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4
Core Mathematics
A "stateoftheart" report for core mathematics will not be
attempted here. It would be almost impossible to prepare a report
meaningful to nonmathematicians, and a report of value to the
mathematical community would require an extraordinary effort
by a very large number of people. Instead we attempt in this chapter
to describe in rather general terms some aspects of core mathe
matics as it appears to some of its practitioners. The essays on
selected topics in mathematics, which appear in a separate volume,7
serve as an elaboration and illustration.
We organize our remarks around the traditional subdivisions of
core mathematics. It should be remembered, however, that the
boundaries between these fields have become much less precise than
they used to be, and that some of the most important developments
are occurring precisely at the interfaces between different fields or
in areas that cannot be accommodated within the existing classifica
tion. As a matter of fact, the confluence of mathematical ideas and
the interpenetration of the various fields are the most character
istic and most rewarding features of contemporary core mathematics.
A sketchy presentation couched in nontechnical language can con
vey only a pale reflection of the excitement felt by those who par
ticipate in these developments.
MATHEMATICAL LOGIC
Logic is one of the most ancient mathematical sciences, since it is
at least as old as Aristotelian philosophy and Euclidean geometry.
57
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The State of the Mathematical Sciences
However, mathematical logic, also called symbolic logic, really
developed in the second half of the ninteenth century, when it was
observed that familiar logical operations can be expressed by alge
braic formulas, and when it seemed possible to define such basic
mathematical concepts as the natural numbers 0, 1, 2, 3, . . . in
purely logical terms.
A major reason for the flowering of mathematical logic lies in the
~ ~;~ ~£ ^~ ~ ^~ ~ r ~
If =' ~11~ L`~ly ~' ~1111~11~e ants, During tne same period, by
Georg Cantor. This theory influenced all mathematics. It also led
to the discovery of paradoxes, nonsensical conclusions reached bv

seemlngly 1egltlmate reasoning processes. Some of these new para
doxes~ are similar to ones encountered by ancient Greek philos
ophers,l which Aristotle disposed of by what we now consider
insufficient arguments.
Since unfettered reasoning had led to paradoxes, mathematicians
recognized that the processes of logical thought require regulation.
Thus, it should be possible to reduce any legitimate piece of mathe
matical reasoning to manipulation of symbols using only stated
axioms and stated rules of inference. So reduced, the reasoning is
said to be "formalized." Of course, formalization in itself provides
no warranty that paradoxes will not arise (as indeed they might if
the axioms and rules have been badly chosen). But without formal
ization we cannot deal in a precise way with the problem of suitably
regulating mathematics.
For many years it was the hope of mathematicians that all mathe
matical reasoning could be formalized in one logical system. A
remarkable discovery by Godel in 1931 shattered this hope, how
ever, and had farreaching repercussions among the philosophers.
What Godel established was "incompleteness" for certain particular
formal systems ("Principia Mathematica" of Whitehead and Rus
sell, and related systems). But the incompleteness was soon recog
nized as applying to every formal system that is correct (only true
Cat + ^ + ~ ~ ~ ~ ^ ^ 1_ A ~ ~ 1 _ \ _ __ 1 _ ~ · _ . ~ · ~
;)LO~L~111~11~ OC1118 ~lUV~Ol~J ana sumclently rlcn to include ordinary
arithmetic. In any such system, there are statements, even about
whole numbers, that are true but cannot be proved within the sys
tem.
~ Here is a sample. The expression, "the least natural number not nameable in
fewer than twentytwo syllables," names in twentyone syllables a natural number
which by its definition cannot be named in fewer than twentytwo syllables.
(G. G. Berry, 1906)
tX says, "I am lying." Is X telling the truth? (Eubulides, fourth century B.C.)
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59
This generalized form of Godel's theorem rests on accepting a
precise analysis of what an "algorithm" or "preassigned computa
tional procedure" is in general. Such an analysis, accepted by most
logicians, was given in 1936 by Church and independently (in an
equivalent form) by Turing. Previously, mathematicians had dealt
only with many particular examples of algorithms, some discovered
by Euclid or before.
Church and Turing each used their analysis to exhibit an infinite
class of questions with the property that no algorithm suffices to
answer correctly all the questions of the class. No matter how long
and hard mathematicians may have worked at devising methods for
answering questions of the class, there will be some questions that
can be answered only by inventing still further methods. Most re
cently (since 1948), it has been shown that such classes of questions
(sometimes called "unsolvable problems") arise in other parts of
mathematics.
One of Cantor's first achievements was the proof that there are
"more" points on a straight line than there are natural numbers; it
is impossible to match up all the points on a line with the natural
numbers 0, 1, 2, 3, .... Cantor conjectured that any set of points on
the line either can be rearranged into a simple sequence or else
contains "as many" points as there are on the whole line. This con
jecture, called the continuum nypotnes~s, was considered one or one
main unsolved problems in mathematics. In 1939, Godel proved
that the continuum hypothesis cannot be disproved by the methods
used in contemporary mathematics and formalized in existing sys
tems of set theory. About a quarter of a century later, P. Cohen
showed that the continuum hypothesis cannot be proved by such
methods either. For further information see Smullyan's essay in
reference 7.
