Is There Life After Calculus?
If you are a student who is taking a standard undergraduate calculus sequence, you may be wondering what comes next. Have you seen the best that mathematics has to offer? Or, as our title asks, is there (mathematical) life after calculus?
In fact, mathematics is a vibrant, exciting field of tremendous variety and depth, for which calculus is only the bare beginning. What follows is a brief overview of the modern mathematical landscape, including a key to the Cornell Mathematics Department courses that are scattered across this landscape. While current mathematics is organized into numerous disciplines and subdisciplines — The official Subject Classification Guide of the American Mathematical Society is almost 100 pages long! — most subjects fall into a modest number of major areas. For upper level and graduate courses, we use the middle digit of our course numbers to identify the area of mathematics to which the course belongs:
- 1 and 2 for analysis;
- 3 for algebra;
- 4 for combinatorics;
- 5 and 6 for geometry and topology;
- 7 for probability and statistics;
- 8 for mathematical logic;
- 9 for reading courses.
The digit 0 is used for various purposes not related to mathematics subject classification, such as mathematics education, the history of mathematics, and some elementary courses. But what are these areas about? The following descriptions will help you navigate the Mathematics section of the Courses of Study catalog and choose courses in mathematics that will serve you well.
Analysis is the branch of mathematics most closely related to calculus and the problems that calculus attempts to solve. It consists of the traditional calculus topics of differentiation, differential equations and integration, together with far-reaching, powerful extensions of these that play a major role in applications to physics and engineering. It also provides a solid theoretical platform on which applied methods can be built. Analysis has two distinct but interactive branches according to the types of functions that are studied: namely, real analysis, which focuses on functions whose domains consist of real numbers, and complex analysis, which deals with functions of a complex variable. This seems like a small distinction, but it turns out to have enormous implications for the theory and results in two very different kinds of subjects. Both have important applications.
The study of differential equations is of central interest in analysis. They describe real-world phenomena ranging from description of planetary orbits to electromagnetic force fields, such as, say, those used in CAT scans. Such equations are traditionally classified either as ordinary differential equations (if they involve functions of one variable) or partial differential equations (if they involve functions of more than one variable). Each of these two corresponds to an active subfield of analysis, which in turn is divided into areas that focus on applications and areas that focus on theoretical questions.
Algebra has its origins in the study of numbers, which began in all major civilizations with a practical, problem-set approach. In the West, this approach led to the development of powerful general methodologies. One such methodology, which originates with Euclid and his school, involves systematic proofs of number properties. A different methodology involves the theory of equations, introduced by Arab mathematicians ("algebra" itself has Arabic etymology). Modern algebra evolved by a fusion of these methodologies. The equation theory of the Arabs has been a powerful tool for symbolic manipulation, whereas the proof theory of the Greeks has provided a method (the axiomatic method) for isolating and codifying key aspects of algebraic systems that are then studied in their own right. A notable example of such fusion is the theory of groups, which can be thought of as a comprehensive analysis of the concept of symmetry. Group theory is an area of active research and is a fundamental tool in many branches of mathematics and physics.
The simplest and most widely known example of modern algebra is linear algebra, which analyzes systems of first-degree equations. Linear algebra appears in virtually every branch of applied mathematics, physics, mathematical economics, etc. Even though the theory of linear algebra is by now very well understood, there are still many interesting areas of research involving linear algebra and questions of computation.
If we pass to systems of equations that are of degree two or higher, then the mathematics is far more difficult and complex. This area of study is known as algebraic geometry. It interfaces in important ways with geometry as well as with the theory of numbers.
Finally, number theory, which started it all, is still a vibrant and challenging part of algebra, perhaps now more than ever with the recent ingenious solution of the renowned 300-year old Fermat Conjecture. Although number theory has been called the purest part of pure mathematics, in recent decades it has also played a practical, central role in applications to cryptography, computer security, and error-correcting codes.
Combinatorics is perhaps most simply defined as the science of counting. More elaborately, combinatorics deals with the numerical relationships and numerical patterns that inhere in complex systems. For a simple example, consider any polyhedral solid and count the numbers of edges, vertices, and faces. These are not random numbers; combinatorial analysis reveals their interrelationships. Practical applications of combinatorics abound from the design of experiments to the analysis of computer algorithms. Combinatorics is, arguably, the most difficult subject in mathematics, which some attribute to the fact that it deals with discrete phenomena as opposed to continuous phenomena, the latter being usually more regular and well behaved. Until recent decades, a large portion of the subject consisted of classes of difficult counting problems, together with ingenious solutions. However, this has since changed radically with the introduction and effective exploitation of important techniques and ideas from neighboring fields, such as algebra and topology, as well as the use by such fields of combinatorial methods and results.
