Topics in Ergodic Theory and Probability (592)


In a nutshell

Rogawski

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.

TuesdayJan 19 The discrete Fourier transform[TV], [Gr1]
ThursdayJan 21 Meshulam's theorem[Gr1]
TuesdayJan 26 Freiman's theorem[N], [R], [Gr2]
ThursdayJan 28 Plunnecke's inequality[N], [R], [Gr2]
Tuesday Feb 2 Bonami-Beckner inequality and Chang's theorem[O'D], [Gr3]
ThursdayFeb 4 Student presentation and examples classProblem 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 classProblem 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]
TuesdayMar 16 ***SPRING BREAK***
ThursdayMar 18 ***SPRING BREAK***
TuesdayMar 23 The inverse theorem revisited[W2], [T2]
Thursday Mar 25 Student presentation and examples classProblem 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 classProblem Set 4

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Preliminary reading

I recommend Chapter 1 of [TV] and Sections 1-3 of [Gr1] in the bibliography below.

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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)

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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.

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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.

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Assessment

There will be no written exam, but homework, attendance, class participation and one presentation are compulsory and will be evaluated.

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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.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-)Balog-Szemeredi theorem.dvi
[M] A. Magyar, The Balog-Szemeredi 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 Host-Kra structure theorem, 2010.sakai

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This page was last updated 8th January 2010.