of Science and Technology of China listings.
• Gerd Faltings arrived and told Scholze and Stix that IHES had put Yi Xiao’s paper online.
• Scholze planned to invite the author to a mathematics conference, and Faltings revealed that Yi Xiao was seventeen, had just completed the college entrance exam, and would soon attend the University of Science and Technology of China.
• Scholze and Stix were stunned by Yi Xiao’s age and circumstances.

Peter Schultz, Gerd Faltings, and Jakob Stix discussed seventeen-year-old Xiao Yi’s mathematical paper and hoped to persuade Shinichi Mochizuki to leave his work on the abc conjecture. Schultz then wrote Xiao Yi an invitation to a Max Planck Institute mathematics conference on September 13.

• Faltings told Schultz that seventeen-year-old Xiao Yi had published a remarkable paper in IHES, astonishing Schultz.
• Schultz said he welcomed younger mathematicians who might surpass him, and Faltings hoped they would produce work he could discuss with Grothendieck.
• Faltings criticized his former student Shinichi Mochizuki’s work on the abc conjecture but hoped Schultz and Stix could convince him to return to Far Abelian Geometry.
• Schultz announced that the conference would take place on September 13 and decided to invite Xiao Yi.
• Schultz wrote Xiao Yi an invitation praising his paper and offering to reimburse all expenses for attending the conference.

After receiving Peter Scholze’s invitation to a mathematics conference about the abc conjecture, Xiao Yi traveled to USTC to register for the new semester. There, Professor Liu Bin showed him a letter from Scholze confirming the invitation, leaving the graduate students amazed.

• Xiao Yi read Peter Scholze’s email inviting him to a September 13 conference in Germany and decided to attend, despite the trip coinciding with freshman military training.
• Xiao Yi spent the following days studying the abc conjecture and Shinichi Mochizuki’s paper so he could follow the conference discussions.
• On September 4, Xiao Yi arrived at USTC to register and learned that Professor Liu Bin wanted to see him.
• The Mathematics Department upperclassmen recognized Xiao Yi as a scholar, told him his registration had been arranged, and asked him for autographs.
• Professor Liu showed Xiao Yi a letter from Peter Scholze inviting him to the conference, and the graduate students were astonished.

Xiao Yi accepted an invitation to a mathematics conference in Germany and discussed his studies with Liu Bin before settling into his dorm at USTC. When classes and military training began, his classmates learned he had taken leave to travel for the conference.

• Xiao Yi received Peter Schultz's invitation to the Max Planck Institute for Mathematics conference on the abc conjecture and Far Abelian Geometry, and agreed to attend.
• Liu Bin offered to arrange for Hu Guangde to accompany Xiao Yi to Germany and suggested Xiao Yi finish his undergraduate studies in one year.
• Xiao Yi chose to pursue graduate study at USTC rather than go abroad, explaining that he wanted to attend the conference because he was interested in the abc conjecture.
• Xiao Yi met his roommates Ye Cheng, Luo Qiao, and Song Ziyang, then returned home to study Shinichi Mochizuki's papers.
• When USTC's classes and military training began, the instructor told Xiao Yi's classmates that he had taken leave to get a visa and would leave for Germany on September 11.

Xiao Yi received his German visa and traveled with Hu Guangde to Bonn for the mathematical discussion. There, Shinichi Mochizuki claimed Xiao Yi’s paper had helped him complete his proof of the abc conjecture, while Peter Scholze remained unconvinced ahead of the next day’s debate.

• The visa officer approved Xiao Yi’s application and invited him to consider studying in Germany.
• Hu Guangde accompanied Xiao Yi on the trip to Frankfurt and Bonn, where they checked into a hotel near the Max Planck Institute for Mathematics.
• Shinichi Mochizuki praised Xiao Yi’s paper and asked him to become his student.
• Peter Scholze challenged Mochizuki’s approach and warned Xiao Yi against joining his group.
• Mochizuki said he had used Xiao Yi’s paper to make the final patch to his proof and expected to convince the mathematical community that he had proved the abc conjecture.
• Scholze maintained that the abc conjecture remained unproved and led Xiao Yi away before the discussion scheduled for the next day.

