John Horton Conway

John Horton Conway died on April 11 of COVID-19. He was 82 years old. In the midst of social distancing measures to fight the coronavirus pandemic, a common refrain is “life goes on”.  But sometimes it doesn’t.

Conway was an emeritus professor at Princeton University.  Among mathematicians, he was known for his breadth and cleverness, as well as his personality and his seemingly infinite curiosity.   In MathSciNet, a bit over one quarter of his papers are in number theory, about a sixth in group theory, and a tenth in convex or discrete geometry.  The rest are dispersed about 20 other classes in the MSC.  Conway managed to make lasting contributions in those other 20 areas, such as his work in algebraic topology and knot theory, where he has an invariant named after him: the Alexander-Conway polynomial.  Meanwhile, behind the scenes, Conway was frequently contributing puzzles, games, and ideas to Martin Gardner, who would write about them in his famous column in Scientific American.

In the Mathematical Reviews database, Conway has 73 coauthors.   A lot of them are famous, but a lot of them not.  Conway seemed driven by curiosity, not by reputations when working with people.

Many people know of Conway because of the Game of Life (also here).  Like many neophyte programmers, Life was one of the first things I programmed.  In my case it was in Fortran on a Hewlett-Packard machine that used punch cards.  It was a great project for beginning programming because it was also a very simple model of a biological system.  For a young math major, this example was notable because all the models and applications in classes so far were based on calculus.  This clearly was not.

A lot of people also know Conway as one of the coauthors of the impressive and classic book Winning Ways for your Mathematical Plays.  (Sadly, the other two coauthors, Elwyn Berlekamp and Richard Guy, also died in the past year or so.)  The beginning  and end of our review of the first edition of the book quickly tell you the truth about it: “The two volumes are crammed to the brim with information, colored illustrations and examples … The thrust of the book lies in the direction of formulating exact or suboptimal polynomial strategies for very broad classes of combinatorial games. It is likely to remain an eminent leader in this field for many years to come.”  His book on sphere packings with Neil Sloane is another classic.

Conway was one of the authors of the monumental Atlas of Finite Groups.  When I first heard about the Atlas, I thought, “That’s crazy.”  I turned out to be half right, it was crazy brilliant.  The core of the book is a compilation of the character tables of all the finite simple groups known at the time.  By some miracle, the authors were able to convince the publisher to produce the book in a large format (42 x 31.6 x 2.9 cm), which was helpful for bigger groups with bigger tables.   It was bulky to carry and had a tendency to bend and to curl at the edges.  Our reviewer, Robert Griess, suggested that the tables be made available on tape.  [Note: See http://brauer.maths.qmul.ac.uk/Atlas/.] The Atlas was clearly an inspiration for the Atlas of Lie Groups and Representations, which is completely online.

Conway had a knack for naming things, as in his famous paper “Monstrous moonshine” with Simon Norton.  (The complete review is below.)  The paper conjectures remarkable correspondences between conjugacy classes of the finite simple group called the Monster and congruence subgroups of the modular group, PSL(2,$\mathbb{Z}$).  The conjecture was proved in MR1172696 by Richard Borcherds, who was a Ph.D. student of Conway.

Conway’s influence was wide, as was his renown.  Shortly after Conway’s death, nice posts quickly began springing up, a few by mathematicians who knew Conway only slightly, but had strong memories of their handful of interactions with him.  Here are links to just a few of them.

• Peter Cameron
• Terry Tao
• The Aperiodical post includes social media posts
• Scott Aaronson
• See also the “Conway’s lesser-know results” thread on MathOverflow.

Siobhan Roberts wrote a nice profile of Conway in the Guardian in 2015.  She also published an engaging biography of him, titled Genius at play.

There is a quote from Isaac Asimov about science:  The most exciting phrase to hear in science, the one that heralds new discoveries, is not “Eureka!” (I found it!) but “That’s funny …”  John Conway was the embodiment of that.

