Contemporary Mathematics — Perelman, Mochizuki, Tao, and the Millennium Problems

21st-Century Mathematics — Three Extreme Lives

The final article of the Lineage of Mathematics series follows three representative figures of 21st-century mathematics. Each has made an extreme life choice:

• Grigori Perelman (1966–) — Russia; proved the Poincaré conjecture and declined both the Fields Medal and the million-dollar prize, then withdrew
• Mochizuki Shinichi (1969–) — Japan; claims to have proved the ABC conjecture in 500 pages of his own theory, but international consensus is still unreached
• Terence Tao (1975–) — Australia; the “modern von Neumann”, a universal mathematician driving AI and mathematics

The “solitary work” lineage of the previous article (Turing and Wiles) appears in more dramatic forms in the 21st century. And 2500 years of history close here.

Grigori Perelman — The Man Who Solved a Century-Old Problem and Vanished

Grigori Yakovlevich Perelman (Григо́рий Я́ковлевич Перельма́н, 1966–) was born in Leningrad (now Saint Petersburg) into a Jewish family. He proved the Poincaré conjecture — open for 100 years — in 2002–2003.

He declined both the 2006 Fields Medal and the 2010 Clay Millennium Prize of one million dollars, and today reportedly lives quietly in Saint Petersburg with his mother. He strongly disliked “media attention” and “unfairness in the mathematical community,” and has retreated entirely from public life.

Poincaré Conjecture (1904) and Ricci Flow

The Poincaré conjecture from an earlier article states:

“Is every simply connected closed 3-manifold homeomorphic to the 3-sphere S³?”

A geometric question, for 100 years attacked from a topological angle without essential progress.

• Dimension ≥ 5 — Smale (1961), Fields Medal
• Dimension 4 — Freedman (1982), Fields Medal
• Dimension 3 — the only remaining case

Perelman used Ricci flow, an analytic method that evolves the metric of a manifold according to its curvature tensor. Started by Richard Hamilton in the 1980s, Ricci flow had faced deep technical and conceptual obstacles; Perelman overcame them completely and resolved the stronger Thurston geometrization conjecture as well.

2002–2003 — Three Preprints

Between November 2002 and July 2003, Perelman posted three preprints to arXiv:

1. “The entropy formula for the Ricci flow and its geometric applications” (2002-11)
2. “Ricci flow with surgery on three-manifolds” (2003-3)
3. “Finite extinction time for the solutions to the Ricci flow on certain three-manifolds” (2003-7)

No journal submission — arXiv only. The papers are extremely condensed, and filling in details required long expositions.

Three independent groups each verified the argument:

• Cao and Zhu
• Kleiner and Lott
• Morgan and Tian

All confirmed Perelman’s argument was correct. A pioneering case of 21st-century open science.

Declining the Fields Medal (2006)

At the 2006 International Congress of Mathematicians in Madrid, the Fields Medal was announced for Perelman. He became the first person to decline (no prior refusal exists).

His reason:

“The prize has nothing to do with whether my proof is correct. If the proof is correct, no additional recognition is needed.”

• Disappointment at unfairness in the community (notably the Cao–Zhu paper presenting itself as having “completed” his proof)
• Aversion to excessive media attention

Declining the Million Dollars (2010)

The Clay Mathematics Institute set seven Millennium Prize Problems in 2000, each with a million-dollar reward:

1. The Riemann hypothesis
2. The Poincaré conjecture ← the only one solved
3. Yang–Mills theory and mass gap
4. Navier–Stokes equations
5. The Hodge conjecture
6. The Birch and Swinnerton-Dyer conjecture
7. P vs NP

In 2010 the prize was awarded to Perelman for the Poincaré proof. He declined that as well:

“My contribution is at most equal to Hamilton’s. It would be unfair for me alone to accept the reward that should go to him.”

He chose “mathematical justice over a million dollars.” An unprecedented choice in the history of mathematics.

Perelman’s Character

• Plays a string instrument; loves literature and philosophy deeply
• Open to mathematical discussion but absolutely refuses media interviews
• Simple diet — rye bread, yogurt
• Russian and Jewish, lived through the end of the USSR and the Russian collapse
• Close bond with his mother Lyuba, still lives with her
• 1992–1995 in the United States (NYU, SUNY, Berkeley) where he learned Ricci flow, then returned to Russia. He chose not to stay in the US

Asked “Have you given up mathematics?”, Perelman replied:

“I have not given up mathematics. I have only been disappointed in the community.”