What has been said up to now may convey the impression that
mathematical logic is a highly abstract subject, related to philosophy
and much too esoteric even for the taste of most mathematicians.
This is indeed the case. Yet this most austere of all mathematical
.
· ~ .~ · · ~ ~ ~ .~
disciplines has made a contribution to America's fastestgrowing
industry. By reducing mathematical reasoning, and in fact all
deductive reasoning, to purely mechanical manipulation of symbols
according to definite rules, mathematical logic prepared for the
development of digital computers. Indeed, a mathematical theory
has been completely formalized if any argument in it could in prin
ciple be checked by computer. It is no accident that two mathe
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The State of the Mathematical Sciences
maticians (don Neumann and Turing) steeped in mathematical
logic participated in the development of modern computers. The
recognition, by Von Neumann, that Turin~'s work implies the
t~fic~:i~ilit~r ^ ~1~;~r=~1 ~.^A ~Am_ JO
'~ ~ A A. _ ~ ~ ^ _
' ~ up computer saved years and millions of
dollars in development.
The interplay between mathematical logic and computing con
tinues. Training in mathematical logic is often a way to creative
work in computer sciences, and the problems on which computer
scientists and logicians work sometimes have certain features in
common. Logicians, for instance, are interested to know whether
certain problems can be solved by algorithms. Computer scientists
ask the same question, except that they want to know whether cer
tain problems can be solved by algorithms in a reasonable time on
existing or projected machines, and, if so, how this can be done in
the most economical way.
The difficulties in the foundations of mathematics that led to
the development of mathematical logic have not yet been wholly
resolved. There are various schools and points of view. For example,
in formulating existence proofs one school, the constructivists,
recognizes as valid only proofs in terms of algorithms, while another
school, the formalists, will accept a proof by contradiction.
NUMBER THEORY
Number theory is perhaps the oldest mathematical discipline. Posi
tive integers appear to have been invented along with language,
and the most ancient civilizations that left written records were
fascinated by properties of integers. Problems in the theory of num
bers can often be stated in a surprisingly simple manner, under
standable to every schoolboy. Many of them deal with prime
numbers, those integers greater than 1 that are divisible only by
themselves end by 1 (2, 3,5, 7, 11, 13, 17, 19, 23, 29, 31, 37,.. .~.
Here is a famous problem: Is every positive even integer a sum of
two primes? The evidence seems to be all for it, as was first observed
by Goldbach in the early eighteenth century. Yet the Goldbach
problem is unsolved. It has been proved only that every integer is
the sum of, at most, 19 primes, and that there is a large number
. .
such that every odd integer exceeding it is the sum of three primes.
The fascination that number theory exerts on mathematics is
twofold. First, there is the deep though seldomverbalized conviction
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61
that in studying natural numbers we are investigating something
imperishable and eternal. We cannot conceive of intelligent thought
anywhere in the universe that would not arrive at natural numbers
as we know them, and an unshakable evidence of an extraterrestial
intelligence would be a communication containing the sequence
2, 3, 5, 7, 11,
Another reason for the fascination of number theory is that its
problems, simple as they sound, often require for their solution the
most elaborate and sophisticated techniques available to mathe
matics. A characteristic phenomenon of recent decades is the appli
cation of probability theory to number theory and, very recently,
the applications of mathematical logic.
Modern number theory is concerned not only with integers but
also with socalled algebraic Integers, that is, with numbers that
can be roots of algebraic equations with integer coefficients and
highest coefficient 1.48 In studying algebraic integers one often uses
a geometric language hardly distinguishable from that of algebraic
geometry (see below).
Number theory has always had an experimental aspect. Interest
ing properties of numbers can be discovered by experimentation,
though the truth of a proposition about infinitely many numbers
can never be established by verifying individual instances, however
many. The celebrated primenumber theorem, for instance, asserts
that the number of primes between ~ and a larger number x is
approximately
x
1 + 2 + 3 + 4 +
x
This was perceived by Gauss, who had computed long tables of
primes. The proof came much later and required, in its original
form, arguments from the theory of functions of complex variables.
Its experimental aspect is one reason why Lumber theory is amon
those disciplines of pure mathematics that feel the impact of the
computer revolution. The computer opened up new possibilities
for numerical experimentation. Number theorists use computers
not only to guess theorems but also to establish results. Numerous
plausible conjectures have been refuted by computer work, and
~ For instance, 1  ~ and 1 + ~ are algebraic integers since both numbers
are roots of the equation x" 2x  1 = 0.
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The State of the Mathematical Sciences
D. H. Lehmer used computers to prove theorems requiring too great
a ramification into cases for conventional methods of proof.
COMBINATORIAL ANALYSIS
Combinatorial analysis is concerned with a variety of "combi
natorial" problems, many of which amount to: "In how many ways
can one perform a certain task?" The problems are usually easy to
state and may be very difficult to solve. They often have important
consequences for physical theory. We give an example:
Consider a checkerboard with n squares on each side, where n is
an even number. We ask for the number In of different ways in
which the board can be covered by dominoes, a domino covering
two squares. A complete solution to this problem is not known at
the present time; but it was recently found that, for large n, No is
equal to (1.338515 . . .~`n2~. This result has significant implications
for statistical mechanics.
