Graduate Mathematics

An equation without a formula

Here is an equation of degree five. x to the fifth, minus four x, plus two, equals zero. It has five solutions, and a computer gives them in an instant. And yet no formula will ever write them with square roots, cube roots or fifth roots. Not because nobody has found it yet. Because it cannot exist. Articulated Course in Mathematics. Volume three: from functional analysis to Galois theory.

Who it is for

The third volume makes a promise: to move from tools to theories, and to change perspective without losing precision. It is for anyone taking on master’s-level mathematics. It assumes a working command of the analysis and algebra of volume two. And it names the real change. At the undergraduate level, the path was marked: definition, theorem, proof, exercise. At the graduate level, you must often find the path yourself. That change is not about theorems: it is about the way of seeing. Volumes four and five both start from here.

The map of the volume

Here is the map. Twenty-five parts. Eighty-six chapters. Four hundred and fifty-five articles, some two thousand six hundred pages. Analysis first: function spaces, measure, Fourier, holomorphic functions. Chance: probability, processes, statistics. Equations: differential, partial differential, dynamical systems. Algebra and geometry: groups, fields, Galois, manifolds, algebraic geometry. And computation: numerical analysis and simulation.

Four ideas

Four ideas hold the whole together. The first: changing space. Finite dimension is no longer enough, and functions become points. Stir a cup of coffee: at least one point of the liquid comes back to where it was. That is a fixed-point theorem, Brouwer’s. The second: giving a meaning to what had none. The Lebesgue integral, and distributions, which differentiate functions that have no derivative. The path of Brownian motion is continuous everywhere and differentiable nowhere: however far you zoom in, it never straightens out. The third: seeing hidden symmetries. Galois theory reads in an equation the group of its symmetries. The fourth: curved spaces. Manifolds replace the plane and space with curved spaces that can only be observed locally, chart by chart. And beneath all four runs a thread. The same gesture, the quotient, creates the integers modulo n in arithmetic, the quotient topology in geometry, and the L p spaces in analysis. At this level, the great themes of the collection are no longer intuitions: they are tools of proof.

Why no formula?

Back to our equation. For degrees two, three and four, formulas exist: Cardano’s and Ferrari’s. For degree five, Abel shows in eighteen twenty-four that the general equation holds out. Galois says why, and for each equation. Let’s draw the curve. It crosses the axis three times: three real roots. The other two are complex, and conjugate. The Galois group gathers the permutations of the roots that algebra cannot tell apart. Conjugation swaps the two complex roots and fixes the other three: a transposition. The polynomial is irreducible, so the group also contains a cycle of length five. A transposition and such a cycle are enough: you get all one hundred and twenty permutations. The group is S five, all of it. But a formula in radicals would require a solvable group, one that can be taken apart in abelian layers. S five is not. So the formula is impossible, for this very equation. The roots exist: we can compute them to any precision. They only escape every formula. As the book puts it, the limit is not in the numbers, but in the language.

Working with the volume

How do you work with this volume? With the same three kinds of exercises, all with solutions. And with two parts written for the reader: reading and working through a master’s course, then three chapters of transition exercises. A practice exercise, in chapter fifty-two: prove that an angle of sixty degrees cannot be trisected with ruler and compass. The cosine of twenty degrees has degree three, and three is not a power of two. A study teaches the criterion that settles the problems of antiquity: a constructible number has degree a power of two. And an expedition retraces the path of Gauss, who showed at eighteen that the regular seventeen-sided polygon can be constructed. Alongside the volume, the three companions. The abridged course, to learn or relearn. The compendium, to review. The formula handbook, to check. At one article a day, the volume takes about fifteen months.

What makes the difference

What makes the difference comes down to a few rules. One article, one idea. Echoes from one field to another, pointed out: you move from knowing the pieces to seeing the building. Every proof has a visible status: complete, sketched or admitted. And the volume exists in French and in English. “Reading an article, following a seminar, identifying the right question: these skills, harder to teach than theorems, are what make a mathematician.”

To remember

To remember. Volume three: master’s-level mathematics. Twenty-five parts, four hundred and fifty-five articles, every exercise with its solution. And an equation whose roots exist, but escape every formula. Volume three opens for free: the opening pages, the table of contents and all of chapter one. The link is in the description.