Mochizuki Shinichi — 500 Pages of “Inter-Universal” Theory

Mochizuki Shinichi (望月新一, 1969–) is a Tokyo-born mathematician, professor at the Research Institute for Mathematical Sciences (RIMS), Kyoto University. He constructed Inter-universal Teichmüller theory (IUTT) and announced a proof of the ABC conjecture through it in 2012.

The proof rests on his own extremely difficult 500-page theory, and the world’s arithmetic-geometry community has not fully been able to verify it. In 2020 it was peer-reviewed and published in the Kyoto journal PRIMS, but with Peter Scholze (Fields 2018) and others voicing strong dissent, as of 2026 international consensus has not been reached.

The ABC Conjecture — “Number Theory’s Universal Theorem”

The ABC conjecture (Oesterlé–Masser, 1985) concerns integers a + b = c (coprime) and their radical (product of distinct primes):

For every ε > 0, there is a constant K_ε such that c ≤ K_ε · rad(abc)^(1+ε).

Intuitively: “a + b = c with only small prime factors (like a, b, c being high prime powers) rarely happens.”

Consequences

If true, in one line it implies enormous swaths of 20th-century number theory:

• Fermat’s Last Theorem (large exponents)
• The Catalan conjecture (consecutive powers)
• An effective version of Mordell’s conjecture (finiteness of rational points on curves)
• Infinitude of Wieferich primes

Nicknamed “number theory’s universal theorem.” Resolving it would sweep away a great deal of 20th-century number theory at once.

Mochizuki’s Career

• 1969 — Born in Tokyo; his father was an economist
• 1985 — Moves to the US at 16, enters Phillips Exeter Academy
• 1992 — PhD from Princeton at 23 (advisor: Faltings)
• 1994 — Assistant professor at RIMS; later full professor
• 1990s — Constructs Hodge–Arakelov theory
• 1996–2000 — Major work in anabelian geometry (special cases of Grothendieck’s conjecture)
• 2000s — Constructs inter-universal Teichmüller theory alone
• August 2012 — Posts four IUTT papers on the RIMS website
• April 2020 — Published in PRIMS after peer review

He is an heir to the algebraic-geometric tradition of Faltings, Drinfeld, and Grothendieck, but known for a bold departure from it in style.

Inter-Universal Teichmüller Theory

IUTT has these features:

1. Multiple mathematical “universes” (Galois categories) are considered in parallel — different systems run alongside each other
2. Theta link — a symmetry-broken correspondence between two universes
3. Separation of addition and multiplication — the additive and multiplicative structures, ordinarily fused in algebra, are separated and reconstructed
4. Frobenioid — an algebraic structure distinct from rings and groups, of Mochizuki’s design

Then the ABC inequality is obtained as a “size” comparison between two universes via the theta link — that is the outline.

The Verification Dispute — Scholze in Kyoto (2018)

In 2018, Peter Scholze (then 25 and a leading Fields candidate) and Jakob Stix visited Kyoto and debated directly with Mochizuki for five days. They produced a report claiming that IUTT’s “Corollary 3.12” (the central inequality) has a fatal gap — the memo Why abc is still a conjecture.

Mochizuki replied that “they misunderstand IUTT,” and released long responses. The paper was published in 2020 after Kyoto’s peer review, but Scholze, Kiran Kedlaya, Brian Conrad, and other principal members of the international arithmetic community still do not accept the proof.

This is an unusual event in modern mathematics, discussed as a case of “a paper that peer review cannot close,” “linguistic and cultural distance,” and “the sociology of the mathematical community.”

Katō Fumiharu’s Mathematics Connecting Universes

In Japanese, Katō Fumiharu (Tokyo Tech → Univ. of Tokyo) played a decisive role in explanation with [Uchū to Uchū o Tsunagu Sūgaku] (KADOKAWA, 2019), a fine popular introduction to the essence of IUTT.

Though sometimes labeled a “Japan-local” matter, Tamagawa Akio, Hoshi Yuichiro, Yamashita Go, Yamashita Tsuyoshi and other mathematicians are said to fully understand IUTT and are located across the world. A rare case of the sociology of contemporary science: “mathematical universes with different languages.”

Terence Tao — “The Modern von Neumann”

Terence Chi-Shen Tao (陶哲軒, 1975–) is a Hong Kong–Australian mathematician, born in Adelaide.