One of the most famous unsolved problems of mathematics, the
fourcolor problem, is combinatorial. This conjecture states that
any planar map can be colored with four colors in such a way that
any two adjacent countries have different colors. In spite of the
vigorous efforts of many talented mathematicians, this conjecture
has not yet been settled; a relatively easy proof shows that five
colors suffice.
Many of the developments of modern science and technology are
stimulating a renewed interest in combinatorial analysis. Problems
are arising from genetics, biochemistry, statistics, coding of infor
mation transmitted to and from orbiting satellites, efforts to design
efficient circuits for electronic computers, and the development of
programming languages. (See Rota's essay in reference 7.)
ALGEBRA
The three major traditional subdivisions of mathematics are alge
bra, analysis, and geometry. Most people, upon hearing the word
"algebra," remember the subject studied in high school and think
of problems involving the solution of linear and quadratic equations.
Those parts of algebra are actually 4,000 years old, and the language
of algebra, that is, the use of variables (letters) to represent num
bers, dates from the Renaissance. Modern algebra is largely a devel
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63
opulent of the last 40 years. Yet its roots go back to earlier times.
Some of the essential ideas, for instance, appeared about 1830 in
the work of Evariste Galois, a man who was persecuted by his gov
ernment for his radicalism and by his teachers for his impertinence
and who was killed in a duel at the age of 21. Modern algebra began
to blossom and to reach maturity as a result of the efforts of a
number of algebraists both in this country and abroad, inspired in
part by the work of a brilliant woman mathematician, Emmy
Noether. Its techniques and mode of thought play an important role
not only in other mathematical disciplines but also in various areas
of the biological, physical, and social sciences.
Modern algebra is concerned with the study of socalled "alge
braic structures." Most of these may be considered as generalizations
of the basic structures of elementary mathematics: the system of
integers, the system of rational numbers, the system of real numbers,
the system of complex numbers, and the system of vectors (arrows)
in space.
We begin with an example of a structure. The elements in the
system of rational numbers are positive and negative common frac
tions and integers, and the number 0. Among these elements there
are defined two operations, addition and multiplication. Any set
of elements for which two operations, called addition and multipli
cation, are defined, obeying the same formal rules as in the case of
rational numbers, is called a field. For instance, the real numbers
form a field and so do complex numbers. Galois discovered the
existence of finite fields, that is, fields with only finite numbers of
elements. For instance, if p is a prime number, there is a field Zp
consisting of p elements, the numbers 0, 1, 9, . . ., p1. Addition and
multiplication in this field are defined by the prescription: add and
multiply in the usual way, then divide by p; the remainder is
called the sum or the product.
(For instance, in Z3 we have
1 + 1 _ 2, 1 + 9 _ 0, ~ + 2 1, 2 X ~ _ 1.) There are many
other fields; for instance, all rational functions of one variable with
complex coefficients form a field.
The advantage of considering systems about which we assume
nothing except the laws of addition and multiplication is that any
statement proved about them applies immediately to a wide variety
of fields in various branches of mathematics. The same is true about
other algebraic structures. In each case the central problem is to
describe the structure of the general system in terms of the struc
ture of particular examples of the system.
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We will review briefly some of the subdivisions of modern
algebra.
Field Theory
The definition of a field was given above. The modern theory of
fields is concerned with the properties of a field K formed from a
field F by adjoining to it the roots of polynomial equations with
coefficients in F. This theory, the modern development of Galois'
original work, is one of the most developed and fruitful parts of
algebra.
Ring Theory
A ring is a set of elements in which two operations, addition and
multiplication, are defined, and these operations satisfy the same
conditions as in a field, with two important exceptions: it is not
required that division by an element distinct from O be always
possible, and it is permitted that the product of two elements de
pend on the order in which we multiply them. (The latter means
that, in a ring, ab and ba may be different.)
An example of a ring is the ring of all integers. Here division is
not always possible, but the product of two elements is independent
of their order. Such a ring is called commutative. Another example
is the ring of all matrices with n rows and n columns. This is a
noncommutative ring since, if a and b are two matrices, it may
happen that ab and ba are different.
Rings arise naturally in many areas of mathematics. In analysis
(see below) many common sets of functions form rings, and the
results of ring theory contribute useful theorems in analysis.
Linear Algebra
During the nineteenth century, the theory of systems of linear
algebraic equations was codified into socalled linear algebra. This
part of algebra probably has more applications in other disciplines,
including engineering, physics, statistics, numerical analysis, and
social sciences, than any other. It is convenient to clothe linear
algebra in a geometric language. The basic concept is that of a
vector space over a field.
The standard example of a vector space is the set of ordinary
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65
geometric vectors, as used in physics. The elements may be thought
of as arrows in ordinary space drawn from a fixed point. Two vec
tors can be added by the "parallelogram law." A vector can be
multiplied by a real number: the product of a vector by a positive
number a is obtained by leaving the direction of the vector un
changed and multiplying the length of the vector by a; the product
of a vector by a negative numbera is obtained by reversing the
direction of the vector and multiplying the length by a; the product
of a vector by 0, finally, is the socalled null vector of length 0. This
vector space is said to be "over the real numbers," since vectors are
multiplied by real numbers.
