BOOK REVIEWA SingularMathematical Promenade
Reviewed by Rafe Jones
A Singular Mathematical
Promenade
By Étienne Ghys

After fifteen seconds with Étienne
Ghys’s A Singular Mathematical
Promenade, I knew that it was unlike any other book I had read. My
first casual flip-through revealed akaleidoscope of photos, drawings,
and diagrams on nearly every page,and yet this visual brilliance occurred under chapter headings containing mathematicalphrases from the deep trenches of graduate-level mathematics, like “singular operads” and “resolution of singularities.” I saw immediately the scale of the book’s ambition:
to appeal to almost everyone in the mathematical worldfrom the undergraduate level up while still presenting arich tapestry of advanced mathematical ideas. In the end,
it comes remarkably close to achieving this lofty vision.
The journey begins with polynomials. Stuck in a particularly boring meeting, the author’s colleague Maxim
Kontsevich passes him a Parisian metro ticket with a scribble and the single word “impossible!” (see Figure 1). Aftersome whispering, the author figures out that Kontsevich has
shown that there cannot be four real polynomials satisfying
P1(x) < P2(x) < P3(x) < P4(x) for small x < 0 and P2(x) < P4(x)< P1(x) < P3(x) for small x > 0. Yet there are no such “forbidden permutations” for three polynomials. Surprised byRafe Jones is professor of mathematics at Carleton College. His email address
is rfjones@carleton.edu.
Communicated by Notices Book Review Editor Stephan Ramon Garcia.
For permission to reprint this article, please contact: reprint-permission@ams.org.
DOI: https://dx.doi.org/10.1090/noti1987December 2019

this constraint on the relative positions of the graphs of four
real polynomials, the author tries to understand all such
constraints, then to place the problem in the more generalcontext of singularities of real algebraic plane curves, thatis, curves defined implicitly by equations of the form F(x,y)= 0 for some real polynomial F. See Figure 2 for an example
studied by Newton, F(x,y) = y6−5xy5+x3y4−7x2y2+6x3+x4.
The singularities of such curves have a rich structure,and if the promenade has a goal, it is to understand thisstructure. On the way, many mathematical vistas open,
some that will be familiar to nearly all readers, and others
that will be familiar to nearly none. The author warns usthat we will get lost fromtime to time but gives
these words of encouragement: “as d’Alembertsaid once, ‘go on…andfaith will catch up to you!’
You may see every now
and then a beautiful panorama emerging from themist…” (p. 3). To illustrate, in a way, he directs
us to the gorgeous painting Wanderer Above the Seaof Fog, by Caspar DavidFriedrich, reproduced atthe very beginning of thetext (see Figure 3).
Although its sui generis
quality makes the bookhard to describe precisely,
one can easily say what it
is not, as in this sentencefrom the back cover: “This Figure 1. Kontsevich’s theorem
is neither an elementary on a metro ticket (p. 1).

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Book Review
introduction to the theory of singularities, nor a specialized treatise containing many new theorems.” It is a series ofmathematical-historical vignettes, eachessentially self-contained but placed
thoughtfully so as to resonate withthose around it. The word “promenade”
from the title is well chosen, as the booktakes a roughly chronological pathbut freely explores side avenues when
they arise. A highly useful “roadmap”(see Figure 4), superimposed on a sixteenth-century Chinese landscape drawing, gives a sense not only of the contentof the book but of the progression andinterconnectedness of the promenade.
Some chapters are detours, followed
solely for their own interest (one example is a delightful chapter on divergent
series on p. 87). Others emerge fromthe mist, only to have their relevance
become clear later on, such as a sudden
chapter on the Möbius band (p. 111).
Still others add texture and richness tothe mathematical ideas, such as a chapter on the local topology of singularities
of complex algebraic curves (p. 157).
The roadmap shows two main
strands, both leading towards a description of a combinatorial structure
called a chord diagram, which canbe used to classify the essential features of a singularity of a real algebraic
plane curve. Starting from Kontsevich’sFigure 2. Zooming
observation about polynomials, fourin on a singularity
chapters discuss which permutations(p. 57).
are allowed—they have already beenstudied by combinatorists under thename “separable permutations”—and how to count them.
An exercise from Donald Knuth’s The Art of Computer Programming is the inspiration for one chapter, and the author
traces the roots of the counting problem back 2000 years
to Hipparchus.
The next strand generalizes the problem from intersections of polynomial graphs to singularities of real algebraic
plane curves. Newton gave us the first significant results inthis vein, and two marvelous chapters take us through partsof the original manuscript where he introduced what are
now known as Newton’s method and Newton polygons.
Newton shows (without proof) that in a neighborhood ofa singularity, a real algebraic plane curve may be decomposed into a finite number of branches, each of which is the“graph” of a formal power series with rational exponents.
(You can see an illustration of this in Figure 2.) We then pay
a visit to Gauss, who in his dissertation gave a topological1868