Xiao Yi attended the symposium on the abc conjecture at the Max Planck Institute for Mathematics, where he took a seat among leading mathematicians and reviewed doubts about Shinichi Mochizuki's proof. The symposium began with a presentation titled “Why abc is still a conjecture.”

• Xiao Yi and Hu Guangde arrived at the Max Planck Institute for Mathematics for Peter Scholze's symposium on the abc conjecture.
• A staff member recognized Xiao Yi's handwritten invitation and escorted him to a seat at the main table with leading mathematicians.
• Xiao Yi recognized Gerd Faltings, Jacob Stix, David Mathers, and Joseph Oesterlé among the attendees, then took a photo with their permission.
• Gerd Faltings praised Xiao Yi's paper and said he looked forward to his next paper.
• Xiao Yi read a packet of points questioning Shinichi Mochizuki's proof before Peter Scholze opened the symposium with a presentation titled “Why abc is still a conjecture.”

Peter Schultz challenged Shinichi Mochizuki’s proof of the abc conjecture at a conference, but Mochizuki answered nearly every objection with equations. As Schultz fell silent, Xiao Yi nervously raised a new question linking Far Abelian Geometry to the equivalence Mochizuki had described.

• Peter Schultz criticized Mochizuki’s proof and began presenting formal objections to it.
• Shinichi Mochizuki answered Schultz’s first objections by writing equations on the blackboard, resolving most of them to the satisfaction of those who understood the debate.
• After more than five hours, Schultz ran out of further questions, leaving the conference hall silent.
• Xiao Yi nervously asked Schultz and Mochizuki to reconsider an earlier issue, arguing that their assumptions about Hodge Theaters and Far Abelian Geometry might contradict Mochizuki’s theorem.

At a mathematical discussion, Xiao Yi suggested that Shinichi Mochizuki’s work in Far Abelian Geometry might contain a contradiction with his own Mochizuki Theorem. Schultz asked him to explain, and Xiao Yi began with Mochizuki’s Hodge Theater theory.

• Xiao Yi told the audience that Mochizuki’s explanations could lead to a contradiction within Far Abelian Geometry and conflict with the Mochizuki Theorem.
• Pierre Deligne and Enrico Bombieri watched Xiao Yi with interest as the audience considered his claim.
• Shinichi Mochizuki tried to answer Xiao Yi but found himself unable to respond immediately.
• Schultz asked Xiao Yi to explain his point in more detail, sensing it might break the deadlock.
• Xiao Yi accepted the microphone and began his explanation with Mochizuki’s Hodge Theater theory.

Xiao Yi used the Mochizuki Theorem to show that the conclusion of Hodge Theater required changing the theorem’s conditions, so IUTT theory could not hold. Shinichi Mochizuki accepted Xiao Yi’s argument, and the conference hall applauded.

• Xiao Yi explained the distinction between étale-like and Frobenius-like objects and connected the debate to his own conjecture.
• Xiao Yi asked to use the blackboard, wrote out his argument, and erased Schultz and Shinichi Mochizuki’s notes.
• Faltings and Shigefumi Mori realized that Xiao Yi’s reasoning used the Mochizuki Theorem to show that either it or IUTT theory had to be wrong.
• Xiao Yi concluded that the Mochizuki Theorem could not change, so the conclusion of Hodge Theater did not hold.
• Shinichi Mochizuki admitted that IUTT theory did not hold and told Xiao Yi, "You were correct."
• Faltings began applauding, and the conference hall joined him.

Xiao Yi received a standing ovation after identifying a flaw in Shinichi Mochizuki’s IUTT work and ending the eight-year dispute. Mochizuki acknowledged Xiao Yi’s insight and returned to his seat, leaving the conference to celebrate Xiao Yi’s achievement.