MR0554399
Conway, J. H.; Norton, S. P.
Monstrous moonshine.
Bull. London Math. Soc. 11 (1979), no. 3, 308–339.
20D08 (10D12)

This extremely informative paper details numerous, apparently systematic, coincidences between genus-0 subgroups of the modular group and subgroups of the “Monster” finite simple group conjectured by Fischer and Griess. (See also J. G. Thompson’s companion papers [20029 and 20030 below].)
The authors begin with a brief history of the increasing observations of such coincidences, prior to their own work. Section 2 then describes the main conjectures. The authors give a correspondence between conjugacy classes of the Monster and congruence subgroups of the modular group—roughly, an element of order $n$ in the Monster will correspond to a subgroup above $\Gamma_0(n)$. Expand a hauptmodul for each such subgroup by Fourier coefficients in powers of $q=e^{2\pi i\tau}$. For fixed $k$, the $q^k$-coefficients of these functions provide a class function, which is conjectured to be an actual character of the Monster (the $k$th “head character”). Sections 3–7 discuss technical details of the correspondence, including relations among the classes, and relations with the Leech lattice for certain Monster elements. Section 8 describes “replication” formulae among the head characters, and other “expansion” and “compression” among the functions and the head characters. Section 9 describes similar work (“moonshine”) for groups other than the Monster. In Section 10 the authors ask why only genus-0 subgroups of congruence type arise—and propose the determination of all such subgroups. (There are apparently 300–400, of which 171 arise in connection with the Monster itself.)
The paper includes tables giving: the irreducible degrees of the Monster (from the character table determined by Fischer, Living-stone and Thorne); sample decompositions of a few head characters in terms of the irreducibles; a class list for the Monster, indicating the correspondence with a subgroup of the modular group; the first 10 head characters in full; and various other information on the classes and the formulae mentioned above.
Since the appearance of the paper, R. Griess has constructed the Monster [“The friendly giant”, Invent. Math., to appear]. The head-character conjecture can be verified by a finite but lengthy computation—and this has been essentially completed by A. O. L. Atkin, P. Fong, and the reviewer. An actual module affording the head characters, at least for the centralizer in the Monster of a 2-central involution, has been constructed by V. G. Kac [Proc. Nat. Acad. Sci. U.S.A. 77 (1980), no. 9(1), 5048–5049]; Griess is now working to extend the action to the full Monster group.
This subsequent work seems to provide at best only a partial answer to the many questions about the interrelations of this finite simple group with the (infinite discrete) genus-0 subgroups of analytic number theory.

Reviewed by Stephen D. Smith

MR0827219  
Conway, J. H.(4-CAMB); Curtis, R. T.(4-CAMB); Norton, S. P.(4-CAMB); Parker, R. A.(4-CAMB); Wilson, R. A.(4-CAMB)
Atlas of finite groups.
Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, Eynsham, 1985. xxxiv+252 pp. ISBN: 0-19-853199-0
20D05 (20-02)

