MSc Pure Mathematics and Mathematical Logic / Course details

In this course we will learn about the abstract theory of measures and the theory of integration that sits on top of it. Then we will cover examples and properties of measure-preserving transformations and the pointwise ergodic theorem before applying ergodic theory to other parts of mathematics.

The unit aims to: Introduce the abstract theory of integration with respect to a measure, introduce measure-preserving transformations, and apply ergodic theory to other parts of mathematics.
 

• Recognise, deduce and apply properties of sigma-algebras and measures.
   
• Construct measures using Caratheodory’s extension theorem and the Riesz representation theorem. 
   
• Compute integrals of measurable functions.
   
• Define Lebesgue spaces and deduce whether a given function belongs to a specific Lebesgue space.
   
• Determine whether transformations are measure-preserving or ergodic.
   
• Interpret applications of the pointwise ergodic theorem to measure-preserving transformations. 
   
• Distinguish measure-preserving transformations via their dynamical properties. 
   
• Describe applications of ergodic theory to other areas of mathematics. 

• Measures and sigma-algebras [4 lectures]
   
• Integration and Lebesgue spaces [4 lectures]
   
• Irrational rotations and Bernoulli shifts [4 lectures]
   
• Measure-preserving transformations and ergodicity [2 lectures]
   
• The pointwise ergodic theorem [2 lectures]
   
• Spectral properties and entropy [4 lectures]
   
• Applications of ergodic theory [2 lectures]

In addition to delivery of content in the two lectures per week, feedback will be given on the weekly problem sheet assignments. Tutorials will provide an opportunity for students' work to be discussed and for feedback on their understanding to be given. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour. 

For weekly problem sheets, feedback will be returned scripts, within a week of submission 

Bartle, R. G. The Elements Of Integration And Lebesgue Measure Wiley 1995

Folland, G. B. Real Analysis: Modern Techniques and Their Applications Wiley 1999

Walters, P. An Introduction to Ergodic Theory Graduate Texts in Mathematics, Springer 1982

Einsiedler, M. and Ward, T. Ergodic Theory with a view towards Number Theory Graduate Texts in Mathematics, Springer 2011