## MATH 254A : Hilbert's fifth problem and related topics

• 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: