You can find details of what is covered in the course on the course syllabus page.

All timetable information for the unit can be found on https://my.manchester.ac.uk.

## Succeeding in the Course Unit

What do I need to do to succeed in this course unit?

There are 8 learning outcomes. These are:

- define total variation distances between probability distributions on the group ZN, calculate and estimate these distances for various distributions in ZN
- define entropy of probability distributions in ZN, compute and estimate entropy for various examples in ZN and relate entropy to the total variation distance
- define and compute convolutions of probability distributions on ZN, model random walks as iterated convolutions and estimate probabilities of events using iterated convolutions,
- define Fourier transforms on the group ZN and estimate Fourier transforms of probability distributions and their convolutions on ZN,
- prove fundamental theorems in harmonic analysis such as convolution theorem and Plancherelâ€™s theorem in ZN
- outline the calculations and estimates of finding the total variation distances of convolutions of probability distributions to the uniform distribution in ZN and alter these proofs in other examples with different constants or parameters,
- explain the key ideas of the theorems and methods presented in the course and describe how each component (harmonic analysis, random walks and group theory) come into play,
- apply the methods presented in the course and prove similar results for analogous contexts such as random walks on higher dimensional lattices (hypercube Z2d and the torus ZNd), matrix groups (GL(ZN)), models for card shuffling in the symmetric group or models for Rubik's cube scrambling as subgroups of the symmetric group

During this course unit you need to acquire the knowledge and develop
the skills to demonstrate that you can achieve these outcomes.

## How do I acquire the knowledge and develop the skills to succeed in this course unit?

Course home page. To see the feedback
opportunities, go to the Assessment &
Feedback page.

## How will I know I have the acquired the knowledge and skills to
succeed in this course unit? (Self-assessment)

There are opportunities throughout the course unit to assess your own
learning. For example, working through the course Exercises and
checking your understanding against the Solutions. You should have
a serious attempt at the exercises *before* referring to the
solutions. In addition, within lectures and tutorials there will be
opportunities for you to have your questions answered and discuss
your progress on a regular basis.
## How will I demonstrate that I have the acquired the knowledge and skills to succeed in this course unit? (Assessment)

You can find full details of the assessment on this course unit on the course syllabus page and coursework information and details about course unit feedback on the

Assessment & Feedback page.