John Charles Fields: A Sketch of His Life and Mathematical Work

John Charles Fields: A Sketch of His Life and
Mathematical Work

Marcus Emmanuel Barnes
Simon Fraser University

Thursday, November 29, 2007

Marcus Emmanuel Barnes Simon Fraser University

John Charles Fields (1863-1932)

Figure: John Charles Fields (1863-1932)


Early Years

Born May 14th, 1863, in Hamilton, Ontario (then Canada
West), the son of the leather shop operator J. C. Fields and
his wife Harriet Bowes.


Early Years

Fields had at least one brother


Early Years

Fields had at least one brother
Attended Hamilton Collegiate where he showed early talent in
mathematics.


Undergraduate Years

Continued his studies at the University of Toronto in 1880,
studying mathematics.


Undergraduate Years

Typical mathematical requirements for a BA degree in Canada
in the second half of the nineteenth century:


Undergraduate Years

Euclid’s Elements books 1-4, 6


Undergraduate Years

algebra to the binomial theorem


Undergraduate Years

trigonometry


Undergraduate Years

trigonometry
basic mechanics and hydrostatics


Undergraduate Years (Continued)

Honours students could go beyond the basic material to study:
conic sections



conic sections
diﬀerential and integral calculus



conic sections
diﬀerential equations



conic sections
diﬀerential equations
various topics in applied mathematics


Undergraduate Years (Continued 2)

Fields had a distinguished undergraduate career, winning a gold
medal upon graduating in 1884 with a BA.


Undergraduate Years (Continued 2)

Fields had a distinguished undergraduate career, winning a gold
medal upon graduating in 1884 with a BA.

What next?


The State of Mathematical Education in N.A. in the 1880s

One option for further study was the PhD, but it was not
possible to obtain a PhD in mathematics in Canada in the
1880s



1880s
Possibilities:



1880s
Possibilities:
a handful of places in the United States or go to Europe.



1880s
Possibilities:
a handful of places in the United States or go to Europe.
Fields chose the newly established Johns Hopkins University


Johns Hopkins University

established in 1876



established in 1876
the mathematics departments was put together by J. J.
Sylvester (1814 - 1897)



established in 1876
Sylvester (1814 - 1897)
From the start, the research productivity of the faculty was of
high importance (something rather unique in North America
at the time).



established in 1876
Sylvester (1814 - 1897)
From the start, the research productivity of the faculty was of
high importance (something rather unique in North America
at the time).
Even though Sylvester left before Fields attended Johns
Hopkins, he would influence Fields’ early mathematical work,
specifically with regards to the “symbolical method in
analysis”. We see this influence in his early papers on
differential equations and differential coefficients.


Fields at Johns Hopkins

At Hopkins Fields took courses on the theory of functions,
linear diﬀerential equations, elliptic and Abelian functions,
among other topics. As well, he participated in several topic
seminars. Most of the courses Fields took were either taught
by W. Story or T. Craig.


Fields at Johns Hopkins

At Hopkins Fields took courses on the theory of functions,
linear diﬀerential equations, elliptic and Abelian functions,
among other topics. As well, he participated in several topic
seminars. Most of the courses Fields took were either taught
by W. Story or T. Craig.
Fields received his PhD in 1887 with a thesis entitled
Symbolic Finite Solutions and Solutions by Deﬁnite Integrals
of the Equation d n y /dx n = x m y .


Fields’ PhD thesis I

n
The equation d y = x m y , which is the focus of Fields’ thesis, is
dx n
similar to certain Riccati equations. Jacopo Riccati (1676-1754),
an Italian nobleman and mathematician, studied certain second
order diﬀerential equations. The diﬀerential equation
dy
= Ay 2 + Bx n ,
dx
A and B constant, became known as Riccati’s equation. In
generalized form it is usually written as
dy
= a0 (x) + a1 (x)y + a2 (x)y 2 .
dx


Fields’ PhD thesis II
Using “symbolic methods” Fields’ was able to give the following
n
general solutions to d y = x m y :
dx n

−9i
Case 1: m = 3i+1
„ «i „ «2i
3 d 3i 3 d 2
y =x x 1− 3i+1 x 1+ 3i+1 x 1− 3i+1 x − 3i+1
dx dx
1 1 1
„ «
3i+1 3i+1 3i+1
× C1 e −(3i+1)λ1 x + C2 e −(3i+1)λ2 x + C3 e −(3i+1)λ3 x ;

−3(3i+1)
Case 2: m = 3i+2
„ «i „ «2i+1
3 d 3i 3 d 1
y =x x 1− 3i+2 x 1− 3i+2 x 1− 3i+2 x − 3i+2
dx dx
1 1 1
„ «
−(3i+2)λ1 x 3i+2 −(3i+2)λ2 x 3i+2 −(3i+2)λ3 x 3i+2
× C1 e + C2 e + C3 e .


