The topics of the course form an introduction to a currently a very exciting and emerging field, using ideas involving the combinatorial properties of groups, analysis and probability theory. Hence the course suits well for anyone with any interest to any of these topics even if weaker in the others. We will revise all the topics in the beginning of the course.

The course starts with the basics by reviewing first year probability notions and introducing probabilistic tools such as convolution, which are natural in the context of groups. For simplicity we will first concentrate on the circle group Zp, but many of the core ideas are similar in more complicated groups such as the symmetric group. We will introduce fundamental topics from Harmonic Analysis such as Fourier transform and demonstrate how they can be applied here.

As prerequisites it helps to be comfortable with probabilistic language of "probability of an event", "expectation", "independence", which are presented in the first year probability course, but we will revise these notions in the beginning. On group theory it helps to be familiar with basic examples such as the symmetric group. From analysis it helps to be comfortable with complex numbers and sequences and series.

If you have any questions do not hesitate in contacting me:

- Dr Tuomas Sahlsten (website)
- tuomas.sahlsten@manchester.ac.uk
- Alan Turing Building, Office 2.120

Your course assessment information will appear on the

**Course materials and resources** can be found below.

- The lectures will be held on every
**Tuesday at 3-4 pm**and every**Thursday 9-10 am**. - The
tutorial of the course will be on every
**Thursday at 10-11 am**starting on the 2nd week at Hum Bridge St Hanson Room. The Hanson Room is located on the ground floor of the Humanities Bridgeford Street Building.

- Lecture notes (pdf, dated 2 Jan 2020, version 1.2)

- Week 1: Introduction and group Z_p
- Week 2: Probability distributions on Z_p, total variation distance and entropy
- Week 3: Convolution and sumsets
- Week 4: Ergodicity, concentration on subgroups and mixing
- Week 5: Fourier transform in Z_p
- Week 6: L^2 theory and convolution theorem for Fourier transform, Bounding distance to uniform using harmonic analysis (Upper Bound Lemma)
- Week 7: Explicit computations of the mixing time using the Upper Bound Lemma using spectral gap, starting theory of random walks on general groups. Beginning theory on the examples Z_2^p, Z_d^p, the unit circle group S^1 = R/Z and torus T^d = R^d/Z^d
- Week 8: More on the random walks on general finite groups, difficulties with the non-Abelian case. Developing harmonic analysis in Z_2^d (Plancherel's theorem) and proving mixing time results for Z_2^d (Upper Bound Lemma in Z_2^d). Interpretation of these using the Ehrenfest Urn model.
- Week 9: Basics of representation theory. Random walks on the symmetric group and back to card shuffling examples. Applying the general group formalism in the symmetric group to connect mixing time to ``sufficiently random'' order of cards
- Week 10: Overview of the general methods presented in the course and how to apply them beyond the examples presented here
- Week 11: Further applications
- Week 12: Revision for the final exam

It is highly recommended that you attend the tutorials as they can provide extra examples and strategies for proofs not found in the lecture notes.

Tutorial questions for Weeks 1-4 can be found from here.

Week 5: Exercises (pdf), Solutions (pdf).

Week 6: Exercises (pdf) Solutions (pdf)