# MATH2701 Abstract Algebra and Fundamental Analysis

MATH2701 is a Mathematics Level II course. See the course overview below.

Units of credit: 6

Prerequisites: MATH1231 or MATH1241 or MATH1251 or DPST1014 with at least a CR, and enrolment in an Advanced Maths or Advanced Science Program.

Cycle of offering: Term 3

Graduate attributes: The course will enhance your research, inquiry and analytical thinking abilities.

More information: The Course Outline (pdf) contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains up-to-date timetabling information.

If you are currently enrolled in MATH2701, you can log into UNSW Moodle for this course.

#### Course Aims

This course aims to introduce advanced mathematics students to a selection of themes of pure mathematics in order to motivate further abstraction and give practice in mathematical rigour.

#### Course Description

Mathematics went through quite a revolution around the turn of the 20th century. In particular, an axiomatic approach infiltrated the mathematical paradigm, both as a tool to ensure mathematical rigour and to abstract common principles working in a variety of different settings.

First year mathematics emphasizes computation over abstraction and rigour. Later year courses (and Pure Mathematics in general) reverse this, so students need to learn some new skills and some new ways of thinking about mathematical objects.

This course is designed to help you develop the ability to write rigorous mathematical proofs in a setting where the level of abstraction is still quite modest. As such it will serve as an excellent preparation for the third year Pure Mathematics courses.

The course consists of two halves, algebra and analysis, each taught for 6 weeks.
Analysis: Most of the calculus you have seen involves equalities. Mathematical analysis however, is largely about inequalities, about suitably bounding quantities that cannot be calculated precisely. Many nice examples come from geometry and we will frequently use these to motivate our discussion in the first part of the analysis section. In the latter part we will look more closely at some aspects of the real numbers, such as how well one can approximate π by a rational
p/q (in terms of how large q is).

Algebra: We will investigate various transformations on the plane and projective plane. We will first study several types of transformations such as translations, reflections, rotations etc. in terms of groups. We will then look at symmetries, i.e. transformations of geometric figures that preserve some property (such as distance or angles between lines), and projective geometry. Projective transformations can change a conic section of one type to another, e.g. an ellipse to a hyperbola.

Mathematics went through quite a revolution around the turn of the 20th century. In particular, an axiomatic approach infiltrated the mathematical paradigm, both as a tool to ensure mathematical rigour and to abstract common principles working in a variety of different settings.

First year mathematics emphasizes computation over abstraction and rigour. Later year courses (and Pure Mathematics in general) reverse this, so students need to learn some new skills and some new ways of thinking about mathematical objects.

This course is designed to help you develop the ability to write rigorous mathematical proofs in a setting where the level of abstraction is still quite modest. As such it will serve as an excellent preparation for the third year Pure Mathematics courses.

The course consists of two halves, algebra and analysis, each taught for 6 weeks.
Analysis: Most of the calculus you have seen involves equalities. Mathematical analysis however, is largely about inequalities, about suitably bounding quantities that cannot be calculated precisely. Many nice examples come from geometry and we will frequently use these to motivate our discussion in the first part of the analysis section. In the latter part we will look more closely at some aspects of the real numbers, such as how well one can approximate π by a rational
p/q (in terms of how large q is).

Algebra: We will investigate various transformations on the plane and projective plane. We will first study several types of transformations such as translations, reflections, rotations etc. in terms of groups. We will then look at symmetries, i.e. transformations of geometric figures that preserve some property (such as distance or angles between lines), and projective geometry. Projective transformations can change a conic section of one type to another, e.g. an ellipse to a hyperbola.