**Course description**: The structural theory of locally compact groups. Haar measure. Lie groups and Lie algebras. The Peter-Weyl theorem. Local groups. The Gleason-Yamabe theorem, and the solution to Hilbert's fifth problem. The structural theory of approximate groups. Connection with ultraproducts. Gromov's theorem on groups of polynomial growth; connections with approximate groups and Hilbert's fifth problem. Connections with fundamental groups of Riemannian manifolds.

**Announcements:**

- There will be no class on Fri Dec 2.

**Instructor**: Terence Tao,__tao@math.ucla.edu__, x64844, MS 6183**Lecture:**MWF 1-1:50, MS5137**Quiz section:**None**Office Hours**: Tue 1-2**Textbook**: None. I will post lecture notes on my blog site.**Prerequisites**: 245AB, as well as understanding of all core undergraduate mathematical topics (e.g. point-set topology, group theory, set theory, linear algebra, etc.). Knowing what a Lie group, an ultrafilter, or a Riemannian manifold is would be helpful, though we will review these topics in the course.**Grading**: This is a topics course, so I am planning a fairly informal grading scheme. Basically, the base grade will be B provided you actually show up to a significant number of classes, and adjusted upwards according to whether you turn in any homework. There is no final or midterm for this course.**Reading Assignment**: Lecture notes will be provided on my blog site. Students are encouraged to comment on these posts. Note that the notes will cover more material than the lectures.**Homework**: Homework assignments will be assigned from the lecture notes, and will be posted on this page.- Homework 1: (Due Mon, Oct 10)
- Notes 0, Exercise 4 (O_n does not contain arbitrarily dense finite subgroups)
- Notes 0, Exercise 7 (A non-Lie locally compact group with no small normal subgroups)
- Notes 0, Exercise 9 (Progressions in the Heisenberg group)
- Notes 0, Exercise 12 (Finite index subgroups and polynomial growth)
- Homework 2: (Due Mon, Oct 17)
- Notes 1, Exercise 9 (Estimate in C^{1,1} local groups)
- Notes 1, Exercise 12 (Classification of one-parameter subgroups)
- Notes 1, Exercise 15 (Local Lie implies Lie)
- Homework 3: (Due Mon, Oct 24)
- Notes 1, Exercise 22 (Lie brackets and the exponential map)
- Notes 2, Exercise 7 (Norm on L(G))
- Notes 2, Exercise 8 (Local injectivity of the exponential map)
- Homework 4: (Due Mon, Oct 31)
- Notes 3, Exercise 3 (Reducing topological groups to the Hausdorff case)
- Notes 3, Exercise 6 (Modular function)
- Homework 5: (Due Mon, Nov 7)
- Notes 3, Exercise 12 (Compactness of integral operators)
- Notes 3, Exercise 20 (One-dimensionality of irreducible abelian representations)
- Notes 3, Exercise 21 (Fourier analysis on compact abelian groups)
- Homework 6: (Due Mon, Nov 14)
- Notes 4, Exercise 12 (Uniqueness of escape metrics)
- Notes 4, Exercise 13 (Compactness in the Hausdorff metric)
- Notes 5, Exercise 8 (Local structure of finite-dimensional locally compact groups)
- Homework 7: (Due Mon, Nov 21)
- Notes 6, Exercise 13 (Ultraproducts and quantifiers)
- Notes 6, Exercise 19 (Nonstandard construction of the reals)
- Notes 7, Exercise 4 (Nonstandard proof of compactness of Hausdorff distance)
- Homework 8 (Due Mon, Nov 28)
- Notes 7, Exercise 22 (A neighbourhood base for ultra approximate groups)
- Notes 7, Exercise 23 (A locally compact topology for ultra approximate groups)
- Notes 7, Exercise 31 (Modeling the lamplighter group)

**Online resources**: