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:
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
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
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
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
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.