A vector space over a field F is a set of elements called vectors for
which one has defined two operations: addition of two vectors and
multiplication of a vector by an element F. It is required that the
same formal rules should hold as in the example above.
A vector space over F is said to have dimension n if, roughly
speaking, a vector can be specified by giving n elements of F. The
space of geometric vectors, for instance, has dimension 3 since, if
we introduce a coordinate system, each geometric vector can be
expressed by three real numbers. Finite dimensional vector spaces
occur in many parts of engineering, physics, statistics, numerical
analysis, and the social sciences.
Linear algebra is the study of finite dimensional vector spaces.
It is particularly concerned with linear transformation, those map
pings of a vector space into itself that preserve addition and multi
plication. Any linear transformation can be represented by a matrix
(square array of numbers, or of elements of F), so that matrix alge
bra can be used to study the theory of linear transformations. The
results of linear algebra that do not depend on the finite dimension
of the vector space have had wide applications in functional analysis
(see below).
A 1lge bras
An associative algebra is a finite dimensional vector space which is
also a ring. The study of such algebras began with the quaternions
of Hamilton (1843~. The first definitive results were the funda
mental structure theorems proved by Wedderburn in 1907. In recent
years, these results have been subsumed in results on the structure
of rings. An associative algebra is called a division algebra if divi
sion by nonzero elements is always possible. Division algebras are
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.
The State of the Mathematical Sciences
the building blocks in the theory of algebras. Their structure has
been extensively studied, but a complete structure theory has not
yet been given.
A nonassociative algebra is a structure obeying the same rules as
an algebra, except that the product of three elements may depend
upon which two one multiplies first. Two classes of such algebras
are of main importance. The first are Lie algebras, in which any
three elements x,y,z satisfy the identity xy + yx x(yz) + you)
+ zany)  0. Such algebras are fundamental for the study of con
tinuous groups and, in recent years, have played art important role
in theoretical physics. The second class is that of the Jordan alge
bras, originally suggested by quantum physics. These are character
ized by the identities xy yx _ x~yx2) (xylem  0.
Group Theory
Groups are among the simplest arid most important algebraic struc
tures. One of the most elementary examples of a group is the set
of all positive rational numbers ire which one considers only the
operation of multiplication. A group is a set of elements in which
one operation, called group operation or group multiplication, is
defined; this operation must obey the same formal rules as ordinary
multiplication of rational numbers, except that we do not require
the commutative law (that is, ab arid ba need not be the same).
The integers form a group if the group operation is the ordinary
addition; this group is infinite (it contains infinitely many elements)
and commutative. A somewhat more complicated example is the
finite group Sn consisting of all permutations of the first n integers.
The group composition of two permutations is obtained by carry
ing out the first permutation and following it by the second. If n is
greater than 2, Sn is not commutative.
A basic problem of group theory is to determine the structure of
all finite groups. Since the building blocks out of which all fruits
groups are constructed are the socalled finite "simple" groups, an
# We give, for the sake of completeness, the precise definition of a "simple"
group. Let G and H be finite groups and let f be a rule that assigns to every
element g o:E G an element ~ _ f(~) of H. If this rule preserves the group
operation, that is, if f(g~g2) fight foggy, and if every element of [I is assigned
to some element of G. then f is called a homomorphism of G onto H. The group
G is "simple" if every homomorphism is either a homomorphism onto a group [1
having as many elements as G or a homomorphism onto a group G consisting of
a single element.
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67
important part of the problem is the determination of the struc
ture of these groups. There has been a recent breakthrough in the
subject with the proof (by Thompson and Feit) that if the number
of elements in a finite simple group is not a prime, it must be even.
This result has already had farreaching implications and lends
strong support to the hope that in the not too distant future the
structure of all finite simple groups will be explicitly determined.
Groups are of fundamental importance in many mathematical
investigations and in many applications of mathematics. Some
people believe the underlying reason for this is that groups give a
precise mathematical meaning to the important geometric concept
of symmetry. (A brilliant exposition of this point of view is found
in Hermann Weyl's book.~°)
Galois used finite groups to study the solvability of algebraic
equations by radicals, that is, by formulas involving only arithmetic
operations and root extractions. He showed that an equation can
be so solved only if its roots exhibit a certain symmetry. This
symmetry is always present in equations of degree 2, 3, and 4 but
not, in general, in equations of higher degree. In particular, there
is no general formula, involving radicals, for solving a fifthdegree
equation. (This epochmaking result was established before Galois
by Abel, another mathematical genius who died at an early age.)
The theory of groups is of importance in chemistry and physics.
The theory of finite groups has applications in crystallography, in
the theory of inorganic complex ions, and in spectroscopy. During
the 1920's and 1930's infinite groups were used in quantum
mechanics. Recent developments at the forefront of physics research
are intimately tied with the concept of symmetry and hence with
groups. As a matter of fact, physicists now use quite sophisticated
results from group theory. For more details, see Dyson's essay in
reference 7.