• 10 — Bronze at IMO
• 11 — Silver at IMO
• 12 — Gold at IMO (a record held until 2023)
• 14 — Bachelor’s degree
• 16 — Master’s degree
• 17 — Enters Princeton PhD (advisor: Stein)
• 21 — PhD
• 24 — Full professor at UCLA
• 31 — Fields Medal (2006)

His first-rate work spans harmonic analysis, additive combinatorics, PDEs, probability, analytic number theory, random matrices, and more. Called “the universal mathematician of our time” and a universalist unseen since von Neumann.

Tao’s Style — “Translation”

Tao’s style is to freely combine methods from multiple fields. In his own words:

“Mathematical maturity is not deep immersion in one area — it is the ability to translate between multiple areas.”

Since Poincaré (the last universal mathematician) of the earlier article, he is the first “surveyor of all territories” to appear in 100 years.

The Green–Tao Theorem (2004)

In the paper with Ben Green, “The primes contain arbitrarily long arithmetic progressions,” they proved:

“The primes contain arithmetic progressions of every length.”

Van der Corput had done the length-3 case in 1939, and no progress had been made for decades.

The core of the proof:

• Uses Szemerédi’s theorem (a set of integers with positive density contains arbitrarily long arithmetic progressions), via Furstenberg’s ergodic-theoretic proof
• Approximates the primes as an “almost random” subset (pseudorandomness of primes)
• Uses the Goldston–Yıldırım sieve

A canonical example of integrating multiple fields (combinatorics, analysis, ergodic theory, number theory). It earned Tao the 2006 Fields Medal.

Tao’s Work

Result	Content	Coauthors
Green–Tao theorem (2004)	Arbitrarily long arithmetic progressions in the primes	Ben Green
Kakeya and restriction	Central problems of harmonic analysis	Alone / Bourgain and others
Navier–Stokes	Possibility of finite-time blow-up	Alone (2016)
Compressed sensing	Recovery of sparse signals	E. Candès
Random matrices	Universality; eigenvalue distributions	Van Vu and others
Horn’s conjecture	Eigenvalues of Hermitian matrices	Knutson and others

Compressed sensing underlies modern MRI acquisition, signal processing, and feature selection in machine learning.

Tao’s Blog What’s New

Since 2007 Tao has run his blog What’s New (terrytao.wordpress.com), sharing research progress, educational explanations, and discussion of the mathematical community. Over 1000 posts in total — a rare resource that communicates modern mathematics at the highest level, openly.

His textbooks Analysis I, II are used worldwide as rigorous introductions to real analysis.

“A great mathematician talking about contemporary mathematics in a public blog” — a new 21st-century mode of scholarly communication. A modern counterpart to Euler’s Letters to a German Princess from the earlier article.

AI and Mathematics (2023–2026) — Tao’s New Direction

From 2023, Tao has actively participated in the frontier of formal theorem-proving with Lean, combined with LLMs (Claude, ChatGPT, GitHub Copilot):

• 2023 — Leads a project to formalize the PFR conjecture (Polynomial Freiman-Ruzsa) in Lean
• 2024 — Cooperates with many mathematicians to trial “AI-assisted proof” workflows
• 2025–2026 — Advocate of a model of human–AI collaboration in mathematical research

Tao has said:

“AI will not replace mathematicians, but it will fundamentally change how they work.”

His views strongly shape the direction of the mathematical community. Turing’s Turing Test prediction from the previous article is coming true on the ground of 21st-century mathematical research.

The Millennium Problems Today (2026)

The Clay Institute’s seven Millennium Problems (set 2000, million-dollar reward each):

Problem	Status	Related article
1. Riemann hypothesis	Open	Euler’s zeta function
2. Poincaré conjecture	Solved by Perelman in 2003 ✓	This article
3. Yang–Mills and mass gap	Open	—
4. Navier–Stokes	Open; Tao has partial results	This article
5. Hodge conjecture	Open	—
6. Birch and Swinnerton-Dyer	Open	—
7. P vs NP	Open	—

One problem solved in 26 years — evidence that frontier mathematics is still with us. Mathematics is not “a finished discipline.” Room to discover remains, well into the 21st century.

