| 2010 Mathematics Handbook
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2010 Mathematics Handbook |
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Branches of Mathematics
Pure Mathematics
Pure mathematics is mathematics which is
studied because of its intrinsic beauty and
usefulness within the subject, rather than
mathematical techniques (sometimes called
applied mathematics) which are developed to
attack specic problems arising outside the eld
of mathematics. Much pure mathematics was
developed completely without regard to its
applicability outside mathematics, but has since
proved to be absolutely indispensable in many
and varied applications, and underlies all applied
mathematics.
A degree with a focus on pure mathematics is an
excellent qualication for a career in teaching or
research, but also in many other domains. Taking
additional courses in applied mathematics,
computer science and statistics can open career
opportunities in government, insurance, banking
and communications. A degree grounded in pure
mathematics provides a good base for further
study towards a masters degree or PhD in
mathematics, or in other branches of the
mathematical and information sciences.
Pure mathematics may be classied broadly into
the areas of Algebra, Analysis, Combinatorics,
Geometry, Logic, Number theory and Topology.
There are many interconnections between these
areas and this adds to their beauty and strength.
Analysis is the subject that grew out of Newton’s
discovery of calculus, although concepts as
convergence and limit can be traced back to
Greek mathematicians of Antiquity, while the rst
works on innite series are due to Indian
mathematicians of the Middle Ages. Analysis
studies such topics as continuity, integration,
differentiability, including the study of ordinary
differential equations, partial differential
equations and probability theory. All these
subjects are critical to the applications of analysis
to physics, engineering, nance, statistics, biology,
genetics and almost anything that has a
quantitative component.
Algebra is concerned with the study of structure,
relation and quantity. It is a pure eld but has a
wide variety of applications, from understanding
the Rubik’s cube to classifying crystal structures
and designing algorithms. A recent powerful
application is to communications security: How
do you communicate securely over an insecure
network (eg. the Internet)? This problem has been
around in a simpler form for centuries and its
solution (found in the late 1970s) is used every
time you use your browser for secure
transmission, such as banking transactions. The
solution, part of what is now called public-key
cryptography, is described completely using
mathematical ideas which are presented in
MATHS 328. You can even easily make your own
code.
Topology is sometimes called rubber sheet
geometry, because it concerns itself with the
spatial properties that are preserved after shapes
are stretched or deformed without breaking. It
does not distinguish between a square and a
circle (as a rubber band circle can be stretched
into a square) and it ignores distances (so that
two different sized circles are equivalent in the
topological universe). Topology studies global
characteristics of shapes and surfaces and
quanties the differences algebraically, then uses
those algebraic tools to further explore these
characteristics and related ideas. The Poincaré
Theorem (a long standing conjecture whose last
case - in 3-dimensions - was proved by Grigori
Perelman) is one of the most famous topological
results. In a simplied version (from 1904) it
states that if any loop on the surface of a
3-dimensional shape can be shrunk to a point (as
a loop can do on the 3-D sphere), then the shape
is just a 3-D sphere. This theorem has
implications in a variety of elds such as
astronomy and relativity theory. Topology has
strong connections to abstract algebra, analysis
and geometry, and has applications to physics,
genetics (eg. understanding the knotting and
unknotting of DNA) and computer science. A
recent development, the topological quantum
eld theory, can be used for breaking
cryptographic systems based on integer
factorisation, widely used in banking encryptions.
“People say pure mathematicians are just playing
games with a bunch of rules“, says Prof. D.
Gauld, whose research topic is topology. ”The
amazing thing is that, so often, 10 or 50 years
later, these great applications arise. When I rst
heard about topological quantum eld theory, in
1994, there was no mention of their connection
with banking encryptions.”
Geometry arose as the eld of knowledge
dealing with spatial relationships. It was one of
the two elds of pre-modern mathematics, the
other being the study of numbers. It appeared
(more than 2500 years ago) as a collection of
techniques dealing with the lengths, angles,
areas, and volumes of physical objects, both on
earth and in the sky. Greek mathematicians
made it into a tool for developing logical
arguments, abstract reasoning and investigating
the nature of space and time. Euclid’s Elements is
the most famous geometry book of the Antiquity,
since it presents geometric knowledge of that
time through a set of axioms, which later came to
be known as Euclidean geometry. Geometric
thinking became a means to nd the most
efcient way to model a given phenomenon, after
abstracting it from its particular instances. After
the development of the calculus and the theory of
differential equations, geometry was expanded to
cover situations in which the classical lines,
planes, and spheres were replaced by ‘shortest
paths on a surface’ (or higher dimensional
objects), ‘minimal surfaces’ (like soap lms), and
‘constant mean curvature surfaces’ (like soap
bubbles). In fact, all sorts of problems in which
the solution was a conguration that minimized
some quantity (such as mass, energy, volume,
etc.) were seen to be special cases of a new
‘differential’ geometry and this launched a
revolution in the study of partial differential
equations that is continuing today. Einstein’s
theory of relativity and modern quantum theory
(including string theory and its generalizations)
are all part of differential geometry’s wide scope.
Its applications include not only theoretical
physics, but computer modelling of shape (eg.
computer models of the brain), graphical
representations, heat ow, optimization and
control theory, and understanding properties of
partial differential equations and their
transformation rules.
The four courses MATHS 150, 250, 253 and
255 form a core that should normally be taken
by students wishing to advance to courses in
Pure Mathematics at Stage III or beyond.
Applied Mathematics
Modern science relies absolutely on applied
mathematics. Any student interested in physics,
biology, Earth sciences, engineering, medicine,
chemistry, economics, or many other areas, will
nd the study of applied mathematics not only
useful, but vitally important.
It is the job of an applied mathematician to show
how mathematical techniques can be applied to
science and technology to answer interesting
questions. Our goal is usually to use
mathematical equations to study real-world
problems rather than to study equations for their
own sake. In our department we use mathematics
to study such diverse areas as physiology, ice
ow, oating runways, astronomy, quantum
chemistry, nonlinear systems, the human genome
and many other areas. Elements of these
research areas are incorporated into our courses
wherever possible.
The rst year course MATHS 162 provides an
introduction to applied mathematics, and it is
strongly recommended that all students with
interests in applied mathematics take this
course. Pure mathematics courses are also very
important for applied mathematics and should
be included in any course of study in applied
mathematics.