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4 Core Mathematics A "state-of-the-art" 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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58 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 far-reaching 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 twenty-two syllables," names in twenty-one syllables a natural number which by its definition cannot be named in fewer than twenty-two syllables. (G. G. Berry, 1906) tX says, "I am lying." Is X telling the truth? (Eubulides, fourth century B.C.)

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Core Mathematics 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 fastest-growing 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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60 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 seldom-verbalized conviction

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Core Mathematics 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 so-called 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 prime-number 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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62 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 four-color 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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Core Mathematics 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 so-called "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, . . ., p-1. 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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64 The State of the Mathematical Sciences 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 so-called 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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Core Mathematics 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 number-a 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 so-called 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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66 . 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 so-called 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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Core Mathematics 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 far-reaching 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 fifth-degree equation. (This epoch-making 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 ever-increasing 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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Core Mathematics 73 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 high-energy-accelerator 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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74 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 (Ehrenpreis-Malgrange). 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 so-called 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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Core Mathematics 75 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 machine-hours 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 n-m. 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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76 The State of the Mathematical Sciences 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 so-called 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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Core Mathematics 77 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 non-Euclidean geometry by Gauss, Bolyai, and Lobachev- sky (and in a somewhat different context by Riemann). This showed that the axioms of Euclid are not self-evident 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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78 ~- 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 three-dimensional 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 number-theoretic 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 so-called 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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Core Mathematics 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 (so-called 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 so-called Kleinian groups initiated by Poincard him

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80 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 one-to-one correspondence between their points in which nearby points in one complex correspond to

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Core Mathematics 81 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 one-to-one 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 three-dimensional 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 26-year-old 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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82 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 well-behaved 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 so-called "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 OCR for page 57
Core Mathematics 83 teenth-century 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 six-parameter Lie group.) This param- eterization has immediate and far-reaching 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 finite-group 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.