• The conference attendees applauded Xiao Yi, and Pierre Deligne and Enrico Bombieri praised his mathematical talent.
• Hu Guangde recalled Xiao Yi studying Mochizuki’s papers and realized that Xiao Yi had understood them well enough to find the flaw.
• Schultz congratulated Xiao Yi, who modestly credited luck and the questions raised during the discussion.
• Mochizuki left the blackboard and returned to his seat as the applause continued for Xiao Yi.
• Japanese mathematicians tried to comfort Mochizuki, but he stopped them and admitted Xiao Yi had found the flaw before he had.

After Shinichi Mochizuki apologized to the mathematical community under Professor Faltings’s supervision, Faltings thanked Xiao Yi for helping him and offered to grant him a request. Xiao Yi asked for advice on his Twin Prime Conjecture research, and Faltings invited him to discuss it and hinted that he could become his student.

• Professor Faltings spoke privately with Shinichi Mochizuki, who returned to the crowd and apologized for the trouble he had caused over the past eight years.
• Shinichi Mochizuki promised to reflect and improve his work under Professor Faltings’s supervision, and Bart Brunn and other mathematicians accepted his apology.
• Schultz said he hoped Shinichi Mochizuki would return to meaningful work, while Xiao Yi stayed silent.
• Professor Faltings thanked Xiao Yi for helping Shinichi Mochizuki and offered to fulfill a request.
• Xiao Yi asked Professor Faltings for advice on his Twin Prime Conjecture research, and Faltings invited him to discuss it and suggested he could ask to become his student.

At a mathematics conference, Deligne tried to recruit Xiao Yi to Princeton while Faltings argued with him, but Schultz led Xiao away before either could take him as a student. After the conference announced its mathematical news and hosted a banquet, Xiao Yi discovered that his social-media post about meeting Faltings had attracted widespread attention.

• Deligne and Bombieri interrupted Faltings as he tried to recruit Xiao Yi, and Deligne invited Xiao Yi to study at Princeton.
• Schultz pulled Xiao Yi away from the arguing mathematicians, believing Xiao Yi had no interest in becoming their student or studying abroad.
• Shinichi Mochizuki announced that the abc conjecture remained open, welcomed himself back to the mathematical community, and praised Xiao Yi as a young genius.
• At the banquet, Xiao Yi checked his phone and found his post about meeting Faltings had drawn many messages and comments, including an invitation from Professor Liu Min to attend Tsinghua for graduate school.
• Xiao Yi replied to his eldest uncle that Faltings was a distinguished mathematician, then saw Chen Muhua joking in the group chat that Faltings had locked him in a dark room until he agreed to become his student.

After the Max Planck Institute for Mathematics conference, Xiao Yi explained his whereabouts in a social media post and agreed to an interview with European News Television. News spread worldwide that he had disproved Shinichi Mochizuki’s IUTT theory and convinced Mochizuki himself.

• Xiao Yi told his group chat he would explain his conference photo in a social media post and teased that major news was coming.
• Xiao Yi posted that he had attended a conference in Germany, denied rumors that he had been confined, and asked people not to call him about upcoming news.
• Reporter Sophia approached Xiao Yi for an interview, and Hu Guangde encouraged him to accept.
• Bojielun learned that Shinichi Mochizuki had admitted his IUTT theory was wrong, then watched conference footage showing Xiao Yi presenting the decisive argument.
• Evans clarified to Bojielun that Xiao Yi, not Schultz, had convinced Mochizuki.
• European News Television released its interview footage of Xiao Yi, spreading the news of the conference result around the world.

After Xiao Yi’s challenge to Shinichi Mochizuki drew international attention, a televised interview showed Xiao Yi discussing mathematics and his approach to Far Abelian Geometry. Faltings, Deligne, and Bombieri praised Xiao Yi’s ability and said his work had freed the abc conjecture from controversy, leaving the problem open for further research.

• International media published headlines about Xiao Yi’s mathematical achievement as attention around him grew.
• Xiao Yi told a European News Television reporter that he admired Shinichi Mochizuki and saw mathematics as a shared search rather than a contest.
• Xiao Yi said he had understood Far Abelian Geometry by reading books, and viewers reacted with astonishment to the interview.
• Faltings praised Xiao Yi’s independent thinking and said he saw in him the potential to become a great mathematician.
• Deligne and Bombieri also praised Xiao Yi, with Bombieri saying his work had freed the abc conjecture from dispute and that the mathematical community should thank him.

