Interested in connections between probability, geometry, and group theory? We will do a survey of probabilistic and geometric techniques and their connections to modern group theory, with a focus on random walks on graphs. The course is for Gabriela Moisescu-Pareja, Agnes Totschnig, Tasmin Chu and given by Louigi Addario-Berry, but anyone is welcome to sit in the lecture room. Presentations are done via iPad but displayed on the screen. This course will follow Gabor Pete's notes *Probability and Geometry on Groups*. **When and where:** Burnside 1234, on Wednesdays, from 9 am to 10:30 am. The room is booked until 11 am. **Join the email list**: Interested in auditing? Email [email protected]. **Overview:** We will attempt to cover Chapters 5, 6, and 12 of Gabor Pete's notes in detail, with some diversions into Chapter 3, Chapter 5, Chapter 7, and Chapter 9. Our current trajectory is Chapter 6 $\rightarrow$ Chapter 3 $\rightarrow$ Chapter 5 $\rightarrow$ Chapter 7 $\rightarrow$ Chapter 12 + material from Chapters 7 and 8 of Lyons and Peres. **** > [! Tentative Schedule]- > - **Week 1** (Tasmin): Chapter 6.1 of PGG. Basic examples of random walks. Recurrence, transience, strong Markov property, theorem 1.2, random walks, basic electrical network theory. > - **Week 2** (Gabriela): 6.1 continued: electrical networks. Hitting times, conductances, resistances, recurrence and transience of Markov chains, electrical networks for analyzing Markov chains on graphs. Polya's theorem on $\mathbb{Z}^d$. > - **Week 3** (Agnès): 3, asymptotic geometry of groups. Space of ends. Maybe some parts of Chapter 4. > - **Week 4** (Tasmin): 5, Isoperimetric inequalities. Von Neumann definition of amenability. Folner definition of amenability. Examples of amenable and non-amenable graphs. Proof that the two are equivalent for countable finitely generated groups. > - **Week 5**: (Gabriela) 6.2. Electrical network theory. Kanai's theorem and proof that recurrence and transience of random walks are quasi-isometry invariants. >- **Week 6**: (Agnès): Chapter 7. Mohar's generalization of Kesten's theorem. Characterization of the amenability of group through the spectral radius of a simple random walk which is proved in chapter 7. > - **Week 7**: (Tasmin): Chapter 7. Finitary analogues of infinite Markov chains and spectral radius. Mixing times. Bounds for the spectral gap. >- **Week 8**: Chapter 9. Poisson boundary. Connection between non-trivial Poisson boundary and non-amenability. >- **Week 9**: Chapter 12 of PGG. Percolation, Bernoulli(p) bond percolation. Insertion and deletion tolerance. A lot of the early machinery in this chapter is fairly independent from other chapters. Galton-Watson processes. Discrete Chung-Fuchs theorem. Branching number of trees. Uses isoperimetric inequalities. A lot of this machinery is also covered (in more depth) in Chapter 3 of Lyons and Peres. >- **Week 10**: More of Chapter 12 on percolation. Unimodularity of graphs. Ergodicity, percolation clusters, cluster indistinguishability. >- **Week 11**: More of Chapter 12 on percolation. Critical percolation: the plane, trees, scaling limits, critical exponents. Warning: Page 187 turns into Hungarian. >- **Rest of the weeks**: Leave them free to spend extra time on chapters where we need it. Otherwise, if we have extra time, we should spend it on: 13.5 measurable group theory, or chapter 7, which connects the isoperimetric inequalities of Chapter 5 with the Markov operator theory in Chapter 6. **Textbook:** The following may be useful resources. Gabor Pete's notes will be the main reference, but Lyons' book *Probability on Trees and Networks* covers a lot of the same material in greater