Features of 21st-Century Mathematics

Looking back over the series, the defining features of 21st-century mathematics:

• Hyper-specialization — mastering one field takes 20+ years
• International collaboration — arXiv, online reviewing, worldwide conferences
• Cooperation with AI — Lean, LLMs, automated proof
• Fusion with computers — the Four-Color Theorem (1976), Kepler’s conjecture (1998), the PFR conjecture (2023)
• “Universes of language” — theories like Mochizuki’s that are hard to share even inside one community
• Anonymous geniuses — Perelman’s choice to withdraw completely from the media
• Diversity — more women, Asian, and non-Western mathematicians

2500 Years — Summary

The Lineage of Mathematics series has followed 2500 years, from Pythagoras in the 6th century BC to Tao in 2026:

• 6th–3rd c. BC — Ancient Greece — Pythagoras, Euclid, Archimedes; mathematics as demonstration
• 3rd–16th c. — Middle Ages and Early Modern — Diophantus, Arabic mathematics, Fibonacci, Seki; non-European mathematics
• 17th c. — First half of the Scientific Revolution — Descartes, Fermat; coordinate geometry and 358-year homework
• Late 17th c. — Calculus — Newton, Leibniz; 30-year priority dispute
• 18–19th c. — Giants — Euler, Gauss; half of all mathematics and the prince of mathematicians
• Early 19th c. — Rigor — Cauchy, Abel, Jacobi; ε-δ and elliptic functions
• 19th c. — Tragic geniuses — Galois, Ramanujan; the duel at 20 and the self-taught mystic
• Early 20th c. — Foundations — Hilbert, Poincaré, Gödel; the 23 problems and incompleteness
• 19–20th c. — Women — Germain, Noether; breaking institutional walls
• 20th c. — Solitary work — Turing, Wiles; computer science and Fermat’s Last Theorem
• 21st c. — Contemporary — Perelman, Mochizuki, Tao; declining a million dollars, inter-universal theory, AI and mathematics

Starting from “counting,” humanity invented “infinity,” “continuity,” “structure,” and “proof,” and thereby learned new ways of grasping the world. From Pythagoras to Tao, mathematicians have been explorers of an abstract country.

12 Scenes from the Drama of People and Discoveries

The 12 scenes we have traced:

1. 6th c. BC — Pythagoras and the discovery that √2 is irrational
2. 3rd c. BC — Euclid’s Elements axiomatizes geometry
3. 1637 — Fermat’s “the margin is too narrow”
4. 1665–66 — Newton’s “annus mirabilis”
5. 1801 — Gauss, at 24, publishes the Disquisitiones
6. 1832 — Galois, at 20, writing on the eve of his duel
7. 1913 — Ramanujan’s letter to Hardy
8. 1931 — Gödel’s incompleteness theorems at 25
9. 1994 — Wiles proves Fermat’s Last Theorem
10. 2003 — Perelman posts three preprints on arXiv; the Poincaré conjecture falls
11. 2012 — Mochizuki’s IUTT / ABC papers appear
12. 2023– — Tao opens the future of mathematics with Lean and LLMs

Mathematics Is Not Finished

Room to discover remains, well into the 21st century. Six of the Millennium Problems are open. Consensus on IUTT is not yet reached. Cooperation with AI has only just begun.

After Poincaré (1854–1912), “the last mathematician who could survey the whole of mathematics,” the field has walked the path of specialization and diffusion. But 21st-century exceptions like Tao, and new forms of collaboration with AI, are once again making a whole-field view possible.

From Pythagoras 2500 years ago to contemporary mathematicians in 2026 — mathematics, the discipline in which discovery continues indefinitely, continues quietly on.

The Series Ends

This series has followed 2500 years of mathematical history in overview + 11 articles = 12 contents.

For readers:

• Introductions — Takagi Teiji’s Kinsei Sūgakushi-dan; E. T. Bell’s Men of Mathematics
• Films — A Beautiful Mind, The Imitation Game, The Man Who Knew Infinity
• Simon Singh — Fermat’s Last Theorem, The Code Book, Big Bang
• Tao’s blog — terrytao.wordpress.com — direct communication from a world-class mathematician
• Online courses — MIT OpenCourseWare, Coursera, mathematical YouTube channels

“Spending a week understanding one theorem” — the most luxurious way to enjoy mathematics. A quiet pleasure known to every mathematician from Pythagoras to Tao.

Theorems 2500 years old, and open problems today, stand open in front of us as the same drama of people and discoveries.

The Lineage of Mathematics series ends here. A quiet epic of numbers and logic, over 2500 years, comes to its close.