Geometry and Topology
These two branches of mathematics are often mentioned together because they both involve the study of properties of space. But whereas geometry focuses on properties of space that involve size, shape, and measurement, topology concerns itself with the less tangible properties of relative position and connectedness.
Nearly every high school student has had some contact with Euclidean geometry. This subject remained virtually unchanged for about 2000 years, during which time it was the jewel in the crown of mathematics, the archetype of logical exactitude and mathematical certainty.
And then in the seventeenth century things changed in a number of ways.
Building on the centuries old computational methods devised by astronomers, astrologers, mariners, and mechanics in their practical pursuits, Descartes systematically introduced the theory of equations into the study of geometry. Newton and others studied properties of curves and surfaces described by equations using the new methods of calculus, just as students now do in current calculus courses. These methods and ideas led eventually to what we call today differential geometry, a basic tool of theoretical physics. For example, differential geometry was the key mathematical ingredient used by Einstein in his development of relativity theory.
Another development culminated in the nineteenth century in the dethroning of Euclidean geometry as the undisputed framework for studying space. Other geometries were also seen to be possible. This axiomatic study of non-Euclidean geometries meshes perfectly with differential geometry, since the latter allows non-Euclidean models for space. Currently there is no consensus as to what kind of geometry best describes the universe in which we live.
Finally, the eighteenth and nineteenth century saw the birth of topology (or, as it was then known, analysis situs), the so-called geometry of position. Topology studies geometric properties that remain invariant under continuous deformation. For example, no matter how a circle changes under a continuous deformation of the plane, points that are within its perimeter remain within the new curve, and points outside remain outside. For another example, no continuous deformation can change a sphere into a plane. So they are topologically distinct.
Topology can be seen as a natural accompaniment to the revolutionary changes in geometry already described. For, once one recognizes that there is more than one possible way of geometrizing the world, i.e., more than just the Euclidean way of measuring sizes and shapes, then it becomes important to inquire which properties of space are independent of such measurement. Topology, which finally came into its own in the twentieth century, is the foundational subject that provides answers to questions such as these. It is a basic tool for physicists and astronomers who are trying to understand the structure and evolution of the universe. Indeed, recent astronomical observations, together with basic results of topology, offer the exciting prospect that we will soon be in possession of the global topological structure of the cosmos.
Probability and Statistics
Everyone has had some contact with the notion of probability, and everyone has seen innumerable references to statistics.
The science of probability was developed by European mathematicians of the eighteenth and nineteenth century in connection with games of chance. Given a game whose characteristics were known, they devised a way of assigning a number between 0 and 1 to each outcome so that if the game were played a large number of times, the number — known as the probability of the outcome — would give a good approximation to the relative frequency of occurrence of that outcome. From this simple beginning, probability theory has evolved into one of the fundamental tools for dealing with uncertainty and chance fluctuation in science, economics, finance, actuarial science, engineering, etc.
One way of thinking about statistics is that it stands probability theory on its head. That is, one is confronted with outcomes, say, of a game of chance, from which one must guess the basic rules of the game. So, statistics seeks to recover laws or rules from numerical data, whereas probability predicts (within some margin of error) what the data will be, given certain rules.
The elementary theories of probability and statistics usually involve discrete models and make substantial use of combinatorics. More advanced parts of each subject rely heavily on real analysis, particularly the theory of integration and its offshoot, measure theory.
Mathematical logic has ancient roots in the work of Aristotle and Leibniz and more modern origins in the early twentieth century work of David Hilbert, Bertrand Russell, Alfred North Whitehead, and Kurt Gödel on the logical foundations of mathematics. But it also plays a central role in modern computer science, for example in the design of computers, the study of computer languages, the analysis of artificial intelligence.
Mathematical logic studies the logical structure of mathematics, ranging from such local issues as the nature of mathematical proof and valid argumentation to such global issues as the structure of axiom-based mathematical theories and models for such theories. One key tool is the notion of a recursive function, pioneered by Gödel and intimately connected with notions of computability and the theory of complexity in computer science.
In addition to its contribution to mathematical foundations and to computer science, mathematical logic and its methods have also led to the solution of a number of important problems in other fields of mathematics such as number theory and analysis.
For reasons of space, and because we wished to describe areas that are well-represented by the Cornell Mathematics Department, the foregoing has had to omit major aspects of mathematics, for example many important areas of applied mathematics. Nevertheless, our sketches do describe most of the significant areas of basic mathematics. We hope that they give you a helpful overview in your explorations of this exciting field.