Fields’ PhD thesis III

In the second portion of his thesis, Fields gives a generalization of
results obtained by E. Kummer (1810 - 1893) and Spitzer for the
n
solution of d y = x m y by definite integrals. Furthermore, by using
dx n
some of the methods from the first part of his thesis, Fields is able
to give particular solutions in definite integrals for given values of
n
m. For example, Fields shows that the equation d y = x −m y has a
dx n
solution given by
Z ∞ Z ∞ m−n m−n
1 +···+un−1 +(xu1 ···un−1 )n−m )
y = x n−1 ··· u2 u3 · · · un−1 e n−m (u1
2 n−2
du1 · · · dun−1
0 0

where m is any positive quantity greater than n.


Immediately after Fields’ PhD

Fields became a fellow at Hopkins (which entailed a certain
amount of undergraduate teaching) until 1889.



In 1889 Fields became professor of mathematics at Allegheny
College in Meadville, Pennsylvania.



In 1889 Fields became professor of mathematics at Allegheny
College in Meadville, Pennsylvania.
During this time, Fields published several papers, including
proofs of the fundamental theorem of algebra, the elliptic
function addition theorem, as well as some papers on number
theory and some further work on diﬀerential coeﬃcients
(using the symbolic method). Many of these papers appeared
in the young American Journal of Mathematics.


Heading to Europe

Fields’ resigned from Allegheny in 1892 as a result of coming into
his modest inheritance from his father and mother who had passed
away when Fields was a teenager. He decided to use the money to
continue his mathematical studies in Europe.


Post-Doctoral Studies in Europe

The standard obituary by J. L. Synge (1897-1995) states that
Fields spent 5 years in Paris and 5 years in Berlin.



There is only documentary evidence regarding Fields’ stay in
Germany.



There is only documentary evidence regarding Fields’ stay in
Germany.
Fields enrolled in G¨ttingen in November of 1894 where he
o
had the opportunity to attend lectures by Felix Klein
(1849-1925) on number theory, as well as an introductory
course on the theory of functions of a complex variable oﬀered
by the Privatdozent Ritter.


Post-doctoral studies in Berlin

Fields stayed at G¨ttingen until May of 1895, at which point
o
he travelled to Berlin.


Post-doctoral studies in Berlin

Fields stayed at G¨ttingen until May of 1895, at which point
o
he travelled to Berlin.
Berlin was a natural choice for Fields given his early interest in
linear diﬀerential equaitons and the fact that L. Fuchs
(1833-1902) and G. Frobenius (1849-1917) were there.


Post-doctoral studies in Berlin II

Fields’ notebooks contain notes from lecture courses by:
G. Frobenius: 2 courses on number theory; 1 on analytic
geometry; 2 on algebraic equations.



L. Fuchs: 9 courses including the theory of hyperelliptic and
Abelian functions, and topics in diﬀerential equations.



L. Fuchs: 9 courses including the theory of hyperelliptic and
Abelian functions, and topics in diﬀerential equations.
K. Hensel (1861-1941): 6 courses, including algebraic
functions of one and two variables, a course on Abelian
integrals, and a course on number theory


Post-doctoral studies in Berlin III

H. A. Schwarz (1843-1921): 15 courses. Topics included
elliptic functions, variational calculus, theory of functions of
one complex variable, synthetic projective geometry, number
theory, and integral calculus.



Hetner: a course on deﬁnite integrals and a course on Fourier
series; Knoblauch: 2 courses on curves and surfaces; Steinitz:
a course on Cantor’s set theory.



Hetner: a course on deﬁnite integrals and a course on Fourier
series; Knoblauch: 2 courses on curves and surfaces; Steinitz:
a course on Cantor’s set theory.
In addition, there is almost the entire series of lectures by M.
Planck which would evolve into his famous course on
theoretical physics, two courses on inorganic chemistry, and
one on the history of philosophy.


Some Images from Fields’ Berlin notebooks


Some Images from Fields’ Berlin notebooks II


Fields leaving certiﬁcate


Research 1894-1900

Fields did not publish any papers during the years 1894 to 1900,
though he seems to have continued to do research, presenting a
talk at the meeting of the American Mathematical Society held in
Toronto in 1897 on the reduction of the general Abelian integral.
He would publish a paper based on this talk in 1901. It is not
really surprising that Fields’ failed to publish during this time given
the number of courses he apparently attended, as can be
ascertained from large number of notebooks full of lecture notes he
accumulated.


Professor Fields I

Fields took up a position as special lecturer at the University
of Toronto in 1902. At that time the mathematics department
had roughly ﬁve members including Fields.