Homological A [ge bra and Category Theory
Modern mathematics is characterized by an everincreasing range of
applications of algebra to other mathematical subjects (see Eilen
berg's essay in reference 7~. A particularly striking example is
topology, a branch of geometry concerned with qualitative rather
than quantitative aspects of shapes of geometric figures. In the early
1920's it was recognized, under the influence of Emmy Noether
especially, that the methods used by topologists are basically alge
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depends on the stimulus in an arbitrary way. In many applications
one develops a linear theory as a first approximation, and one is
then forced to proceed to a nonlinear theory. For instance, differen
tial equations describing the propagation of small disturbances in
air density (sound waves) are linear, while the equations describing
the propagation of strong waves are nonlinear. From the mathemati
cal point of view, the linear theory is always simpler.
The modern theory of ordinary differential equations includes
control theory, which is closely related to technical and engineering
applications. The fundamental problem is to control a system, such
as a satellite, by means of certain control mechanisms. It is often
desired to bring the system into a certain state (for example, to
guide a missile to a target) with a minimum amount of some quantity
(such as fuel>. This is a problem in "Optimal Control Theory."
Although control theory has developed mainly during the past ten
years, it has attracted many research workers in both mathematics
and engineering; there have been over 400 publications in this
field, including a number of books. This is a field to which the
Russians, led by Pontryagin, originally a topologist, have con
tributed a great deal.
One of the more classical developments in ordinary differential
equations is stability theory, on which much pioneering work was
done many years ago by Poincard and BirkhofI. Many of the prob
lems raised by them have recently been solved by mathematicians in
Russia (Kolmogorov, Arnold) and in this country (Moser).
Although the problem has its origin in astronomy (stability of
planetary orbits), the methods developed for its solution have appli
cations to highenergyaccelerator design and to the study of charged
particles in magnetic fields, like that of the earth. A typical problem
is to ensure the containment of charged particles for long times in
the narrow tubular vacuum chambers of an accelerator without
hitting the wall. Some of the results of these studies have been
applied to the orbit theory of artificial satellites near the ablate
earth.
Of all the subfields of ordinary differential equations, the theory
of structural stability of vector fields (systems of differential
equations) on manifolds (i.e., surfaces and their generalization to
higher dimensions) is perhaps of interest to the greatest number of
mathematicians in other fields. A vector field is said to be struc
turally stable if the general shape of the solution curves remains
unchanged if the vector field is perturbed slightly. In 1959, all
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The State of the Mathematical Sciences
structurally stable systems on ordinary surfaces were classified,
and it was proved that any vector field on such a manifold can be
approximated arbitrarily closely by a structurally stable one. But
Smale showed that this approximation theorem is not true on mani
folds of dimension greater than three. Whether it holds on mani
folds of dimension three is unknown at present. It is hoped that
continued work will lead to a general theory of the classification of
vector fields.
The recent theory of linear partial differential equations profited
from the creation of a new language, the language of "distributions"
or generalized functions introduced by L. Schwartz. An example of
a distribution is the famous delta function, 8(x), of Dirac, which
takes on the value O for all values of x 7~ 0, but is such that
r+oo
~00
8(x~dx 1.
Mathematicians originally did not recognize this as a legitimate
function, but in the late 1940's Schwartz developed a theory of
generalized functions, or distributions, in which not only the Dirac
function but also much weirder entities occupy a legitimate place.
A new generation of mathematicians learned to look at partial
differential equations from the viewpoint of distributions, and they
have created a new branch of this old discipline that abounds in
beautiful and powerful results. Here is an example. Every linear
partial differential equation with constant coefficients, of the form
Lu f, where f is any function or generalized function, has solu
tions (EhrenpreisMalgrange). This theorem cannot be generalized
any further. If the equation does not have constant coefficients, a
solution need not exist. In fact, a few years ago, the experts were
surprised by Hans Lewy's discovery of a simple linear partial differ
ential equation that has no solution whatsoever.
Why was this surprising? Because traditionally mathematicians
took their partial differential equations from physics, and the
existence of the solution was more or less assured by socalled
physical intuition. The task of the mathematician was to justify the
intuition by a rigorous proof and to produce a method for actually
finding the solution. This work is certainly nearing completion as
far as linear problems are concerned.
But nature is not always linear. Compressible fluid flow, viscous
flow, magnetohydrodynamics and plasma physics, general relativity,
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and other disciplines challenge the mathematicians to solve non
linear problems. The theory of nonlinear partial differential
equations is at present a very active discipline. Important break
throughs have been achieved, but an immense amount of work
remains to be done.
Much of the modern work in partial differential equations looks
highly esoteric, and only a few years ago such work would have horn
~ A: ~ ~ ~_ ~, ~
J ~^ ~. FJ ~ ~ A
`;ons~aerea or no Interest for applications, where one wants a solu
. ~ . ~ . ~
r . me.
lion expressed In a workable form, say by a sufficiently simple
i:ormuia. l he advent of the modern computing machines has
changed this. If a problem involving a differential equation is
sufficiently understood theoretically, then, in principle at least, a
numerical solution can be obtained on a machine. If the mathe
matics of the problem is not understood, then the biggest machine
and an unlimited number of machinehours may fail to yield a
solution.
