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MATH3281: TOPOLOGY III

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Type Open
Level 3
Credits 20
Availability Available in 2023/24
Module Cap
Location Durham
Department Mathematical Sciences

Prerequisites

  • Complex Analysis II (MATH2011) AND Algebra II (MATH 2581).

Corequisites

  • None.

Excluded Combinations of Modules

Aims

  • To provide a balanced introduction to Point Set, Geometric andAlgebraic Topology, with particular emphasis on surfaces and knots.

Content

  • Topological Spaces and Continuous Functions: Topology on aset, open sets, closed sets, limit points and closure, examples oftopologies.
  • Compactness and Connectedness.
  • Topological groups and group actions.
  • The Orthogonal groups. The Fundamental Group: calculationfor circle, homotopy type, homotopy equivalence.
  • Generators and relations of groups, Tietze theorem, VanKampen's theorem.
  • Compact surfaces, classical knots, basic knotinvariants.

Learning Outcomes

Subject-specific Knowledge:

  • By the end of the module students will: be able to solvenovel and/or complex problems in Topology.
  • have a systematic and coherent understanding of theoreticalmathematics in the field of Topology.
  • have acquired a coherent body of knowledge of these subjectsdemonstrated through one or more of the following topic areas:Topological spaces.
  • Topological Groups and group actions.
  • Fundamental group, homotopy type.
  • Group presentations and Van Kampen's Theorem.
  • Surfaces and Knots.

Subject-specific Skills:

  • In addition students will have specialised mathematicalskills in the following areas which can be used with minimal guidance:Spatial awareness.

Key Skills:

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

  • Lectures demonstrate what is required to be learned and theapplication of the theory to practical examples.
  • Assignments for self-study develop problem-solving skills andenable students to test and develop their knowledge andunderstanding.
  • Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
  • The end-of-year examination assesses the knowledge acquiredand the ability to solve predictable and unpredictableproblems.

Teaching Methods and Learning Hours

ActivityNumberFrequencyDurationTotalMonitored
Lectures422 per week for 20 weeks and 2 in term 31 Hour42 
Problems Classes8Four in each of terms 1 and 21 Hour8 
Preparation and Reading150 
Total200 

Summative Assessment

Component: ExaminationComponent Weighting: 100%
ElementLength / DurationElement WeightingResit Opportunity
three-hour end of year examination 100 

Formative Assessment

Eight written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

More information

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