Ellenberg, Jordan

mathematician, University of Wisconsin; author of How Not to Be Wrong and Shape

Jordan Ellenberg: Mathematics of High-Dimensional Shapes and Geometries

Lex Fridman Podcast · Jun 13, 2021

• Jordan Ellenberg
• symmetry and group theory
• topology
• Poincaré conjecture
• Fermat's Last Theorem
• Andrew Wiles
• deformation theory
• twin prime conjecture
• John Conway
• Game of Life
• p-adic numbers
• math communication
• Perelman and the Fields Medal

Core ideas

Ellenberg frames modern mathematics as centrally concerned with the question of "when do we consider two things to be the same" — a question of symmetry in the broadest sense, not just the everyday left-right symmetry of a face, but any well-defined transformation (rotation, stretching, permutation) under which some property is preserved. He traces this to a personal childhood epiphany, staring at a 6-by-8 grid of stereo-speaker holes and realizing that counting them as "6 rows of 8" versus "8 columns of 6" gives the same number for a reason — the geometric fact of the rectangle underlying the algebraic fact that multiplication order doesn't matter. He extends this idea to group theory (the abstract study of collections of symmetries that can be combined, from a two-symmetry human face to the vast permutation group of a shuffled card deck) and connects it to open problems in AI: recognizing handwritten digits, he argues, clearly involves something beyond classical symmetries like rotation, since no simple, well-understood group of transformations captures what makes a "2" different from a "3" — a gap between formal mathematical symmetry and the more ad hoc, opaque "sameness" judgments performed by neural networks.

Much of the conversation traces the influence of Henri Poincaré, whom Ellenberg calls a pioneer of topology (which Poincaré himself called "analysis situs"), and who first showed that understanding physical systems like the three-body problem requires reasoning about abstract higher-dimensional "phase spaces" (position plus velocity) rather than just three-dimensional physical space — a move Ellenberg treats as paradigmatic of how mathematics gets "dragged" toward higher abstraction by concrete physical problems, later vindicated when Einstein's relativity made four-dimensional spacetime physically real. Ellenberg explains the Poincaré conjecture (about which three-dimensional spaces are topologically the "standard" one, tested via loops that can or cannot be shrunk to a point) and its eventual proof by Grigori Perelman using Richard Hamilton's Ricci flow technique — a method that, in a pattern Ellenberg says recurs constantly in geometry, resolves a question about one space by studying the higher-order "space of all such spaces." He discusses at length the philosophical puzzle of "how many holes does a straw have," using it to introduce homology — the modern topological insight (developed by Poincaré, Betti, and later formalized by Emmy Noether) that holes have an arithmetic, illustrated by the equation "waist = leg + leg" for a pair of pants.

On Fermat's Last Theorem, Ellenberg argues the popular "lost elegant proof" romance is almost certainly false — Fermat's own later writings suggest he never had a full proof — but that the value of the problem lay not in a hoped-for short proof but in the entire body of number theory it generated over centuries, including Kummer's 19th-century realization (while diagnosing a flawed proof by Lamé) that unique prime factorization, which holds for ordinary integers, breaks down in more general number systems relevant to the problem — a "failure" Ellenberg frames as generative rather than disappointing. He describes Andrew Wiles's eventual proof (with Richard Taylor, building on work by Mazur, Ribet, Frey, and others) as resting on deformation theory: understanding how mathematical objects can be perturbed using an exotic, non-intuitive notion of "distance" (the p-adic metric, where numbers are "close" if their difference is divisible by a large power of a prime) to piece together local information into a global structural argument.

On the practice and culture of mathematics, Ellenberg discusses prime numbers and the twin prime conjecture, noting the counterintuitive but productive habit among number theorists of treating primes — which are fully deterministic — as if they were randomly distributed, because this heuristic reliably predicts true statements even though "no chance is actually involved." He praises John Conway (his colleague at Princeton) as an exemplar of playful mathematics inseparable from personality, discussing Conway's invention of the Game of Life and surreal numbers, and notes Conway was ambivalent about being best known publicly for the comparatively "simple" Game of Life rather than his deeper theorems. He reflects on math communication (praising YouTube creators like Grant Sanderson/3Blue1Brown and the Soviet tradition of treating mathematical and scientific excellence with the reverence Americans reserve for athletes), and on Grigori Perelman's refusal of the Fields Medal for the Poincaré proof, which he declines to fully explain (citing his own uncertainty about Perelman's actual reasoning) but uses to reflect more generally on occasions when following one's principles requires forgoing external validation.

Ellenberg closes on more personal reflections: advice to give the "high self-esteem" choice — acting as a confident, undoubting version of oneself would act, even while still privately holding doubts — and a discussion of Blaise Pascal, noting that despite being famous for "Pascal's Wager," Pascal's own account of his religious belief rested not on a mathematical argument but on a direct mystical experience, which Ellenberg uses to gently sidestep the question of whether mathematics can address the meaning of life.

Excerpts