Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century – Masha Gessen

Thoughts: Perfect Rigor is a biography of Grigory Perelman, the Russian mathematician who solved the Poincaré Conjecture, as well as an outline of the events surrounding his coming up with the proof and its release. The title is a bit unfortunate, and I wouldn't be surprised if it was the publisher's choice rather than the author's – Masha Gessen doesn't lean particularly hard on the ideas of "rigor" or "genius", and even the "mathematical breakthrough of the century" only occupies the last few chapters. I learned a bit about the culture of professional mathematics, and about antisemitism in the Soviet Union (students being denied opportunities and academics being denied positions because they were Jewish was a recurring theme throughout the book), but didn’t come away from the book with any grand insights. Nevertheless, I found Gessen’s writing engaging; those with an interest in contemporary mathematics would likely enjoy this book.

(The notes below are not a summary of the book, but rather raw notes - whatever I thought, at the time, might be worth remembering.)

Gessen, Masha. 2009. Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century. Houghton Mifflin Harcourt.

Prologue

• viii: To look up/learn more about: The Clay Institute’s seven Millennium Problems (of which the Poincaré Conjecture, solved by Grigory Perelman, is one)
• ix: a program: a plan that a mathematician makes for solving a problem.

1. Escape into the Imagination

• 3: “It stands to reason that the Soviet human rights movement was founded by a mathematician” (i.e. logic theorist Alexander Yesenin-Volpin) - mathematicians have a tendency to think in terms of absolute rules and their implications. Yesenin-Volpin and others simply “called on the Soviet authorities to obey the country’s written law. In other words, they demanded logic and consistency.”

2. How to Make a Mathematician

• 25: To learn about: Ramsey theory, used to solve problems such as how many people must be invited to a party such that at least m of them will know each other OR n of them will not know each other.
• 26: Sergei Rukshin, who founded very successful math camps/clubs, identified one of the secrets of his success: " Every child must be heard out on every problem he thinks he has solved." - labor intensive for both instructors and students, but forces students to learn how to communicate their conclusions.

3. A Beautiful School

• 38: The Dalton School / Dalton Plan. Founded in the first decades of the 20th Century, and served as a model for a math clubs Andrey Kolmogorov taught at, “called for an individual instruction plan for every student. Each child would map out his own path for the month and proceed to work independently.”

4. A Perfect Score

5. Rules for Adulthood

6. Guardian Angels

7. Round Trip

8. The Problem

• 131: Henri Poincaré: “How is it that mathematics is not reduced to a gigantic tautology?” “Are we then to admit that… all the theorems with which so many volumes are filled are only indirect ways of saying that A is A?”
• 135: to learn more about Riemannian (a.k.a. elliptic) geometry, in which all straight lines are geodesics, eventually looping back on themselves. All geodesics will, at some point, intersect with each other.
• 139: a hypersurface (topology): an object that has as many dimensions as possible in a given space. E.g. a sphere in 3D space (topologically speaking, it has two dimensions - any point on the sphere can be described using two coordinates). Poincaré was interested in the 3D surface of a 4D ball in 4 dimensions.
• 141-142: “Experts say there is one living man, the American geometer William Thurston, who can imagine four dimensions. Thurston, they say, is posessed of a geometric intuition unlike that of any other human. ‘When you see him or talk to him, he is often staring out into space and you can see that he sees these pictures,’ said John Morgan…”

9. The Proof Emerges

10. The Madness

• 175-176: The research of Simon Baron-Cohen suggests that people who have autism / Asperger’s syndrome have a tendency to systematize, “to analyze and/or build a system (of any kind) based on identifying input-operation-output rules”. Tested Cambridge undergrads: mathematicians were 3-7x more likely to have been diagnosed with autism. Similar correlation found in a study at participants in the British Mathematical Olympiad.

11. The Million-Dollar Question

• 200: book mentioned: “Courant and Robbins’s classic What is Mathematics?” - helped inspire Jim Carlson, eventually the head of the Clay Mathematics Institute, to turn from psychology and physics to mathematics during his studies.