Professor Fields I

By 1905 he had gained a regular position as Associate
Professor and would later become Professor in 1914 and
Research Professor in 1923.


Professor Fields I

By 1905 he had gained a regular position as Associate
Professor and would later become Professor in 1914 and
Research Professor in 1923.
Among the honours that Fields received was being elected to
the Royal Society of Canada in 1909 and to the Royal Society
of London in 1913.


Professor Fields II
On a local level, Fields was active in the life of the university,
often giving talks to the mathematics and physics student
club. He also successfully lobbied the Ontario legislature for
monetary support for scientiﬁc research being carried out at
the University.


Professor Fields II
the University.
Fields was involved with scientiﬁc organization on the national
level. For example, he was President of the Royal Canadian
Institute from 1919 to 1925.


Professor Fields II
the University.
On the international level, Fields was Vice-President of both
the British Association for the Advancement of Science in
1924 and the American Association for the Advancement of
Science, Section A in 1924.


Professor Fields II
the University.
On the international level, Fields was Vice-President of both
the British Association for the Advancement of Science in
1924 and the American Association for the Advancement of
Science, Section A in 1924.
Fields is also well known for organizing the 1924 International
Congress of Mathematicians held in Toronto. It was during a
cross country train trip with the conference delegates that
Fields’ health began to deteriorate.

Fields’ theory of algebraic functions

At the turn of the twentieth century, there were several approaches
to the theory, often categorized as either transcendental,
geometric, or arithmetic. Fields’ approach seems to be an
outgrowth of Hensel’s push for a purely “algebraic” approach (i.e.,
arithmetic) to the theory of algebraic functions. However, Fields’
approach, contrary to that presented by Hensel and Landsberg in
their 1904 memoir seems to have retained some of the
Weierstrassian function theoretic methods, in that it avoids the use
of Riemann surfaces and the theory of divisors entirely.


Fields’ theory of algebraic functions II

One of Fields’ primary goals was to study rational functions subject
to the condition deﬁned by an algebraic function F (x, y ) = 0. We
would now describe these as rational functions on a variety.


Fields’ theory of algebraic functions III

Fields’ theory is built up from the following concept:
Deﬁnition (Order of Coincidence)
The order of coincidence (at a point) of a rational function H(x, y )
with respect to a branch y − P = 0 is the smallest exponent of the
series expansion for H(x, P).


Fields’ theory of algebraic functions IIII

Example
Consider the rational function H(x, y ) = y 2 + x subject to
F (x, y ) = y 3 + x 3 y + x = 0. Then the order of coincidence can by
found by substituting y = Pi into H, resulting in

1 ω 8 2 ω 4 16
H(x, Pi ) = (ωx 3 + x 3 + · · · )2 + x = ω 2 x 3 + x + x 3 + ··· .
3 9
Thus the order of coincidence of H(x, y ) with respect to the
branch y − Pi , i = 1, 2, 3, is 2 , the least exponent in the series
3
expansion of H(x, Pi ).


Fields’ theory of algebraic functions IV

Using the basic machinery provided by the concept of order of
coincidence, Fields is able to build up a theory that recovers the
core results of algebraic function theory, like the Riemann-Roch
theorem.


Fields’ theory of algebraic functions V

In order to give state one version of the Riemann-Roch theorem
that Fields gives, we need to clearify some terminology.
Definition (Adjoint Curve)
A curve C is said to be adjoint to a curve C when the multiple
points of C are ordinary or cusps and if C has a point of
multiplicity of order k − 1 at every multiple point of C of order k.

Definition (Strength of a Set of Multiple Points)
Given a curve F (x, y ) = 0 of order n, the strength of a set of Q
(multiple) points used in determining an adjoint curve of degree
n − 3 is defined to be the number of q conditions to which the
coefficients of the general adjoint curve of degree n − 3 must be
subjected in order that it may pass through these Q points.


Fields’ theory of algebraic functions VI

An algebraic equation F (x, y ) = 0 can be factored into a product
of ρ irreducible factors. In stating the Riemann-Roch theorem,
Fields uses the following notations. He indicates the poles ci of the
ﬁrst order by ci−1 and uses the term “coincidences” to indicate
singularities such as the ci s.


Fields’ theory of algebraic functions VII

The following is one version of the theorem which Fields gives in
his 1906 memoir:
Theorem (Riemann-Roch)
The most general rational function of (x, y ) whose inﬁnities are
−1 −1
included under a certain set of Q inﬁnities c1 , . . . , cQ , depends
upon Q − q + ρ arbitrary constants where q is the strength of the
set of Q coincidences c1 , ..., cQ .


