Topics in Ergodic Theory and Probability (592)
In a nutshell
| Instructor: | Julia Wolf, Email: jwolf137 at math.rutgers.edu |
| Lectures: | Tuesday and Thursday, 3:20-4:40pm, HLL-124 , 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 |
Syllabus | Preliminary reading | Problem sets | Student presentations | Class notes | Assessment | Bibliography
Syllabus
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 | Bonami-Beckner inequality and Chang's theorem | [O'D], [Gr3] |
| Thursday | Feb 4 | Student presentation and examples class | Problem Set 1 |
| Tuesday | Feb 9 | Balog-Szemeredi-Gowers 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 | Furstenberg-Sarkozy theorem | [Gr6] |
| Tuesday | Apr 20 | U^k seminorms and characteristic factors | [K], [W3], [W4] |
| Thursday | Apr 22 | nilmanifolds and the Host-Kra 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 |
Preliminary reading
I recommend Chapter 1 of [TV] and Sections 1-3 of [Gr1] in the bibliography below.
Problem sets
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)
Student presentations
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 square-difference 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.
Class notes
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.
Assessment
There will be no written exam, but homework, attendance, class participation and one presentation are compulsory and will be evaluated.
Bibliography
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. | |
| [Gr2] | B.J. Green, Lecture notes on "Structure theory of set addition", 2002. | |
| [Gr3] | B.J. Green, Lecture notes on "Restriction and Kakeya phenomena", 2003. | |
| [Gr4] | B.J. Green, Lecture notes on "Additive combinatorics", 2009. | |
| [Gr5] | B.J. Green, Montreal lecture notes on "Quadratic Fourier analysis", 2006. | |
| [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. | |
| [GrT1] | B.J. Green and T. Tao, An inverse theorem for the Gowers U^3 norm, 2006. | |
| [GrT2] | B.J. Green and T. Tao, New bounds for Szemeredi's theorem I, 2009. | |
| [K] | B. Kra, Ergodic methods in additive combinatorics, 2006. | |
| [L] | V. Lev, The (Gowers-)Balog-Szemeredi theorem. | dvi |
| [M] | A. Magyar, The Balog-Szemeredi theorem. | |
| [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. | |
| [R] | I.Z. Ruzsa, Lecture notes on "Sumsets and structure", 2008. | |
| [S] | K. Soundararajan, Lecture notes on "Additive combinatorics", 2007. | |
| [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 Host-Kra structure theorem, 2010. | sakai |