Algebraic topology studies properties of topological spaces and maps between them by associating algebraic invariants (fundamental groups, homology groups, cohomology groups) to each space. This course gives a solid introduction to fundamental ideas and results that are employed nowadays in most areas of mathematics, theoretical physics and computer science. This course aims to understand some fundamental ideas in algebraic topology; to apply discrete, algebraic methods to solve topological problems; to develop some intuition for how algebraic topology relates to concrete topological problems.

Topics to be covered include:

Fundamental group and covering spaces; Brouwer fixed point theorem and Fundamental theorem of algebra; Homology theory and cohomology theory; Jordan-Brouwer separation theorem, Lefschetz fixed theorem; some additional topics (Orientation, Poincare duality, if time permits)

This is an Honours Pathway Course. It builds upon the material of MATH3302 and MATH2322 and emphasises mathematical rigour and proof.

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the fundamental concepts of algebraic topology and their role in modern mathematics and applied contexts.
2. Demonstrate accurate and efficient use of algebraic topology techniques.
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from algebraic topology.
4. Apply problem-solving using algebraic topology techniques applied to diverse situations in physics, engineering and other mathematical contexts.

5. Ability to conduct some (limited) independent research under expert supervision.

Assessment will be based on:

• Assignment 1 (20%: LO 1-5)
• Assignment 2 (20%; LO 1-5)
• Assignment 3 (20%; LO 1-5)
• Presentation (10%; LO 1-5)
• Take home exam (30%; LO 1-4)

36 lectures and 10 tutorials.

Requires a mark of 60 or more in both MATH3320 and MATH2322, and basic knowledge of abstract algebra, linear algebra, and point-set topology.