Reception of Fields’ work on algebraic functions I

G. Landsberg, who had also undertaken work to ﬁnd new algebraic
proofs of the Riemann-Roch theorem, in reviews in the Jahrbuch
uber die Fortschritte der Mathematik, a reviewing and abstracting
¨
journal based in Germany, stated that he had reservations about
Fields’ earlier works on algebraic function theory, particularly with
regards to simplifying assumptions that were made based on
geometric arguments — that is, Fields’ approach was not algebraic
enough for Landsberg


Reception of Fields’ work on algebraic functions II

Given that much of Fields’ later writings were reworkings of various
parts of his 1906 monograph, we can surmise that the reception of
his monograph was lukewarm.


Reception of Fields’ work on algebraic functions III

Consider Fields’ paper of 1910 entitled “The Complementary
Theorem” which appeared in the pages of the American Journal of
Mathematics. G. Faber of the University of K¨nigsberg, in his
o
reviewing the paper in the Jahrbuch wrote that “the paper
purports to give a proof of the so-called ‘Weierstrass Preparation
Theorem’ that the author gave in the 11th chapter of his Theory
of Algebraic Functions, by a shorter and simple one,” however “the
proof still seems to me long and hard to understand.”


Reception of Fields’ work on algebraic functions IV

In another review, on Fields’ paper entitled “Direct derivation of
the complementary theorem from elementary properties of the
rational functions,” which was published in the proceedings of the
fifth International Congress of Mathematicians in 1913, Prof.
Lampe of Berlin writes, after quoting Fields’ own introduction to a
paper in the Philosophical Transactions of the Royal Society where
Fields’ claims to have achieved simplification, that “perhaps he
[Fields] could try for even more simplification.”


Decline in research productivity

Fields’ research productivity started to subside as he spent more
and more time as a scientiﬁc organizer in the 1920s and as his
health began to deteriorate.


John Charles Fields, 1863-1932.
Fields’ life ended on August 9th, 1932, apparently from stroke. He
is buried in Hamilton Cemetery which overlooks the western end of
Lake Ontario (at “Cootes Paradise” where McMaster University
now sits).

Figure: Fields’ gravestone:“John Charles Fields, Born May 14, 1863, Died
August 9, 1932.”


Fields’ estate

He left an estate of $45071, a large part of which was used toward
what would become the international medals in mathematics he
was in the process of organizing. He left his brother with a small
annuity, and his maid Julia Agnes Sinclair, widow, a small pension
as long as she remained unmarried.


The impact of Fields’ mathematical work

Fields’ papers are badly written in the sense that they are hard
to follow.



to follow.
So who exactly read his work?



to follow.
So who exactly read his work?
Clearly, his student S. Beatty (1881-1970) read his work;
those who reviewed Fields’ work must have read portions of it;
the American mathematician Bliss mentions Fields’ book in
the preface of his book on algebraic functions, but does not
use Fields’ approach in his book.


The impact of Fields’ mathematical work II

His work was well regarded, as can be seen by his election to
the Royal Society in 1913 and to other societies and
academies.



academies.
Fields’ work was “old fashioned”, did not afford easy
generalization, and would be subsumed by approaches utilizing
the machinery afforded by modern abstract algebra, so the
ultimate influence of his work is small, maybe mostly
expressed through the work of Fields’ student S. Beatty.



academies.
Fields’ work was “old fashioned”, did not afford easy
generalization, and would be subsumed by approaches utilizing
the machinery afforded by modern abstract algebra, so the
ultimate influence of his work is small, maybe mostly
expressed through the work of Fields’ student S. Beatty.

However, consider this...


The impact of Fields’ mathematical work III

The research legitimized Fields as a scientiﬁc authority.



Fields’ authority surely must have played a key role in the his
push, along with others, to get governmental support for
scientiﬁc research, a goal that would eventually come to
fruition, as can be seen in the funding structures that exist
today.



Fields’ authority surely must have played a key role in the his
push, along with others, to get governmental support for
scientiﬁc research, a goal that would eventually come to
fruition, as can be seen in the funding structures that exist
today.
According to S. Beatty, writing around 1930, Fields “by his
insistence on the value of research as well as by the
importance of his published papers, has, perhaps, done most
of all Canadians to advance the cause of mathematics in
Canada.”


Thank you for your attention

Thank you for your attention!


Fields’ love affair

[The Star, August 10, 1932: “Death Claims Noted Savant at
University”] “Dr. Fields was a bachelor, and an amusing story is
told of his ‘love affairs’ abroad. On one occasion, while abroad, he
arranged to meet Professor Love at a certain hotel. When Dr.
Fields arrived there, the girl clerk presented him with a telegram
reading: (Sorry, I cannot meet you, Love). The doctor treasured
this evidence of his ‘love affairs.’”


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