Measure Theory and Ergodic Theory

MATH41021 Measure Theory and Ergodic Theory 15 Semester 1 Undergraduate Level 4 Donald Robertson Unit coordinator Department of Mathematics 1 FHEQ level (Framework for Higher Education Qualifications) ' Masters/Integrated Masters P4 ' European Credit Transfer & Accumulation System Rating : 7.5 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. Measure theory – an abstraction of what it means to assign size to objects – underpins most of modern analysis and probability, serving as a foundation for such diverse topics as harmonic analysis, stochastic differential equations, fractal geometry and ergodic theory. It came to prominence at the beginning of the 20th century when Lebesgue used it to develop an entirely new approach to integration that overcame the deficiencies of older integrals.Ergodic theory lies at the confluence of dynamical systems and measure theory. The abstract nature of measure theory has led to breakthrough applications of ergodic theory in many different branches of mathematics. For example, ergodic theory played a central role in the proof of the Green-Tao theorem on arithmetic progressions in primes, and in Margulis’s resolution of the Oppenheim conjecture.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.   <ul><li>Recognise, deduce and apply properties of sigma-algebras and measures.  </li><li>Construct measures using Caratheodory’s extension theorem and the Riesz representation theorem.   </li><li>Compute integrals of measurable functions.  </li><li>Define Lebesgue spaces and deduce whether a given function belongs to a specific Lebesgue space.  </li><li>Determine whether transformations are measure-preserving or ergodic.  </li><li>Interpret applications of the pointwise ergodic theorem to measure-preserving transformations.   </li><li>Distinguish measure-preserving transformations via their dynamical properties.   </li><li>Describe applications of ergodic theory to other areas of mathematics. </li></ul> <ul><li>Measures and sigma-algebras [4 lectures]  </li><li>Integration and Lebesgue spaces [4 lectures]  </li><li>Irrational rotations and Bernoulli shifts [4 lectures]  </li><li>Measure-preserving transformations and ergodicity [2 lectures]  </li><li>The pointwise ergodic theorem [2 lectures]  </li><li>Spectral properties and entropy [4 lectures]  </li><li>Applications of ergodic theory [2 lectures]</li></ul> 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.  1 Written exam 100% For weekly problem sheets, feedback will be returned scripts, within a week of submission  MATH21111 Metric Spaces Pre-Requisite Compulsory MATH11112 Real Analysis Pre-Requisite Compulsory MATH20101 Real Analysis A Pre-Requisite Compulsory MATH20111 Real Analysis B Pre-Requisite Compulsory MATH41021 Pre-Requisites: MATH11112 and MATH21111 N Bartle, R. G. The Elements Of Integration And Lebesgue Measure Wiley 1995Folland, G. B. Real Analysis: Modern Techniques and Their Applications Wiley 1999Walters, P. An Introduction to Ergodic Theory Graduate Texts in Mathematics, Springer 1982Einsiedler, M. and Ward, T. Ergodic Theory with a view towards Number Theory Graduate Texts in Mathematics, Springer 2011   Lectures 22 Tutorials 11 0 117