Another example from the theory of partial differential equations
is the index theorem. Suppose we want to solve a linear equation
Lu I,
where L stands for a complicated operator that may involve differ
entiations, integrations, etc., u is the unknown, which may be a set
of numbers, a function, or a set of functions, and f consists of given
data. In general, there will be a solution only if f satisfies a certain
number of conditions. Let us assume that this number is finite, and
let us call it A. Also, if the problem is solvable at all. the .~]l,rion
may not be unique; let us assume that the solution depends on B
parameters. The number ~B is called the index of the problem.
The simplest example is a system of n linear algebraic equations
~1 ~T ~At_ ~
~ _ _ _ Tt ~
In m unknowns. it can easily be shown that in this case the index
B is equal to nm. The index problem makes sense also when
the equations are differential equations, but until very recently it
could be solved only in relatively simple cases. This difficult prob
lem was generalized, attacked, and solved about three years ago by
Atiyah and Singer. The resulting formula expresses the difference
A B in terms of certain topological invariants of the region and
of the equations considered. The work involved not only some of
the most refined recent tools of analysis but also results and methods
from differential geometry and topology. It also turned out that
this index formula contained as special cases several important
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results in other fields of mathematics and their generalizations. It
is completely impossible to assign this beautiful achievement to any
one mathematical discipline. It belongs to analysis as much as to
geometry and topology, and it has strong connections with algebra.
Probability Theory
The theory of probability, which arose during the sixteenth and
seventeenth centuries out of rather frivolous questions of computing
odds in gambling, has had an explosive development during the
past few decades. On the one hand, this theory, which deals with
"mathematics of chance," has been put on firm foundations, which
makes it into a science as deductive and as rigorous as geometry
and algebra. On the other hand, its range of applications has grown
immensely. (See Kac's essay in reference 7 for an extended discus
sion of probability theory.)
Some 40 years ago the calculus of probability consisted essentially
of a collection of disconnected problems. Under the influence of
Gauss, undue importance was attached to the socalled theory of
errors, a topic now practically forgotten. Instead, probability has
moved in new directions by developing the theory of stochastic
processes and the modern fluctuation theory. The unexpected re
sults of these theories have introduced entirely new viewpoints and
attitudes and have also opened new avenues of research. However,
for the purposes of this report it is more important to stress that
these developments have broken the traditional isolation of prob
ability theory, with the result that it now plays an important role
in a surprising number of mathematical disciplines. Examples of
these disciplines are measure theory, function spaces, classical poten
tial theory, Hilbert spaces, information theory, and partial differen
tial equations. This list could be extended, but it should suffice to
add that probabilistic methods have been used for proofs in logic.
Some recent work promises to establish further new connections
between logic and probability.
The same intense and fruitful flow of ideas can be observed be
tween mathematical probability and various applications. Important
parts of probability theory were stimulated by technological prob
lems (information and prediction theory, theory of noise), and
modern probability is playing an important role in industry and in
quality control and reliability engineering, as well as in natural
and social sciences. In addition, probability forms the basis of the
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modern theory of statistical inference (see Kiefer's essay in refer
ence 7).
GEO METRY
Three turning points in the history of geometry determined the
course of mathematics and exerted influences beyond the boundaries
of mathematics and even of science in general. The first occurred in
Greece, when the collection of geometric facts and rules assembled
in the ancient civilizations of the East was transformed into a
deductive science. This event signified the birth of deductive reason
ing. The second event was the discovery of analytic geometry by
Fermat and Descartes, who showed how points can be represented
by pairs or triples of numbers and geometric figures by algebraic
equations. This discovery bridged the chasm between geometric
and arithmetic concepts that dominated Greek mathematics and
prepared for the invention of calculus. The third event was the dis
covery of nonEuclidean geometry by Gauss, Bolyai, and Lobachev
sky (and in a somewhat different context by Riemann). This showed
that the axioms of Euclid are not selfevident truths as had been
assumed for centuries, and that consistent geometries built on other
axioms are possible (see Coxeter's essay in reference 7~. The results
of this discovery for human thought were stupendous. Only the
consequences for mathematics are of interest to us here. The
axiomatic method that pervades all modern mathematics is one of
these. So, in a certain sense, is mathematical logic and the use of
formal languages. Sc' is, finally, the whole proliferation of various
geometries in which one begins by assuming only some of the
familiar properties of the space of everyday experience and then
proceeds to develop their logical consequences.
It would seem that by permitting itself this unlimited freedom
and by cutting, as it were, the umbilical cord tying geometry to its
early experimental origins, this science would lose its connection
with reality and would cease to be a tool of science and technology.
It is well known that the opposite has taken place.
We will review briefly some subdivisions of modern geometry.
Algebraic Geometry
Today, most high school students get a taste of this discipline when
they study conies (ellipses, parabolas, and hyperbolas). These
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The State of the Mathematical Sciences
curves, known to the ancient Greeks, are geometric embodiments
of simple algebraic equations. An ellipse, for instance, consists of
all points in the plane whose coordinates x and y satisfy an equation
of the form (xia)2 + (y/b)2 = 1, where a and b are two numbers.
Algebraic geometry is concerned with solutions of systems of
algebraic equations using a geometric language. A basic aim is to
achieve a full understanding of the totality of solutions. In order
to obtain this, it is necessary to extend the scope of the algebra and
the geometry involved. Modern algebraic geometry studies equations
whose coefficients are not only ordinary real or complex numbers,
but also elements of more general fields or rings. It considers the
geometric configurations defined by such equations not only in the
plane or in ordinary threedimensional space but also in higher
dimensional spaces.