depth, at the expense of brevity. - [Gabor Pete's notes: Probability and Geometry on Groups](https://math.bme.hu/~gabor/PGG.pdf) - [Lyons: Probability on Trees and Networks](https://rdlyons.pages.iu.edu/prbtree/book.pdf) - [[MATH 447 (Stochastic Processes)|Stochastic Processes Review Notes by Tasmin]] > [!Lecture Notes] > - [[Notation.pdf|Notation Review: O(n) and o(n)]] > - [[Lec_1_ Review_Of_Markov_Chains.pdf|Lecture 1: 6.1 PGG. Basic review of Markov chain theory. Recurrence and transience. Hitting times. Stationary measures. Markov operators, electrical networks, conductances, Laplacian, harmonic functions. Only pages 1-17 (up to the proof of the lemma) were covered.]] > - [[Lec_2_Gabriela.pdf|Lecture 2: 1.1 PGG. Proof of Polya's theorem. A bit more electrical network theory.]] Also Tasmin's notes on [[447-worked-examples-6.pdf|Poisson processes]] and [[worked-examples-7.pdf|continuous-time Markov chains]] may be useful review. > - [[Lec_3_Agnes.pdf|Lecture 3: chapter 3 of PGG. Definition of quasi-isometry. Proof of Švarc–Milnor lemma.]] The following resources were used as supplements: Coarse Geometry and Randomness, by Itai Benjamini, [Topics in Geometric Group Theory by Sameer Kailasa](https://can01.safelinks.protection.outlook.com/?url=http%3A%2F%2Fmath.uchicago.edu%2F~may%2FREU2014%2FREUPapers%2FKailasa.pdf&data=05%7C02%7Ctasmin.chu%40mail.mcgill.ca%7C77d857ab3c21400f8f4d08dc1d353e20%7Ccd31967152e74a68afa9fcf8f89f09ea%7C0%7C0%7C638417362049979468%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=MV7nRHcwW9v3xo%2F1zKgRyEZF6jbkSh4zhjcQGkwzals%3D&reserved=0), and [A course on geometric group theory by Brian H Bowditch](https://can01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.math.ucdavis.edu%2F~kapovich%2F280-2009%2Fbhb-ggtcourse.pdf&data=05%7C02%7Ctasmin.chu%40mail.mcgill.ca%7C77d857ab3c21400f8f4d08dc1d353e20%7Ccd31967152e74a68afa9fcf8f89f09ea%7C0%7C0%7C638417362049993229%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=KnnmpxkKCGgEr1OgRe5RyH4xRtwQjY6ulqDrrNctWS4%3D&reserved=0). > - [[Lec_4_Tasmin.pdf|Lecture 4: Chapter 5 of PGG. Isoperimetric inequalities. Definition of Folner and von Neumann amenability.]] > - [[Lec_5_Gabriela.pdf |Lecture 5: 6.2 of PGG. Continuation of electrical network theory. Proof of Kanai's theorem that transience is a quasi-isometry invariant.]] > - [[Lec_6_Agnes.pdf |Lecture 6: 7.1 and 7.2 of PGG. Markov operator norm. A proof of Kesten's theorem about the amenability of a group.]] > - [[Lec_7_Tasmin.pdf |Lecture 7: 7.1 of PGG. Spectral gap, expanders, and mixing times. Definition of Property T. ]] > - [[Lec_8_Gabriela.pdf|Lecture 8: All about expander graphs. Examples, definitions. Connections with Property T groups.]] > - [[Lec_9_Agnes.pdf |Lecture 9: Intro to percolation theory. Primer on Bernoulli(p) bond percolation.]] > - [[Lec_10_Tasmin.pdf |Lecture 10: Proof of the Burton-Keane theorem. Intro to Mass Transport Principle and unimodularity.]] PS: I have heard my site is somewhat buggy when displaying PDFs, especially on mobile. For troubleshooting, always try opening the PDF on a computer, and try Safari if Chrome is not working. Apologies, I think it's an issue with the platform my site is run on. You can email me for the notes if my site is not working. >[!Recommended exercises with partial solutions] > - Lecture 1: [[Exercise 6.2]], [[Exercise 6.3]], 6.5, [[Exercise 6.10]]. > - Lecture 2: [[Exercise 1.1]], [[Exercise 1.2]], [[Exercise 1.3]], [[Exercise 1.4]], [[Exercise 1.5]], [[Exercise 1.6]], [[Exercise 1.7]]. > - Lecture 3: Exercise 1.27, 1.28, 1.47, 1.34 from [Itai Benjamini's Coarse Geometry and Randomnness](https://www.wisdom.weizmann.ac.il/~itai/stflouraug24.pdf#page9).