At last, an official collection of character tables and related information about many finite simple groups has appeared in book form. This information is important to specialists in finite group theory and the volume contains neatly presented instructional material which the nonspecialists can appreciate. For years, the authors have used the material at a very high level. It has been reworded and refined by experience. At the month-long 1979 Santa Cruz conference on finite groups, Simon Norton carried a shopping bag of tattered printouts and character tables to deal with urgent questions about simple groups. Now, we can all have the power of such rapid access, but in a classier format!
The “classic” character table of a finite group $G$ is by definition a $k\times k$ matrix of complex numbers, whose rows are indexed by the $k$ irreducible characters and whose columns are indexed by the $k$ conjugacy classes; of course, it is not unique because there is no generally accepted way to order the index sets, though the principal character (corresponding to the trivial homomorphism $G\to {\rm GL}(1,{\bf C}))$ is always listed first. The $(i,j)$ entry is $\chi_i(g_j)$, the value of the $i$th irreducible character on a representative of the $j$th conjugacy class, and this algebraic number is always a sum of $d\ |g_j|$th roots of unity, where $d=\chi_i(1)$ is the degree of $\chi_i$.
The efforts of the last 25 years to classify finite simple groups created a greater need to have numerical and combinatorial information about the known groups. The occasional tables produced by R. Brauer or J. S. Frame or J. Todd years ago were followed by a flood of tables in the 1960s and 1970s. Generally, these were distributed informally, often with no name or source written on them and always without proof. Referring to a character table in a research article was awkward at times. The general theory of Brauer gave many arithmetic conditions on the character table which in “easy” cases allowed one to fill in many blank entries for the table of a particular group. This was not always the case. For instance, David Hunt’s work on the tables for the Fischer $3$-transposition groups took an especially long time and involved extensive computer work and a study of induct-restrict tables for subgroups with known character tables.
In sum, the five authors have collected some of this early and unpublished work, then greatly extended it and put it in a form suitable for easy modern applications.
The book is organized as follows: (I) Introduction and explanations (28 pages), (II) The character tables (235 pages), (III) Supplementary tables (6 pages), (IV) References (8 pages) and Index (1 page).
(I): Sections 1, 2 and 3 contain a rapid introduction to the families of finite simple groups. It is clear and telegraphic in style and not intended for someone who is looking for full discussions and constructions.
Sections 4 through 7 discuss the multiplier, automorphism groups, isoclinism and the group extension theory which is relevant to interpreting the blocks (and broken-edge blocks) in the tables, notation for conjugacy classes, algebraic numbers and algebraic conjugates of these two concepts. We comment on the tables themselves in (II). The authors’ notations for algebraic integers are very successful for character tables, e.g., $z=z_N=\exp(2\pi i/N)$, $b_N=\frac12\sum^{N-1}_{t=1}z^{t^2}$, $c_N=\frac13\sum^{N-1}_{t=1} z^{t^3}$ (for $N\equiv 1$ (mod 3)), etc.
One fault with the exposition is that the authors use terms and notation without explanation, then define them later. In the above sequence of definitions, for $z_N$, $b_N$, $c_N,\cdots$, one finds “$n_2$”, but not a definition until further down the column. The notation $^*k$ is used in Section 7.3 but no hint is given for where to look for the definition. It would help if an index of notations and definitions were included to help the reader who starts reading in the middle.
The authors discuss the several existing systems of notation for the simple groups. Parts of the system used in the Atlas make the reviewer uncomfortable.
The most glaring item is the use of “O” for the simple composition factor of the $n$-dimensional orthogonal group of type $\epsilon$ over ${\bf F}_q$. In other systems, this group would be ${\rm P}\Omega^\epsilon(n,q)$ or one of $D_m(q)$, $^2D_m(q)$ (when $n=2m$) or $B_m(q)$ when $n=2m+1$. The authors reject these notations because they want one letter for the basic name of all these simple groups.
The second comment is about names assigned to sporadic groups; see Table 1, page viii. The principle generally used by group theorists has been to name a sporadic group after its discoverers and use a symbol related to these names. The sometime exceptions to this have been the Conway groups (denoted by $.0,\;.1,\;.2$ and $.3$ since 1968 but by ${\rm Co}_0,\; {\rm Co}_1,\;{\rm Co}_2$, and ${\rm Co}_3$ in this volume), the Fischer groups (denoted by $M(22),\;M(23)$ and $M(24)’$ originally, but later by ${\rm Fi}_{22},\;{\rm Fi}_{23}$ and ${\rm Fi}’_{24}$) and the Monster (the group discovered by Fischer and the reviewer in November 1973; the Atlas symbols are $M$, FG and $F_1$) and the Baby Monster (the $\{3,4\} ^+$-transposition group discovered by Fischer earlier in 1973; the Atlas symbols are $B$ and $F_2$) and the Harada group (called the Harada-Norton group in the Atlas; the Atlas symbols are HN and $F_5$).
The system of $F$‘s with subscripts has several nice group-theoretic features. However, there seems to be no natural systems covering all sporadics. Why not keep the names and remember the history, at least? Perhaps later developments will suggest a good solution.
Finally some comments about notation for other finite groups. Several recommendations in 5.2 really are at variance with general usage. The authors mention $C_m$ for a cyclic group of order $m$ but not ${\bf Z}_m$! Their term “diagonal product” $A\triangle B$ is otherwise known as a pullback or a fiber product. The most common notation for an extraspecial group is $p^{1+2n}$ or $p_\epsilon^{1+2n}$. Since notation for an extension $A\cdot B$ reads left-to-right along an ascending series, it would be more appropriate to write $(A\times B)\frac12$ than $\frac12(A\times B)$.
(II): The organization of the individual tables is discussed in Section 6. See page xxiv for a well-diagrammed example. Let $G$ be the simple group. The tables come in blocks with each block corresponding to an extension of the form $m.G.a$, where $m$ is a cyclic quotient of the Schur multiplier and $a$ is a cyclic subgroup of the outer automorphism group; for reaons why these cases suffice (nearly), see 6.5 and 6.6.