One of the attractions of algebraic geometry is its intimate con
nection with number theory; intricate algebraic and geometric
constructions are sometimes used to obtain numbertheoretic re
sults.~ Algebraic geometry is now flourishing and attracting some
of the most talented young men entering mathematics. We mention
two recent developments.
An outstanding problem was whether, given k n independent
algebraic equations in k unknowns, it is possible to represent all
solutions of this system by a smooth geometric figure. (More pre
cisely, can one transform the variety defined by this system into a
smooth figure by using socalled birational transformations? The
affirmative answer for n 1 was found in the nineteenth century.
For n I_ ~ the first purely algebraic proof for systems with coe~
cients in the complex numbers was given by Zariski about 30 years
ago. Ten years later, he solved the case n _ 3. Recently, the gen
eral case (any n) was settled affirmatively by Hironaka.
Another achievement of a quite different nature is the systematic
rebuilding of the foundation of algebraic geometry now led by
Grothendieck in France. His work, which also leads to solutions of
important concrete problems, has influenced many young mathe
maticiarls, including those working in other fields.
~ For instance, "Fermat's last theorem" (a statement made in the seventeenth
century and thus far proved only in certain special cases) can be interpreted as
follows: let It be an integer greater than 2 and consider the algebraic surface
x. + y" = zip in the x,y,z space; are there points on this surface with integral
coordinates other than the three points (O. O. 0), (1, O. 1), (0, 1, 1~? Fermat
claimed there are not.
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Differential Geometry
79
lust as the analytic geometry of Fermat and Descartes and its sub
sequent development into algebraic geometry may be described as
applications of algebra to geometry, so differential geometry may
be described as the application of calculus to geometry. It reached
maturity only when mathematicians learned to live in spaces of
more than three dimensions. At this level of abstraction one can
not draw figures but must work with formulas, and the formulas
~nemse~ves become so complicated that the invention of a con
venient notation is a major task. This task was accomplished by
Italian geometers toward the turn of the nineteenth century. The
"tensor analysis" or "Ricci calculus" created by them seemed at
that time an example of monastic mathematics at its most esoteric.
But when Einstein was searching for a mathematical language to
express ideas of general relativity, he was led, after much inner
resistance, as he himself stated, to tensor analysis.
The unexpected application of differential geometry in relativity
led to a flowering of this discipline that continues today. The
original work in differential geometry was primarily "local"; one
studied a limited, usually very small, part of the space under con
sideration. In modern differential geometry one studies "global"
properties, that is, properties of the space as a whole, and one is
particularly interested in the connection between local and global
properties.
. ~ ~ ~
Point Set Topology
This branch of geometry is first of all the study of very general
geometric figures in ordinary space. Classical geometry, from the
Greeks until the nineteenth century, confined its studies to rela
tively simple figures straight lines, triangles, circles, ellipses, and
the like. Set theory led mathematicians to consider also "wild"
figures curves without tangents, curves that pass through every
point in a square (socalled Peano curves), and so forth. When this
kind of mathematics was born, some mathematicians, including
even as great a man as Poincard, dismissed it as a study of freaks
_ · ,
it appeared, however, that the freaks that Poincare thought were
invented by people looking for ugliness appear by themselves in
connection with beautiful classical objects (this was observed in
the theory of socalled Kleinian groups initiated by Poincard him
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The State of the Mathematical Sciences
self) and that further investigations reveal beauty and harmony in
what first appeared to be a world of chaos.
Another aspect of point set topology is the logical analysis of our
most basic ideas about space.
Point set topology played a special role in the development of
mathematics in this country, since the unique pedagogical work of
R. L. Moore (University of Texas), which is responsible directly or
indirectly for so much research activity in mathematics, has been
centered about point set topology. (For further information about
point set topology, we refer to the essay by Bing in reference 7.)
Algebraic Topology
Topology is concerned with the study of shapes of geometric figures
and with qualitative properties of continuous transformations be
tween such figures. The original names of the discipline were
analysis situs and later combinatorial topology. In studying quali
tative aspects of geometry, one considered first figures composed of
the simplest conceivable building blocks points, straight segments,
triangles, tetrahedra, for example. Brouwer, one of the founders of
topology, named these building blocks "simplexes." Certain basic
figures studied in algebraic topology are called "complexes." They
are composed of simplexes in such a way that two simplexes that
touch each other have a whole face in common. In studying com
plexes one pays no attention to their size or shape but only to the
way the various simplexes are combined. Hence the name "combi
natorial topology." It was changed to "algebraic topology" when it
became apparent that the progress of. the discipline depended
essentially on the use of sophisticated algebraic tools. Today there
is available a variety of algebraic aids for the study of topology. In
fact, some parts of topology have been completely reduced to alge
bra, and although the resulting algebraic problems have not been
solved completely, they are theoretically solvable in the sense that
their effective computability has been established.
