TO THE EDGE OF THE MAP

BARRY MAZUR

The title—Map and Territory—that Shyam Iyengar chose for this volume is, ofcourse, rich in possible interpretations. The word map suggests any contrivance—perhaps of ephemeral utility—meant to model the geography of any ’territory.’ I’lltake the title to be an invitation to write about the manner in which we fashionstructures of thought—the ‘maps,’— in order to understand, and negotiate our waythrough, whatever realm it is that embraces the objects of our thought—the ‘terri-tories.’

It may be bad strategy to blurt out the point of this essay on the first page, butit is simply this: any faithful map of our thought processes has problems setting itsboundaries: we don’t think in closed systems and it is often at the edge of the mapwhere things begin to get really interesting.

But one can’t get very far without encountering attempts to chart our modes andstrategies of reasoning. Consider the way modus ponens maps out one of our mostelemental moves of thought, as in:

All men are mortal/Socrates is a man/∴ Socrates is mortal.

In effect, to frame this syllogism in the now-ubiquitous vocabulary of sets, it givesus:

A is a subset of B/ a is an element of A/ ∴ a is an element of B.

In the Posterior Analytics, Aristotle dwells on the finer issue—“reasoned fact”versus “fact” already raised by this modest starting point, i.e., by modus ponens.The question he focuses on is the difference between chain of deduction and chain ofcausality as illustrated by the astronomically quite naive syllogism:

The planets don’t twinkle;far stars twinkle;the planets are near.

A chain of deduction would be:

Date: April 16, 2017 .1

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The planets don’t twinkle; therefore—so one deduces that—they are near.

A chain of causality would be:

The planets are near; therefore—i.e., ‘that is why’—they don’t twinkle.

Aristotle calls ’nearness’ the “fact,” and ’not twinkling’ the “reasoned fact.” Hepoints out that, as it is, the above syllogism focuses on the fact. But reverse themajor premiss and the middle of that syllogism and you have a demonstration of thereasoned fact. Aristotle is explicit about the territories to which each of these typeof reasoning is applicable:

As optics is related to geometry, so another science is related to optics,namely the theory of the rainbow. Here knowledge of the fact is withinthe province of the natural philosopher, knowledge of the reasoned factwithin that of the optician, either qua optician or qua mathematicaloptician. Many sciences not standing in this mutual relation enterinto it at points; e.g. medicine and geometry: it is the physician’sbusiness to know that circular wounds heal more slowly, the geometer’sto know the reason why.

Aristotle, Posterior Analytics I.13

Thus begins the project of mapping out the syntax and semantics of logic, contin-ued in relatively modern times by the Port-Royal logicians1, by John Boole’s “TheLaws of Thought” as he mapped out the territory of his “universe of discourse,” andby John Venn’s diagrams, and—of course—by all of modern formal logic.

1La logique, ou l’art de penser, published anonymously in 1662 by Antoine Arnauld and PierreNicole, possibly with contributions of Blaise Pascal.

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Territory—to be sure—doesn’t often mean mere virgin landmass previously un-considered before the cartographer happens to have gotten to it. The dictionarydefinition would have the word mean an area of land under the jurisdiction of aruler or state or possibly a piece of land defended by a ‘territorial animal’ somewhatin the spirit of the above quotation from the Posterior Analytics. A territory, then,is already the embodiment of a story. But, for us, it is enough to think of territory as

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‘a focus of interest.’ And the map as elucidating the lay of the land...or as imposinga specific narrative.

4/16/2017 Minard.png (2003×955)

https://upload.wikimedia.org/wikipedia/commons/2/29/Minard.png 1/1

Maps offer (sometimes quite pointed) narrative for the territory they’re meantto record. For example, here above is Minard’s well known “figurative map” ofNapoleon’s losses of troops at the various stages of his 1812-1813 invasion of Russia(where the width of the colored band is meant to be proportional to the number ofsoldiers still active at that point in the campaign, and—below that—the graph ofthe temperature at that point).

But all maps orient, organize, delimit. What is fascinating is the manner in whicha map frames its limits, and points to the dragons lurking at its ends, or distortsthe perimeter of the onionskin of the globe by flattening it onto the page. Theunproclaimed assumption of any map is that the boundary, the perimeter, makessense: it encloses a meaningful-in-itself territory, a closed system. Precisely becauseof that it becomes particularly interesting to press beyond those borders.