To the left of the block is the downward running list of characters $(\chi_1= 1,\chi_2,\chi_3,\cdots)$ and their indicators (0, $+$ or $-$ as the character is not real-valued, afforded by a real representation, or real-valued but not afforded by a real representation). Across the top is a band with several rows of information about the columns (indexed by the conjugacy classes, $C_i,\;i=1,\cdots,k$). The experience of the last 25 years has shown the importance of enriching the traditional “classic” character table to include power maps (i.e., for $n\in{\bf Z}$, which classes contain the $n$th powers of elements from a fixed class), factorizations (i.e. if $g\in C_i$ and $\pi$ is a set of primes and $g=g_\pi g_{\pi’}$ is the unique commuting factorization of $g$ into a $\pi$-element and a $\pi’$-element, which $C_j$ contains $g_\pi$), and so on. A simple application of this information, which is not possible to execute with a strictly classical table, is to find the dimension of the space of cubic invariants on a module $V$ affording the character $\chi$. The character on the symmetric tensor cube of $V$ is $g\mapsto \frac16\{\chi(g)^3+3\chi(g)\chi(g^2)+2\chi(g)^3\}$ and so its inner product with the trivial character of $G$ gives the answer.
The difficulty of getting these blocks correct increases generally according to the sequence $m=1$, $a=1$; $a=1$; $m,a$ arbitrary. Indeed the authors acknowledge errors which turned up as the book went to press (see page xxxii, bottom). How the notations extend across the several upward and downward extensions is articulated well.
(III): The final part of the Atlas text consists of three tables and a list of references. (1) Partitions and classes of characters for $S_n$, useful, say, in working out particular invariants of the group in question. (2) Involvement of sporadic groups in one another (the single “?” in this Atlas table is now claimed to be “$-$” in recent work of R. A. Wilson). (3) Orders of over 250 simple groups, with orders in base 10 and in factorized forms and with Schur multiplier and outer automorphism group.
(IV) The bibliography is restricted to (i) some very general works on the families of finite simple groups and (ii) lengthy lists of articles on each of the 26 sporadic groups.
Survey articles (no proofs) for absolute beginners are worth mentioning and could go in (i), e.g., a paper by R. Carter [J. London Math. Soc. 40 (1965), 193–240; MR0174655] for groups of Lie type and a paper by the reviewer [in Vertex operators in mathematics and physics (Berkeley, Calif., 1983), 217–229, Springer, New York, 1985; MR0781380] for sporadic groups. Also, references for Schur multiplier and automorphism groups would be of general interest.
Tables of numerical information are notorious for errors and it does pay to compare; for example, the order of McLaughlin’s group is incorrectly given on page 136 of D. Gorenstein’s Finite simple groups [Plenum, New York, 1982; MR0698782]. After the Higman-Sims group, $G$, was discovered in 1968, it was deduced that $G$ must have subgroups $K\leq H\leq G$ with $H\cong {\rm PSU}(3,5)$ and $K\cong {\rm Alt}_7$. Of course, the characters of $G$ must restrict sensibly to characters of $K$ and $H$ but the character tables then at hand produced a contradiction! The error in the tables was found.
Should a researcher, urgently needing to prove a theorem, trust the Atlas? The question is like that of whether to accept the classification of finite simple groups. Both efforts are widely respected, the participants in both have worked at high levels to reach the goal, yet have admitted that errors exist. In both cases, the group theory community feels that probably only local adjustments would be needed in the ambient program to deal with errors. So, the answer is: “Yes, but$\ldots$”.
Only a purist would turn his or her back on either claim of completion. To make progress, we must accept them as essentially correct but pay attention for some time and look for alternate arguments whenever possible. One can treat them as axioms when writing arguments down formally.
Norton has shown a list of errors discovered since publication. One is a nonsquare character table! It is worth mentioning that Chat-Yin Ho recently found a maximal $7$-local subgroup of the Monster not on the Atlas list. There may be a problem with the list of maximal subgroups for ${\rm Co} _1$.
{Reviewer’s remarks: The reviewer is disappointed at the incorrectness of the scholarship in a few instances (notwithstanding the disclaimer on page xxxii, Section 8.5.1). The correctness of the Monster character table is not completely proved (though not doubted). (a) The determination of the conjugacy classes requires sufficient knowledge of centralizers of elements in a subgroup of ${\bf M}$ of the form $2^{1+24}\cdot{\rm Co}_1$; the authors guessed the basic information, then proceeded. (b) The existence of the irreducible character of degree 196883 was taken as a hypothesis (196883 is the smallest number which could be the degree of a nonprincipal character); a proof that such a character exists was claimed by Norton in 1981 but no manuscript has appeared, and its relationship with (a) has not been explicitly stated; existence of such a character is necessary to complete the program devised by J. G. Thompson [Bull. London Math. Soc. 11 (1979), no. 3, 340–346; MR0554400] for proving uniqueness of ${\bf M}$.
{It would have been helpful to have some recent references, e.g. to the reviewer’s recent work on code loops. The reviewer understands that future editions will contain no new references.
{The book is attractive in appearance. The cover is a cherry red with white writing on stiff cardboard. The authors’ names form a neat matrix listed vertically in alphabetical order (which agrees with their respective ages, apparently), each with two initials and a 6-letter last name. The price is extremely fair. The authors are to be commended for their influence on the price and for getting the publisher to replace the originally intended soft binding.
{The book is large—too large for most briefcases. The wire binding on the reviewer’s copy became deformed right away and interfered with easy closing and opening of the book to lie flat on a table. The edges of the pages near the binding have begun to suffer due to struggles with the binding. One idea is to make the tables available on tape, potentially a big saving of effort for the user who intends computer calculations.
{The mathematics community (and physics community) should be grateful to the creators of the Atlas for their extremely fine service. An appreciation and use of the finite simple groups might be expected to spread noticeably faster as a result.}