Figure 3. Caspar David Friedrich, Wanderer Above theSea of Fog.The author proposes it as “a suggestion of themathematical world” (p. ii).

proof of the fundamental theorem of algebra, relying on
the assertion that an algebraic curve cannot stop or retrace
itself—in other words, if it enters a disc, it must get out. But
Gauss neglected to prove this! The fascinating history of
the search for a proof of this claim occupies an additional
chapter, and further chapters complete the proof of Newton’s claims. This establishes that “a small circle arounda singularity of a real algebraic plane curve intersects thecurve in an even number of points and defines a chord
diagram, i.e., 2n points cyclically ordered on a circle andgrouped in pairs” (p. 6).
To go deeper into the topological structure of singularities, one must blow them up. This algebraic technique is
explosive only metaphorically—it is “a kind of microscope
enabling us to look deeply into the singularity” (p. 6).
The process transforms a neighborhood of a singularity
into a Möbius band, and repeated applications lead togeneralizations called Möbius necklaces. Blowing up plays
a crucial role in the crowning chapters on chord diagrams
and in particular in a result that certain chord diagrams
cannot arise from singularities of plane curves. There is anextended, though very interesting, detour into the topological structure of singularities of complex algebraic planecurves; these are the chapters on the bottom of the roadmappicture. Just before the chapters on chord diagrams, thethread of separable permutations returns, leading into theNotices of the American Mathematical Society

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Book Review
theory of operads—a quite modern framework for understanding certain classes of operations like grafting trees orbraids onto one another or perturbing polynomials withother polynomials.
Author Étienne Ghys, a CNRS senior researcher at theÉcole Normale Superiéure de Lyon, lists on his website
research interests in “Geometry, Topology, DynamicalSystems, mathematics in general.” After reading this book,one can only conclude the last phrase is not an overstatement. This sentence from the back cover aptly captures thedazzling mathematical breadth of his Promenade: “We playwith a bit of algebra, topology, geometry, complex analysis,combinatorics, and computer science.”
Equally dazzling is the wealth of mathematical history
woven into the exposition. Its breadth is breathtaking. Thishistory comes not in sidebars but rather as an essential partof the mathematical narrative. For instance, the chapter
on Newton’s method introduces its mathematical ideas
by explicating several very clear reproductions of pages
from Newton’s 1671 unpublished De Methodis Serierum etFluxionium, as well as a few from a 1736 English translation (see Figure 5 for an example). We see reformulationsof these originals into modern mathematical language,
thereby connecting the old ideas to ones that most readers
will have encountered in courses. These primary texts showcase both Newton’s brilliance and (perhaps) laziness—heeschewed formal proof and contented himself with working out examples as illustrations of his general method.
Two chapters later we learn that Cramer read Newton’s
work and furnished complete proofs in his 1750 book
Introduction à l’Analyse des Lignes Courbes Algébriques. However, Cramer was not pleased about this, writing in thepreface: “It is regrettable that Mr. Newton was content toparade his discoveries without including proofs, and thathe preferred the pleasure of making himself admired rather
than of instructing” (p. 67). Here I have translated the quotation into English; this is one of a handful of unfortunate
places where interesting quotations are left untranslated
from the French.
At other places history serves as a welcome digression. At
the end of a reasonably demanding chapter on the Hopf fibration, there comes a marvelous discussion about whether
the universe in Dante’s Divine Comedy is homeomorphic toa 3-sphere (p. 155). The prominent role of history makes
this book more of a narrative than many other math books
and creates a personal feel that makes the abstract ideas
seem more immediate. Nearly every chapter has someelement of history, personal anecdote, or visual embellishment. For example, in the chapter relating to Donald
Knuth’s work, there is a sketch of Knuth (p. 17) and photosof old editions of The Art of Computer Programming (p. 19).
Another example comes at the beginning of the chapter on
Milnor’s fibration (p. 177): one finds a picture of Milnor,
whose avuncular smile warms the page.
The book makes use of the margins of each page in apositively magical way. The margins are home to photos,
stories, quotes, diagrams, examples, jokes, and references.
They function almost as a second text, an internal voiceof the author that pipes up over and around the primary
public voice found in the main text. They are at their bestwhen you most need them—when you are a bit confusedby the main text or you need a moment to work out a test
case to make sure you understand a theorem. Then you willfind an explanatory aside or precisely the test case that you
need to solidify your intuition.
There is a joy to this book that suffuses all of its aspects.
There are jokes, quips, and winks in the text, in the images,
and in the margins. There are flights of fancy, moments ofFigure 4. Landscape of the Four Seasons (Eight Views of the Xiao and Xiang Rivers), by Soami, early 16th century (pp. 8–9). Colors
give an idea of the difficulty of each chapter; in increasing order of difficulty: green, blue, red, and black.

