MATH 254A : Topics in Ergodic Theory

  • 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 Rutgers)

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

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

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

ˇ        Lecture: MWF 4-4:50, MS5117

ˇ        Quiz section: None

ˇ        Office Hours: M 2-3

ˇ        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        Dave Witte Morris’ book.

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.

ˇ        Ben Green’s lecture notes on ergodic theory.

ˇ        The blog for this course.