Reviewed by R. L. Griess

MR0258014
Conway, J. H.
An enumeration of knots and links, and some of their algebraic properties. 1970 Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329–358 Pergamon, Oxford
55.20

In this essential paper (i) a new efficient notation for describing specific knots is expounded, (ii) identities are reported which reflect the behaviour of knot invariants on changing some structure elements coded in the notation, (iii) lists of all prime knots up to 11 crossings and of all prime links up to 10 crossings are given in this notation, a census which checks (and corrects) and enlarges the existing tables still based on Tait’s, Little’s and Kirkman’s work before 1900. The ideas are presented here in expository style, whereas a more technical paper with more complete presentation of the subject is promised.
The first part explains the new notation. This is based on choosing an edge-connected 4-valent planar graph, the simplest of which, marked by 1*, looks like an 8; in fact only eight such graphs are needed within the range of the tables. If for each node of the chosen graph one substitutes a “tangle”, one gets a normed projection of a ($\mu$-component) knot. A “tangle” consists of a normed projection of strings such that there are four free ends pointing to the four compass directions. A few operations are defined on tangles, such that the notation and classes of “integral”, “rational” and “algebraic” tangles can be defined recursively starting with the specific tangle marked by 1. That way the symbol for a knot indicating the graph and the substituted algebraic tangles contains some structural elements of the knot. As H. F. Trotter points out [Computational problems in abstract algebra (Proc. Conf., Oxford, 1967), pp. 359–364, Pergamon, Oxford, 1970; MR0258015], this notation seems to be “much the best both for handwork and (perhaps with some minor modification) for computer representation”.
Then the author describes the interplay between knot equivalences and elements figuring in the notation; for rational tangles substituted into the graph 1*, there are remarkable connections with continued fractions. The next sections contain remarks concerning Alexander polynomial, Minkowski unit and signature; some identities are reported which allowed short computations but which have wider applications, in part not yet fully explored. The last section gives the inferences to draw from the new lists to the open problem of whether every slice knot is a ribbon knot.
The lists (computed by hand) given as an appendix include the 1-component knots up to 8 crossings with symmetries, signature, Minkowski unit, determinant and polynomial, up to 10 crossings with symmetries, determinant and polynomial, and an enumeration of all 11-crossing knots, alternating and nonalternating; furthermore, the links up to 8 crossings with linking numbers, symmetries, signature, Minkowski unit, determinant and polynomial, up to 9 crossings with linking numbers and polynomial, and an enumeration of the 10-crossing links. {Remark: the column heading on p. 345 should read $\delta^0$ instead of $\sigma^0$.}

Reviewed by H. E. Debrunner