Chapter Summaries

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.

After returning home, Xiao Yi emailed James Maynard a detailed proposal connecting Far Abelian Geometry with Automorphic Forms and extending Maynard’s work on prime distribution. Maynard was impressed by Xiao Yi’s arguments and ended the chapter considering a trip to China.

• Xiao Yi finished reading Maynard’s three papers, compiled questions and ideas into a PDF, and emailed it to him.
• Maynard praised Richman for improving the level of distribution result from 0.6 to 0.617.
• Maynard opened Xiao Yi’s email and read his proposal to combine Far Abelian Geometry with Automorphic Forms, potentially raising the level of distribution beyond 0.7 and perhaps to 1.
• Maynard was shocked when Xiao Yi’s first section raised the value of θ to 0.65 in four pages.
• After reading the full document, Maynard stayed at his office contemplating Xiao Yi’s ideas and said he was wondering whether he should visit China.

Maynard told Richman that Xiao Yi had raised the analytic number theory bound to 0.65, prompting Maynard to study Xiao Yi’s work and consider traveling to China. The next morning, Xiao Yi resumed his studies in Shangjing and found an email from Maynard.

• Maynard postponed deciding whether to visit China and began preparing to study Xiao Yi’s attachment on Far Abelian Geometry and Automorphic Forms.
• Maynard scheduled Richman’s defense for a week later and explained that Xiao Yi had improved the θ value to 0.65.
• Richman and Maynard’s other students were stunned by Xiao Yi’s achievement, while Maynard wondered whether to begin studying Far Abelian Geometry.
• Xiao Yi woke early, continued his usual study routine, and noted that his report at the Academy of Mathematics and Systems Science was scheduled for September 26.
• Xiao Yi checked his inbox and saw that Professor Maynard had sent him an email.

After Professor Maynard introduced a mysterious functor connecting p-adic étale and crystalline cohomology, Xiao Yi saw how it could link his research to Far Abelian Geometry and began a major derivation. Meanwhile, news spread that he would give a report in Shangjing on September 26, and students from nearby universities learned they could attend.

• Xiao Yi read Professor Maynard's email and attachment about the Kuznetsov Trace Formula, p-adic Hodge theory, and the mysterious functor.
• Xiao Yi realized that the functor might let him connect p-adic étale cohomology to crystalline cohomology and Far Abelian Geometry through the Frobenius homomorphism.
• Xiao Yi began a lengthy derivation that he hoped would resolve the problems in his research.
• Students in a joint mathematics class group circulated a poster announcing Xiao Yi's September 26 report after his return to China.
• Chen Muhua announced that students from Peking University, Tsinghua, and Beihang could sign up to attend the report, prompting jealousy from students at other universities.

As students packed the hall for Xiao Yi’s upcoming lecture, Xi Lan and Hu Guangde found him asleep at his desk after working through the night. Xiao Yi said he had finished preparing and planned to discuss new material, leaving the academicians unsure how much he intended to cover.

• Students from universities in Shangjing filled the reservations for Xiao Yi’s lecture, prompting the Academy of Mathematics and Systems Science to move it to a larger hall.
• On September 25, Xi Lan and Hu Guangde went to Xiao Yi’s hotel room to discuss the lecture and found he was not answering the door or his phone.
• A hotel attendant opened the room, and Xi Lan and Hu Guangde found Xiao Yi asleep over draft papers covered in mathematical derivations.
• Hu Guangde woke Xiao Yi, who apologized and explained that he had been researching all night and fallen asleep at his desk.
• When Xi Lan asked about the lecture, Xiao Yi said he had prepared and planned to discuss new things beyond what he had presented at the seminar.

After solving the problems he had been researching, Xiao Yi emailed James Maynard with the news, then went to sleep. Maynard eagerly tried to discover which problems Xiao Yi had solved, while Xiao Yi’s lecture on ABC Conjecture, Far Abelian Geometry, and Automorphic Forms was set to begin in China.

• Xiao Yi organized his research papers, emailed James Maynard that he had solved all the problems, and went to bed.
• James Maynard attended Richman’s doctoral defense and wondered why Xiao Yi had not replied to his email.
• Maynard found Xiao Yi’s message but could not tell which problems Xiao Yi had solved, so he searched online for information about him.
• Maynard learned that Xiao Yi would give a lecture in China the next day and called a Chinese professor to ask about watching the livestream.
• On September 26, young people gathered at the Chinese Academy of Sciences Basic Science Park to see the poster announcing Xiao Yi’s lecture on ABC Conjecture, Far Abelian Geometry, and Automorphic Forms.