Topics in Ergodic Theory and Probability (592)
Instructor: 
Julia Wolf, Email: jwolf137 at math.rutgers.edu 
Lectures: 
Tuesday and Thursday, 3:204:40pm, HLL124 , BUS 
Links: 
department course information 
Textbook: 
there is no designated textbook for this course, but reading materials will be referenced below 
Homework: 
there will be five problem sets, due the day before the examples class 
Prerequisites: 
elementary analysis, probability and combinatorics 
Office hours: 
Tu Th at Hill 432, by appointment 

Even though we will be covering some ergodic theory, and will be using probability in hidden form throughout, the title of the course is perhaps not 100 % appropriate  it was chosen for administrative reasons.
Tuesday  Jan 19 
The discrete Fourier transform  [TV], [Gr1] 
Thursday  Jan 21 
Meshulam's theorem  [Gr1] 
Tuesday  Jan 26 
Freiman's theorem  [N], [R], [Gr2] 
Thursday  Jan 28 
Plunnecke's inequality  [N], [R], [Gr2] 
Tuesday  Feb 2 
BonamiBeckner inequality and Chang's theorem  [O'D], [Gr3] 
Thursday  Feb 4 
Student presentation and examples class  Problem Set 1 
Tuesday  Feb 9 
BalogSzemerediGowers theorem  [L], [Gr4], [M] 
Thursday  Feb 11 
Uniformity norms  [TV], [Gr5] 
Tuesday  Feb 16 
Inverse theorem for the U^3 norm, part I  [GrT1], [Gr5] 
Thursday  Feb 18 
Inverse theorem for the U^3 norm, part II  [GrT1], [Gr5] 
Tuesday  Feb 23 
Szemeredi's theorem for progressions of length 4  [GrT1], [GrT2] 
Thursday  Feb 25 
***INSTRUCTOR AWAY***  
Tuesday  Mar 2 
Student presentation and examples class  Problem Set 2 
Thursday  Mar 4 
Bohr sets  [TV], [GrT1] 
Tuesday  Mar 9 
Roth's theorem, part I  [TV], [Gr4] 
Thursday  Mar 11 
Roth's theorem, part II  [TV], [W1] 
Tuesday  Mar 16 
***SPRING BREAK***  
Thursday  Mar 18 
***SPRING BREAK***  
Tuesday  Mar 23 
The inverse theorem revisited  [W2], [T2] 
Thursday  Mar 25 
Student presentation and examples class  Problem Set 3 
Tuesday  Mar 30 
An introduction to measure theory  [Gr6] 
Thursday  Apr 1 
Ergodic theorems  [T1], [Gr6], [K] 
Tuesday  Apr 6 
Khintchine recurrence  [T1], [Gr6], [K] 
Thursday  Apr 8 
Furstenberg's correspondence principle  [T1], [Gr6], [K] 
Tuesday  Apr 13 
Student presentations  
Thursday  Apr 15 
FurstenbergSarkozy theorem  [Gr6] 
Tuesday  Apr 20 
U^k seminorms and characteristic factors  [K], [W3], [W4] 
Thursday  Apr 22 
nilmanifolds and the HostKra structure theorem  [K], [W3], [W4] 
Tuesday  Apr 27 
Special lecture by Madhur Tulsiani (IAS): Connections between additive combinatorics and computer science  
Thursday  Apr 29 
Student presentations and examples class  Problem Set 4 
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I recommend Chapter 1 of [TV] and Sections 13 of [Gr1] in the bibliography below.
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The links below will be activated in due course.
Problem Set 1 (Last updated: January 25)
Problem Set 2 (Last updated: February 7)
Problem Set 3 (Last updated: March 1)
Problem Set 4 (Last updated: April 2)
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Each presentation will last approximately 45 minutes. The following topics are available.
1. (Humberto Montalvan) Lev's proof of Meshulam's theorem, available at http://arxiv.org/pdf/0911.0513
2. (Wei Chen) Behrend's construction for long progressions following Lacey and Laba, available at http://www.math.ubc.ca/~ilaba/preprints/longaps.dvi
3. (Bobby DeMarco) Croot and Sisask's proof of Roth's theorem, available at http://arxiv.org/pdf/0801.2577
4a. (Kellen Myers) Green's proof of Sarkozy's theorem, available at http://www.dpmms.cam.ac.uk/~bjg23/papers/BG2.ps
4b. (Li Zhang) Ruzsa's construction for squaredifference free sets, available at http://www.springerlink.com/content/b08n636647h0121h/
5. (Dennis Hou) A result of Gowers on quasirandom groups, available at http://www.dpmms.cam.ac.uk/~wtg10/quasirandomgroups.pdf
If you are registered for this class, or are interested in giving a presentation, please email me with your preferred two (or three) choices by Monday, January 25.
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Since there is no textbook for this course, students will take turns writing up class notes. These will be available from the Sakai site for this course. If you are not registered but would like to have access to the site, please email me with your Rutgers email address.
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There will be no written exam, but homework, attendance, class participation and one presentation are compulsory and will be evaluated.
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For general reading in this area, you may also wish to refer to A Personal Roadmap.
Much of the material I am referencing in the syllabus is due to Green, whose expositions are hard to improve on. For variety, you may also wish to consult the notes by Soundararajan [S], which cover many of our topics. In addition, the book [TV] will be a handy companion throughout.
[Gr1] 
B.J. Green, Finite fields models in additive combinatorics, 2005.  pdf 
[Gr2] 
B.J. Green, Lecture notes on "Structure theory of set addition", 2002.  pdf 
[Gr3] 
B.J. Green, Lecture notes on "Restriction and Kakeya phenomena", 2003.  pdf 
[Gr4] 
B.J. Green, Lecture notes on "Additive combinatorics", 2009.  pdf 
[Gr5] 
B.J. Green, Montreal lecture notes on "Quadratic Fourier analysis", 2006.  pdf 
[Gr6] 
B.J. Green, Lecture notes on "Ergodic theory", 2009.  website 
[GrS] 
B.J. Green and T. Sanders, Boolean functions with small spectral norm, 2006.  pdf 
[GrT1] 
B.J. Green and T. Tao, An inverse theorem for the Gowers U^3 norm, 2006.  pdf 
[GrT2] 
B.J. Green and T. Tao, New bounds for Szemeredi's theorem I, 2009.  pdf 
[K] 
B. Kra, Ergodic methods in additive combinatorics, 2006.  pdf 
[L] 
V. Lev, The (Gowers)BalogSzemeredi theorem.  dvi 
[M] 
A. Magyar, The BalogSzemeredi theorem.  pdf 
[N] 
M.B. Nathanson, Additive number theory: inverse problems and the geometry of sumsets, Springer, 1996.  
[O'D] 
R. O'Donnell, Lecture notes on "Boolean analysis", 2007.  pdf 
[R] 
I.Z. Ruzsa, Lecture notes on "Sumsets and structure", 2008.  pdf 
[S] 
K. Soundararajan, Lecture notes on "Additive combinatorics", 2007.  pdf 
[T1] 
T. Tao, Blog entries on Ergodic theory, 2006.  blog 
[T2] 
T. Tao, Blog entries on Higher order Fourier analysis, 2010.  blog 
[TV] 
T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press, 2006.  
[W1] 
J.Wolf, Notes on Bourgain's theorem, 2010.  sakai 
[W2] 
J.Wolf, Slides from my Princeton talk, 2010.  sakai 
[W3] 
J.Wolf, Slides from my Paris talk, 2009.  sakai 
[W4] 
J.Wolf, Slides on the HostKra structure theorem, 2010.  sakai 
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