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Unit information: Algebraic Topology in 2020/21

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Unit name Algebraic Topology
Unit code MATHM1200
Credit points 20
Level of study M/7
Teaching block(s) Teaching Block 1 (weeks 1 - 12)
Unit director Dr. Mark Hagen
Open unit status Not open
Pre-requisites

MATH20006 Metric Spaces and MATH33300 Group Theory

Co-requisites

None

School/department School of Mathematics
Faculty Faculty of Science

Description

Unit Aims

The aim of the unit is to give an introduction to algebraic topology with an emphasis on cell complexes, fundamental groups and homology.

Unit Description

Algebraic Topology concerns constructing and understanding topological spaces through algebraic, combinatorial and geometric techniques. In particular, groups are associated to spaces to reveal their essential structural features and to distinguish them. In cruder terms, it is about adjectives that capture and distinguish essential features of spaces.

The theory is powerful. We will give applications including proofs of The Fundamental Theory of Algebra and Brouwer's Fixed Point Theorem (which is important in economics).

Relation to Other Units

This is one of three Level M units which develop group theory in various directions. The others are Representation Theory and Galois Theory.

Intended learning outcomes

Learning Objectives

Students should absorb the idea of algebraic invariants to distinguish between complex objects, their geometric intuition should be sharpened, they should have a better appreciation of the interconnectivity of different fields of mathematics, and they should have a keener sense of the power and applicability of abstract theories.

Transferable Skills

  • The assimilation of abstract and novel ideas.
  • Geometric intuition.
  • How to place intuitive ideas on a rigorous footing.
  • Presentation skills.

Teaching details

Lectures, problem sets and discussion of problems, student presentations.

Assessment Details

50% Exam, 50% Coursework - assessed problem sheets

Reading and References

Recommended

  • Allen Hatcher, Algebraic Topology, Cambridge University Press, 2001, Chapters 0,1,2.
  • James R. Munkres, Topology (2nd Edition), Prentice Hall, 2000
  • W. A. Sutherland, Introduction to Metric and Topological Spaces, Clarendon Press, 2009
  • O. Ya. Viro, O.A. Ivanov, V.M. Kharlamov, N.Y. Netsvetaev, Elementary Topology, American Mathematical Society, 2008

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