“The chance to meet other interested maths students at a similar level to myself, this motivates me to engage in further study and to deepen my understanding of mathematics. It also expands my world view in unexpected and exciting ways.”

Alastair Anderberg, The University of Newcastle

Algebraic Topology: First Steps in Cohomology

Lecturers

Dr David Roberts, The University of Adelaide

Synopsis

Algebraic topology is one of the key areas of pure mathematics to be developed in the middle of the 20th century, with techniques leaking out to many other areas of mathematics aside from its origin in topology. These days it is even showing up in applied mathematics, with topological data analysis becoming a larger field every year. This course aims to give a first treatment of algebraic topology using cohomology, taking both a combinatorial and topological point of view, and treating the basics of homological algebra used to do computations. We will also cover the basic ideas of category theory so as to take advantage of functoriality of cohomology.

This course should be suitable for a first introduction to modern ideas of topology, homological algebra, algebraic topology and the basic language of category theory.

Course Overview

  • Delta-sets as combinatorial models for spaces and their cohomology, homological algebra, topological spaces and constructions, singular cohomology of topological spaces, simplicial sets as an improvement of Delta-sets, the Eilenberg–Steenrod axioms, cup products, applications.

Prerequisites

  • Students taking this course will have ideally seen a first course in topology, for instance using metric spaces, and should be familiar with rings and modules. Familiarity with the basics of manifolds (eg as submanifolds of R^n) would be helpful, but not critical. The course will review the necessary topological background and extend this to a more sophisticated viewpoint.

Assessment

  • 2 small quizzes per week (worth 10% in total)
  • Four assignments (worth 10% each)
  • An exam (worth 50%)

Resources/pre-reading

  • Lecture notes will be provided during the course
  • Hatcher’s book Algebraic Topology is freely available from his website, and I may refer to it from time to time for extra material

Pre-Course Quiz

Not sure if you should sign up for this course?

Take this quiz and look at some of the expected foundational skills in this topic

Dr David Roberts
The University of Adelaide

David is a pure mathematician working at the intersection of category theory and geometry (and other topics). He is currently a postdoc in the Institute for Geometry and its Applications at the University of Adelaide. David’s research largely involves finding the abstract structure behind other parts of mathematics, and also creating new instances of abstract nonsense in as concrete terms as possible.