## MATH 254B : Higher order Fourier Analysis

• Course description: Weyl equidistribution theory; van der Corput lemma.  Roth's theorem.  Gowers uniformity norms; Szemeredi's theorem.  Polynomials in finite fields. Nilmanifolds and nilsequences.  Nilpotent Ratner type theorems.   The inverse conjecture for the Gowers norms.  The Green-Tao theorem; linear equations in primes.  Connections with ergodic theory.

Announcements:

• Instructor: Terence Tao, tao@math.ucla.edu, x64844, MS 6183
• Lecture: MWF 10-10:50, MS5117
• Quiz section: None
• Office Hours: M 3-4
• Textbook: None.  I will post lecture notes on my blog site.
• Prerequisite: A graduate exposure to Fourier analysis (e.g. 245AB or 247A).  Basic knowledge of groups and Lie groups will also be useful.  Some exposure to combinatorics or ergodic theory will be helpful, but not strictly necessary.
• 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.
• 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.
• Homework 1 (Due Fri, Apr 9): Do Exercises 6,7 from Notes 1.
• Homework 2 (Due Fri, Apr 23): Do Exercise 35 (Vinogradov lemma, ultralimit version) from Notes 1.  (Note that early versions of the notes may have a slightly different exercise numbering.)
• Homework 3 (Due Fri, May 7): Do Exercise 7 (Roth's theorem in finite abelian groups) from Notes 2.
• Homework 4 (Due Fri, May 21): Do Exercises 4, 5 from Notes 3.
• Homework 5 (Due Fri, June 4): Do Exercise 6 (Chevalley-Warning theorem) from Notes 4.

Online resources: