**Course description**: Basic ergodic theorems (pointwise, mean, maximal) and recurrence theorems (Poincare, Khintchine, etc.) Topological dynamics. Structural theory of measure-preserving systems; characteristic factors. Spectral theory of dynamical systems. Multiple recurrence theorems (Furstenberg, etc.) and connections with additive combinatorics (e.g. Szemerédi’s theorem). Orbits in homogeneous spaces, especially nilmanifolds; Ratner’s theorem. Further topics as time allows may include joinings, dynamical entropy, return times theorems, arithmetic progressions in primes, and/or orbit equivalence.

**Announcements:**

ˇ
There will
be no class on Friday March 14 (I will be at

ˇ
There
will be no classes on Monday Feb 4 and Wednesday Feb 6 (I will be in

ˇ
The
first class will be on Wednesday, January 9 (I will be at the AMS meeting in San Diego
on Monday January 7).

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**Instructor**: Terence Tao, tao@math.ucla.edu, x64844, MS 6183

ˇ
**Lecture: **MWF 4-4:50, MS5117

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**Quiz section: **None

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**Office Hours**: M 2-3

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**Textbook**: I will use a number of sources,
including Furstenberg’s “Recurrence in ergodic
theory and combinatorial number theory” and Witte Morris’ “Ratner’s
theorems on unipotent flows”. I will post lecture notes on my blog site.

ˇ
**Prerequisite**: Math 245AB is highly
recommended. In particular, familiarity
with measure theory and point set topology is pretty much essential. It will also help if you know what a Lie
group is.

ˇ
**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.

ˇ
**Reading**** Assignment**: Lecture notes will be provided on my blog site.
Students are encouraged to comment on these posts.

ˇ
**Homework**: There are six homework
assignments:

1. **Homework
1: **Do Exercise 1 from Lecture
2. (Due Wed Jan 23.)

2. **Homework
2:** Do Exercise 9 from Lecture
3. (Due Wed Jan 30.)

3. **Homework
3:** Do Exercises 6 and 7 from Lecture
4. (Due Fri Feb 8.)

4. **Homework
4:** Do Exercise 6 from Lecture
6. (Due Wed Feb 20.)

5. **Homework
5:** Do Exercise 10 from Lecture
8. (Due Wed Feb 28.)

6. **Homework
6:** Do Exercises 2 and 3 from Lecture
10. (Due Fri Mar 7.)

**Online
resources**:

ˇ Akshay Venkatesh’s lecture notes cover similar ground to this course.

ˇ A book by Einsiedler and Ward on ergodic theory from a number-theoretic perspective.

ˇ Curt McMullen’s lecture notes on ergodic theory.

ˇ Bryna Kra’s lectures on ergodic theory and additive combinatorics.