In some contexts, it is natural to have a more relaxed view of the borders of one’smap. The biologist C.H. Waddington devised a compelling geographical metaphorfor the various paths an embryo might take in its development: it is as if the embryowere a stone poised on a hilltop, and morphogenesis consists in rolling down the hill,under the force of gravity. Well, there are the natural deep grooves that constitute thepath for ’normal’ development, but for some embryos the ricochet of their downwardride might knock them into a neighboring (but abnormal) pathway. Therefore, tofully understand the repertoire of possible development routes, one must plot out allthose deviant neighboring grooves—i.e., one must map out a significantly larger span

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of what he called the epigenetic landscape. This cartographical spread beyond theprobable to the possible is analogous to the use, in Physics, of Feynman’s diagrams2.

In this essay I will be considering the richness of going to the limit and beyond—inthe ‘maps and territories’ that organize our thought.

Mathematical objects of study often don’t come as singletons, to be examined inisolation. They tend to appear as particular individuals living in a family of like-structured objects. Often the members of such a family can be labeled by continuousparameters, these ‘continuous parameters’ constituting a geometric space of somesort, where two labels are very close to each other if the objects they label are‘very like each other.’ That is, geometric features of the parameter space reflect therelative relationships between the various mathematical objects that are labelled bythese parameters.

The technical term denoting a parameter space that labels the various mem-bers of a species of mathematical object is Moduli Space. (“Moduli” meaning“parameters”—and more specifically, parameters describing the possible ways ofvarying a mathematical object and staying within its species.) The idea of sys-tematically studying mathematical objects in the context of their possible variationis ubiquitous in mathematics (and has achieved the status of a high art in AlgebraicGeometry). A comprehension of the detailed structure of the moduli space of a givenspecies often provides a powerful way to understand the deeper structures of the veryobjects of that species. And a surprise in store for anyone who thinks along theselines is that the moduli space classifying any given species of mathematical object

2For example, in Feynman’s book QED see the classical law: angle of incidence equals angle ofreflection proved by dealing with all possible reflecting paths.

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is often extraordinarily rich in structure—richer in structure than the species that itclassifies.

Continuing the metaphorical reach of map and territory, we can view a modulispace as map and the species it is meant to study as territory. Here, as I mentioned,the map may have more intricate structure than the territory it maps. In AlgebraicGeometry, there is—at times—an interesting reversal of focus, where the modulispace, per se, becomes the primary object of investigation.3

Here are three examples: the first is something of a toy illustration; the secondis an example where the exquisite complexity of the edge was quite a surprise whenit was first appreciated; and the third is where the edge is nothing more than asingle point and yet centering one’s focus around that point makes a deep and ratheramazing connection with a somewhat different field of mathematics.

1. Consider triangles in the Euclidean plane, taken up to similarity

3To allude to an example of this switch of focus, I might mention Shimura varieties, these beingmoduli spaces that classify a certain species of mathematical object interesting enough in its ownright. But the Shimura varieties themselves play a key role in a significantly different project:establishing a bridge between two disparate mathematical fields: representation theory of reductivegroups and algebraic number theory.

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For any similarity-equivalence class of triangles, S, choose a member-triangle ofthat class: ∆ := ABC ∈ S. After possible renaming the vertices, suppose its longestside is AB, noting that ∆ might be isosceles or equilateral, and therefore would havetwo or three ‘longest sides’—if so, just choose one of them to be AB.

Rescale ∆—which doesn’t change its similarity class—so that AB is of length 2.Now place ∆ in the plane R2 with the vertex A at the point (−1, 0), the vertex Bat the point (+1, 0). Flip ∆ around the x-axis if necessary to arrange that its thirdvertex, C is in the upper-half-plane. Flip ∆ about the y-axis if necessary to arrangethat C is in the (closed) upper-right-quadrant of the plane. That is, C = (a, b) witha ≥ 0 and b > 0. Since AB is (one of) the longest side(s) of ∆, C lies in the shadedregion

M := {(a, b) ∈ R2 |a ≥ 0; b > 0; (a+ 1)2 + b2 ≤ 4}

given in the diagram below.

Call this—so arranged—point C in the (closed) upper-right-quadrant of the planethe modulus of the similarity-equivalence class of triangles S:

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C = µ(S) ∈M.

We have a one-one correspondence between similarity-equivalence class of trianglesand their moduli; i.e., the points in M:

S ↔ µ(S)SoM is the moduli space of the species: similarity equivalence classes of triangles

in the Euclidean plane. It is our ‘map.’ The plane geometry of M relates nicely tothe structure of the species that it ‘maps out’ in that points close to each other inM label similarity equivalence classes of triangles that have representatives that areclose. Any real-valued continuous function, for example, f :M can be interpreted asproviding a numerical invariant of any similarity-class of triangles, these invariantsbeing sensitive to the closeness of different similarity-classes. And now, consider itsboundary.

The boundary of M consists of three pieces:

• the vertical piece αγ,• the arc of the circle βγ, and• the horizontal piece αβ.

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The first two pieces are actually in M and (except for the common point γ) theylabel similarity equivalence classes of isosceles triangles. The distinction here beingthat points on αγ label such similarity equivalence classes where the two equal sidesof the triangle are shorter than the third side, while points on βγ label those wherethe two equal sides of the triangle are longer than the third side. The point γ at whichthey meet labels the (unique) equivalence class of equilateral triangles; i.e., trianglessuch that all three sides have equal length. In both of these sides, we encounterthe issue of nonrigidity. Say that a mathematical object is rigid if it admits nonontrivial symmetries. ‘Most’ triangles are rigid but not the ones on these two sidesof the boundary.

It is the third piece of the boundary, the horizontal interval αβ : {(x, 0) 0 ≤ x ≤ 1}to which I want to pay particular attention, even though it is not formally part of ourmoduli spaceM at all: points on the interval αβ correspond—if to anything at all—to degenerate flattened triangles where the third vertex C lies in the line-segmentAB. Studying these objects—a teratology of triangles—might seem bizarre. Yet it isprecisely in the neighborhoods of such regions in many of the moduli spaces currentlystudied where profound things take place. It is often—to push the metaphor—thevery edge of the map that captures one’s attention. Not, perhaps, for this toymodel, the moduli space of similarity-classes of triangles,4 but for other moduli spacesin mathematics—in particular, for the next example to be discussed: namely, theMandelbrot set, viewed as moduli space.

2. The Mandelbrot set

Let c a complex number. Consider the transformation of the complex plane

z 7→ z2 + c

4On the other hand, this moduli space has some interesting dynamics, tending toward thatbottom edge: Curt McMullen, in an interesting essay (unpublished) entitled Barycentric subdivi-sion, martingales and hyperbolic geometry studies the statistics of the operation of performing abarycentric subdivision of a triangle. That is, if ∆ = ABC is our triangle, let D denote thebarycenter of ∆ and decompose ∆ into a union of three triangles

∆1 = ABD, ∆2 = BCD, ∆3 = CAD.

We can think of

∆ 7→ {∆1,∆2,∆3}as a many-valued transformation on similarity classes of triangles, and therefore on points in M.McMullen proves that statistically this transformation produces thinner triangles, i.e., representedby lower points inM. So, iteration of this operation can be thought of as producing gentle cascades,sending points on our moduli space—statistically speaking— towards the bottom horizontal interval.

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If we ask questions about how this (seemingly comprehensible) geometric transformation—call it D(c) —behaves–or ‘performs’—when we iterate it:

z 7→ z2 + c 7→(z2 + c

)2+ c 7→ . . . ,

i.e., when we ask questions about it as a dynamical system—for example what itsorbits look like. The forward orbit of a point z0 ∈ C is the set of points in C thatoccur as the image of z0. One feature of interest is called its Filled Julia Set J(c)which by definition is the set of points z ∈ C whose forward orbits are bounded. Thesimplest (i.e., the most misleading) example of such a Filled Julia Set is when c = 0:J(0) is the unit disc. It is a theorem that there are only two types of topology for any‘Filled Julia set’ J(c). These can be connected ——or they can be homeomorphic toa Cantor set–in which case they are called dust:

Consider, then, our ’territory’, which is this species of mathematical object:

Dynamical Systems D(c) : iteration of z 7→ z2 + c

where c is a complex number for which the Filled Julia Set J(c) is connected–i.e.,is not ‘dust’.

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And consider its corresponding ‘map’—i.e., the moduli space parametrizing thisspecies:

The region in the complex plane consisting of complex numbers c thatcorrespond to such dynamical systems D(c); i.e., those with J(c) connected.

This region is (now) called the Mandelbrot set (as are regions related to analo-gous problems).

Up to the end of the First World War, the foundations of the theory of suchstructures was called Fatou-Julia theory. I’m guessing that Fatou or Julia would nothave been able to make too exact a numerical plot of these regions. Here is a picturedrawn by Julia:

Partly due to the ravages of the first world war, and partly from the generalconsensus that the problems in this field were essentially understood, there was alull, of half a century, in the study of Julia sets and the now-named Mandelbrot sets.

But in the early 1980s Mandelbrot made (as he described it)“a respectful exam-ination of mounds of computer-generated graphics.” His pictures of such sets weresignificantly more accurate, and tended to look like the figure below (which is aneven more modern version of the ones Mandelbrot produced)5:

5I’m grateful to Sarah Koch and Xavier Buff for this elegant picture.