We shall describe one basic problem, on the boundary between
point set topology and algebraic topology. Does the method of repre
senting a geometric figure as a complex really capture all the geo
metric properties we want to capture? More precisely, suppose we
are given two complexes that are topologically equivalent, that is,
such that one can establish a onetoone correspondence between
their points in which nearby points in one complex correspond to
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nearby points in the other. Can one then subdivide the complexes
into finer ones in such a way that the resulting subdivided com
plexes are "combinatorially equivalent," so that, under an appro
pr~ate onetoone correspondence, vertices of a simplex in one
correspond to vertices of a simplex in the other?
The answer may depend on the dimension in which one operates.
It is affirmative for dimensions 1, 2, and 3, but Milnor proved about
five years ago that it is, in general, negative. On the other hand, the
answer is probably affirmative for complexes that have a certain
smoothness property, in the technical language for manifolds. (The
affirmative answer for the case of threedimensional manifolds was
found by Poise in 1952. His proof was ingenious but elementary.
The affirmative answer for certain manifolds of dimension at least 5
has been obtained very recently by the 26yearold mathematician
Sullivan. He draws on many parts of the machinery developed by
topologists during the past 20 years. The case of four dimensions
appears today to be completely inaccessible.)
One of the most important applications of topology is to analysis,
in particular to solving differential equations. One link between
topology and analysis is established by the celebrated "Morse theory"
of critical points. The recent extension of this theory to infinite di
mensional spaces (by Smale and others? promises to he of .si~nifi
cance for solving nonlinear partial differential equations.
, ~
Differential Topology
III algebraic topology one considers complexes, that is, geometric
figures that can be decomposed into simple building blocks, sim
plexes. Another approach is to consider figures that can be described,
near every point, by a system of coordinates, like the ones used in
analytic geometry. An example is the surface of a inhere an which
· , , . ~
i.
1~ red VVAll~
points are 1ocatect by means of the geographic coordinates, latitude
and longitude. It is in general impossible to use one system of coordi
nates for the whole figure. In the case of geographic coordinates on
the sphere, for instance, there are "singularities" at the two poles.
It is required, however, that we should be able to cover the whole
figure by patches in which coordinates can be defined; and it is re
quired that the formulas describing the transition from one coordi
nate system to another should involve only smooth (differentiable)
functions. If a geometric figure can be covered by such coordinate
patches, it is called a smooth manifold; if a system of coordinate
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The State of the Mathematical Sciences
patches has been defined, we say that we have defined on our figure
a differentiable structure.
It had been believed for a long time that whether one does
topology of wellbehaved figures by complexes or by differentiable
manifolds is a matter of technical convenience. It was believed, for
instance, that there is essentially only one way in which a differ
entiable structure can be defined on a reasonable figure. The advan
tage of the approach through complexes lay in the possibility of
using algebra. The advantages of the approach through manifolds
~, ,
lay In the possibility ot using analysts, differentiation, and integra
tion. The celebrated theorems of de Rham, which showed that cer
tain numbers characterizing the shape of a figure from the point of
view of algebraic topology could also be arrived at by integral cal
culus, were an early triumph of this approach.
But differential topology as a fully independent discipline was
born only in 1952, with Milnor's remarkable discovery that even on
a figure as simple from the topological point of view as the seven
dimensional sphere, there is more than one way of defining a
differentiable structure. This discovery shared the fate of other
fundamental "counterexamples." At first it seemed that the socalled
"Milnor spheres" were freaks that their discoverer was ingenious
enough to construct, but which could not possibly occur in a reason
able mathematical context. Recently, however, it was discovered that
Milnor spheres appear inevitably in the theory of functions of sev
eral complex variables. Also, far from being ugly and chaotic, the
existence of different differentiable structures reveals a world of
hidden harmony and symmetry.
Differential topology has now become one of the most active
mathematical disciplines. It is, more than any other branch of
topology, connected with differential geometry, the theory of differ
ential equations, and algebraic geometry. In some sense the appear
ance of differential topology represents the return of topology to its
ordains in problems of mechanics and differential equations. Here
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teenthcentury Norwegian mathematician, Sophus Lie) geometry,
analysis, and algebra are inseparably joined. Lie groups are groups
in the ordinary sense that have the added feature of being param
eterized, at least locally, by real numbers. (An example is the
totality of all displacements of a rigid body. This is a group, since
a succession of displacements is again a displacement and every
displacement can be canceled by another. A displacement of a rigid
body can be described by six numbers, three to tell how some fixed
point in the body has been translated and three to tell how the
body has been rotated about this point. Thus the group of rigid
motions in ordinary space is a sixparameter Lie group.) This param
eterization has immediate and farreaching consequences, because
techniques from analysis become applicable and Lie's original con
cept, the infinitesimal group (now called Lie algebra), assumes a
central place in the theory. Geometric objects led naturally to a
rich supply of examples, and much effort was made to classify all
Lie groups in terms of these examples.
Lie groups play a dominating and unifying role in modern
mathematics; they have stimulated significant research in algebra
and topology, and such widely different fields as finitegroup theory
and differential geometry are today strongly influenced by Lie
group theory. The "representation theory" of Lie groups, which on
occasions has been the source of inspiration of important discoveries
in physics, has also suggested entirely new directions in analysis,
thereby incorporating and revitalizing large parts of classical
analysis.