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Book Review
will enjoy its freewheeling yet mathematically
substantial approachand are almost certain
to learn new mathematics and new historicalcontexts for mathematics they already know.
They will be dazzled by
the lovely images andillustrations as muchas a calculus student,
and the latter’s curiosity may be piqued by
the chapter on Newton’s
method. If a reader enjoys mathematics andhas a modicum of experience with it, she willfind something to enjoy
in this book. Few readers will be drawn into
every last detail, but allFigure 5. A page from Colson’s 1736 translation of Newton’s De Methodis Serierum et Fluxionum. In it,
we see the application of what is now known as Newton’s method to approximate the root of x3 − 2x − 5 will gain some apprecithat is close to 2. For a striking image of the original in Newton’s handwriting, see p. 48 of the book.
ation of the beauty andinterconnectedness ofhigh artistry, and sheer excitement leaping off the page. One
themathematicaluniverse.
example is the previously mentioned opening story of theA PDF of the book is available for free on the author’s
book: of Kontsevich scrawling a theorem on a metro ticket
website,though it is certainly a joy to hold the physicalduring a boring meeting and passing it to the author. Atvolume,
which qualifies as a work of art independent ofanother moment, the reader is ordered to go read Euler’sthemathematicalmeaning of its contents. The pictures andpapers (p. 109). During a strenuous chapter on singularities
diagrams
appearin
outstanding resolution, with crisp andof complex curves, the margin shows road signs indicating
vibrant
colors,andthepage width is pleasingly wider thanincreasing grades. When defining an operad, the authoratypicalbook—helping
to give the margins and their magwrites that he will not “write down the formulas expressing
nificentcontents
theirdue.
It makes a fabulous addition tothese axioms since I would be unable myself to read theamathematicallibrary.
formulas that I wrote. I prefer to give first an example.…”The book begins and ends with reproductions of lush,
contemplative paintings by Caspar David Friedrich, the firstthe already mentioned Wanderer Above the Sea of Fog andthe second a sunset painting called Tree of Crows. The bookis alive with imagination, feeling, and humor.
While reading the book, I wondered at many junctures
who the target audience is. The mathematical level fluctuates enormously over the course of the book, ranging
from elementary and likely familiar (e.g., separable permutations, Newton’s method) to quite advanced and likely
Rafe Jones
unfamiliar (e.g., the chapter on Stasheff’s loop spaces).
Some chapters are complex enough that even mathematiCredits
cal veterans must frequently pause with pencil and paper
Figures 1, 2, and 4 are courtesy of Étienne Ghys.
to work through the arguments. The author addresses theFigure 3 is by Caspar David Friedrich [Public domain], via
question of audience directly in the preface, but in a way I
Wikimedia Commons.
Figure 5 is from the John Adams Library (call number
found somewhat unsatisfying: “I wrote this ‘petit livre’ with2310187), Rare Books & Manuscripts Department, Boston
one specific reader in mind: myself, when I was an underPublic Library.
graduate…” (p. 3). Yet the narrow audience implied by thisAuthor photo is by Laura Chihara.
statement sells the book short. Seasoned mathematicians
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Volume 66, Number 11