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From such pictures alone it became evident6 that there is an immense amount ofstructure to the regions drawn and to their perimeters. Specifically: the perimeters,whose self-similar infinitely laciness7 was captured by Mandelbrot’s then novel—butnow ubiquitous—piece of vocabulary: fractal . This complexity at the edge of suchmaps almost immediately re-energized and broadened the field of research, makingit clear that very little of the basic structure inherent in these Julia sets had beenperceived, let alone understood. It also suggested new applications and Mandelbrotproclaimed—with some justification—that “Fatou-Julia theory ‘officially’ came backto life”8 on the day when, in a seminar in Paris, he displayed his illustrations.

6Computers nowadays (as we all know) can accumulate and manipulate massive data sets. Butthey also play the role of microscope for pure mathematics, allowing for a type of extreme visualacuity that is, itself, a powerful kind of evidence.

7“as the small pool by the elm ices over,” which is a line of Kevin Holden’s poem Julia Set thatappeared in his book Solar (Fence Books, 2016).

8This is from Benoit Mandelbrot’s book Fractals and Chaos.

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Here is a picture9 of the Mandelbot set with sample pictures of the correspondingFilled Julia Sets—color-coded so that a Filled Julia Set J(c) will have the same coloras the portion of the Mandelbrot set containing the point c.

3. The moduli spaces classifying elliptic curves

Elliptic curves play a key role in a surprising number of different mathemat-ical subjects. Moreover, depending on the subject in which you want to considerthem, elliptic curves will have surprisingly different appearances, definitions10, anduses. And mysteries. For the purposes of this essay, we will take these mathemat-ical concepts as they make their first appearance in complex analysis: an elliptic

9I’m grateful to Sarah Koch and Xavier Buff for the diagrams and comments. Xavier Buffmentioned that if you think of the Mandelbrot set as an island in an ocean; and each filled Julia setJ(c) corresponding to a point c of the Mandelbrot set, as an ’inhabitant’ of this island, the mainopen question in the subject is whether there is a hidden landmass (component of the interior of M)where inhabitants are thin (have empty interior. . . but have positive measure). As far as we know,says Xavier, all inhabitants that live inland have some interior. Thanks, as well, to Curt McMullenfor helpful comments.

10One often deals rather with elliptic curves endowed with a bit of extra structure.

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curve—for us—can be thought of as given by an equivalence class of rectangular orparallelogram lattices in the complex plane C:

where, to be more precise, such a lattice, L, is a subgroup of the additive groupC generated by two elements that are linearly independent over the field of realnumbers R (so that L consists of the points of a configuration of the type drawn inthe picture above). The equivalence relation that we will be considering might becalled ‘complex-similarity.’ That is, two such lattices L,L′ are in the same equivalenceclass if there is such a nonzero complex number a such that scaling by a brings L toL′; i.e., a · L = L′.

How can we construct the ‘moduli space,’ that maps out the territory of thismathematical species: lattices in the complex plane taken up to complex similarity?Any lattice L is generated by two complex numbers, but we are allowed to scale ourlattice so we can always arrange it so that one of those complex numbers is the realnumber 1. If we think, then, of L as generated by 1 and some other complex numberτ , since 1 and τ are required to be linearly independent over the field of real numbers,τ is genuinely complex—i.e., not real—and by changing its sign, if necessary, we canarrange it so that τ is in the upper-half of the complex plane. Thus “τ” determinesthe complex similarity class of a lattice–i.e., the lattice L generated by 1 and τ .

But there are many points τ in the upper half plane that generate, when takentogether with 1, this same complex similarity class of lattices. For instance, L isthe same as the lattice generated by 1 and τ + 1; and its similarity class is the sameas the lattice L′ generated by 1 and −1/τ (rescale L′ by multiplying by τ ; so τL′ isgenerated by τ = τ · 1 and −1 = τ · −1/τ).

So we can change τ by any of these two transformations τ 7→ τ + 1 or τ 7→ −1/τor by any combination of these transformations and their inverses and still have a“τ” that together with 1 generates a lattice in the same complex similarity class asL. The group generated by these two transformations ‘tiles’ the upper half plane inquite an intriguing way:

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Here, for any similarity class of lattices, there is a unique τ in any of these tilessuch that the lattice generated by 1 and τ is in that similarity class (if we are carefulabout our description of which of the boundary pieces belong to which tiles). So, forexample, the points of the shaded region (this comprises a single tile) is in one-onecorrespondence with the set of similarity class of lattices.