Xiao Yi presented Faltings with his approach to improving bounds toward the Twin Prime Conjecture using Far Abelian Geometry and Bombieri's Theorem. After Faltings tested his reasoning with questions, he suggested that Xiao Yi also consider the Elliott-Halberstam Conjecture.

• Xiao Yi visited Faltings at the Max Planck Institute for Mathematics to seek advice on his research.
• Xiao Yi explained that he hoped to improve Bombieri's Theorem and strengthen existing results to lower the twin-prime upper bound of 246, possibly proving the Twin Prime Conjecture.
• Faltings questioned Xiao Yi about the Siegel-Walfisz hypothesis, and Xiao Yi explained how incomplete Kloosterman sum estimates addressed it.
• Faltings raised a boundary issue for S1 and then asked Xiao Yi more questions, which Xiao Yi answered with prepared reasoning.
• Faltings said he had little useful help to offer and advised Xiao Yi to consider the Elliott-Halberstam Conjecture.

Faltings suggested that Xiao Yi combine Far Abelian Geometry with Automorphic Forms to investigate the Elliott-Halberstam Conjecture. Xiao Yi filled the blackboard with a new approach, drawing the attention of three visiting Fields Medalists, and Bombieri asked what he planned to do next.

• Faltings proposed that Xiao Yi apply Far Abelian Geometry and Automorphic Forms to the Elliott-Halberstam Conjecture.
• Xiao Yi immediately began developing the idea on the blackboard while Faltings erased space for him.
• Schultz entered with Deligne and Bombieri, and Faltings urged them to stay quiet as Xiao Yi worked.
• Bombieri, Deligne, and Schultz watched Xiao Yi combine Far Abelian Geometry with Automorphic Forms and reacted with surprise.
• Xiao Yi stopped writing, and Bombieri asked him to explain his next line of thought.

Xiao Yi outlined how he might use Far Abelian Geometry, étale cohomology, and sieve methods to advance the Elliott-Halberstam conjecture, and the mathematicians praised his approach and recommended reading. The chapter ended with Faltings and Schultz agreeing to attend the Princeton Mathematical Conference if Xiao Yi went.

• Bombieri asked Xiao Yi how he planned to approach the Elliott-Halberstam conjecture, and Xiao Yi described combining Far Abelian Geometry with étale cohomology and sieve methods.
• Bombieri and the other Fields Medal winners praised Xiao Yi's ideas and discussed how success could make Far Abelian Geometry more widely studied.
• Xiao Yi asked the mathematicians to recommend books and papers, and Bombieri and Schultz urged him to read James Maynard's three papers raising the distribution level to 0.6.
• Bombieri said he would compile further reading materials for Xiao Yi, then Deligne asked Faltings whether he would attend the Princeton Mathematical Conference.
• Deligne invited Xiao Yi to present at the conference, and Faltings and Schultz said they would attend if Xiao Yi went.

Xiao Yi returned to China and was met by reporters before Hu Guangde and Bai Lu brought him to the Academy of Mathematics and Systems Science. He agreed to give a report on the discussion meeting and Shinichi Mochizuki’s theory, then ended the day at his hotel, remembering to email Professor Maynard.

• Xiao Yi and Hu Guangde arrived at Shangjing International Airport, where reporters and onlookers surrounded Xiao Yi.
• Bai Lu picked them up and explained that the Academy of Mathematics and Systems Science wanted Xiao Yi to give a report.
• Xiao Yi agreed to discuss the meeting on the thirteenth and the problems in Shinichi Mochizuki’s theory.
• Academician Xi Lan and the academy’s leaders welcomed Xiao Yi and Hu Guangde, then arranged a hotel room for Xiao Yi.
• Xiao Yi settled into the hotel after a long day and remembered that he needed to email Professor Maynard.