This abundance of different ways of generating the same complex similarity classturns out to be far more manageable than one might first think, thanks to theexistence of something called the elliptic modular function—or, more familiarly,the j-function—“j.”

The elliptic modular function j(τ) is a complex analytic function on the upper half-plane τ 7→ j(τ) that maps each of these tiles in a one-one manner onto the entirecomplex plane, thereby giving us a clean parametrization of this species: complexsimilarity classes of lattices.

That is, if L is generated by 1 and τ , the complex number j(τ) depends onlyon L, and—in fact—only on the similarity equivalence class {L} of L. So renamej(τ) =: j({L}) and call the complex number j({L}) the j-invariant of the similarityequivalence class {L}.

Having done this, we get a one-one correspondence between similarity equivalenceclasses and their j-invariants. Moreover, any complex number is the j-invariant of a(unique) similarity equivalence class of a lattice:

{L} ↔ j({L}) ∈ C.

So the complex plane C plays the role of the ‘moduli space’—our metaphoricalmap—of the species: complex similarity classes of lattices.

What comprises the edge or the end of this map? The answer is that we must passfrom the complex plane to the Riemann sphere, by adjoining the “point at infinity,”

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∞. Here is a picture of the stereographic projection of the Riemann sphere onto thecomplex plane, with the “North pole” acting as this point at infinity:

S := C ∪ {∞}.

The single point ∞ ∈ S comprises the end of our map; i.e., of our moduli space. Itcorresponds, if to anything at all, to a curious degenerate similarity class of lattices:namely, the limit—as y tends to infinity— of the similarity classes of lattices Lygenerated by 1 and τ = iy.

Since j(τ) = j(τ + 1), the value of j at τ depends only on q := e2πiτ so we mayrewrite the elliptic modular function as a function of this new variable q. Note thatthe limiting value of q for τ = iy with y tending to infinity is: q = 0.

Something quite curious happens when we view the elliptic modular function ascentered about this missing point q = 0, i.e, as given by its Laurent series in q = e2πiτ ,

j(τ) =1

q+744+

∑n≥1

cnqn =

1

q+744+196884q+21493760q2+864299970q3+20245856256q4+. . .

Or, if you wish, by this same formula presented as its Fourier expansion :

j(τ) = e−2πiτ + 744 +∑n≥1

cne2πinτ ,

There are two surprising things about these (Fourier, or Laurent series) coefficientscn. First, they are all non-negative integers. But also these numbers c1, c2, c3, c4, c5, . . .lead us to strikingly profound structure in a part of mathematics that one mightimagine to be quite remote for our starting place, lattices in the plane.

Namely, the Monster group M (also known as the Fischer-Greiss Monstergroup). M is a finite simple group that is referred to as ‘sporadic’ because it isn’t

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a member of any of the standard infinite families of finite simple groups connectedwith various types of geometries. It is quite large, having

2463205976112133171923293141475971

elements. As with all finite groups the group M can be ‘represented’ as a group oflinear transformations of complex N -dimensional space for various dimensions N .The dimensions N for which M acts irreducibly on CN–including the 1-dimensionalspace on which M acts trivially—comprise a finite list of numbers:

1, 196883, 21296876, 842609326, 18538750076, . . . .

So, the very smallest dimension N for which this curious group M can be viewedas a ‘group of linear transformations’ on CN is 196883. In 1978 John McKay madethe following puzzling and somewhat amazing observation: the first few Fouriercoefficients of the elliptic modular function j can be expressed as sums—with veryfew summands!—of the dimensions N for which M acts irreducibly on CN . Forexample:

196884 = 1 + 196883,21493760 = 1 + 196883 + 21296876, and864299970 = 2× 1 + 2× 196883 + 21296876 + 842609326,20245856256 = 3× 1 + 3× 196883 + 21296876 + 2× 842609326 + 18538750076.

This extremely arresting purely numerical observation suggested a world of newstructure: it led to the conjecture that there lurked an infinite sequence of ‘naturalin some sense’ complex representation spaces of the Monster group,

V1, V2, V3, . . . , Vn, . . .

where, for n = 1, 2, 3, . . . the Fourier coefficient cn of the elliptic modular function isequal to the dimension of Vn. This seemed, perhaps, so startling at the time that theconjecture was labeled monstrous moonshine. When eventually proved11 it has givenbirth to another profound field in mathematics, and intimate links with physics. Andall this, inspired by the elliptic modular function j; i.e., by contemplating lattices inthe plane.

There seems to be a tenacious unity to mathematics, where ideas trespass theborders of any field, any designated ‘territory’, and any map is merely provisional.

11by the work of . . .