SUMMER SCHOOL
Additive combinatorics

The book "Additive Combinatorics" by Terence Tao and Van Vu is a good survey of the area covered by some of the papers below.

  1. Growth and generation in SL2(Z/pZ).
     by H. A. Helfgott
    http://arxiv.org/pdf/math/0509024v4


    The goal is to prove the main theorem. A brief account on expanders and connections to Cayley graphs is also welcome.
    [presenter: Gautam]

  2. On an argument of Shkredov on two-dimensional corners.
     by Lacey, M. and McClain, W.
    Online J. Anal. Comb. No. 2 (2007), Art. 2, 21 pp. (electronic)


    Additional material can be found in Ben Green's notes

    http://www.maths.bris.ac.uk/~mabjg/papers/corners.pdf
    [presenter: Kovac]

  3. A polynomial bound in Freiman's theorem.
     by Chang, M.
    Duke Math. J. 113 (2002), no. 3, 399-419


    Prove Theorem 1 and Theorem 2. Additional material can be found in Ben Green's notes

    http://www.maths.bris.ac.uk/~mabjg/papers/PFR.pdf

    http://www.maths.bris.ac.uk/~mabjg/papers/icmsnotes.pdf
    [presenter: Stovall]

  4. A sum-product estimate in finite fields, and applications.
     by Bourgain, J.; Katz, N.; Tao, T
    Geom. Funct. Anal. 14 (2004), no. 1, 27--57


    Prove Theorem 1.1. Cover also the applications in sections 7, 8 and if time permits, also section 6.
    [presenter: Fox]

  5. New bounds for Szemeredi's Theorem, II: A new bound for r_4(N).
     by Green, B. and Tao, T
    http://arxiv.org/abs/math.NT/0610604


    Prove Thm. 1.1. The emphasis should be on the first six sections. The material in the Appendix can be briefly mentioned (if time permits).
    [presenter: Maples]

  6. An inverse theorem for the Gowers $U^3(G)$ norm.
     by Green, B. and Tao, T



    Part I: Prove Theorem 2.3. This covers the first six sections. Only focus on the material from these sections which is immediately relevant to the proof of Theorem 2.3. In particular, some of the lemmas on Bohr sets are only relevant to the second part of the presentation. So is Example 2.4. If time permits, also briefly cover Section 7.
    [presenter: Lee]

  7. An inverse theorem for the Gowers $U^3(G)$ norm.
     by Green, B. and Tao, T



    Part II: Prove Theorem 2.7. This covers sections 8, 9, and the relevant lemmas from the first six sections. Example 2.4 which is relevant for this discussion will also be presented. If time permits, mention also the other variants of Theorem 2.7, namely Theorem 10.9, Proposition 12.6 and Theorem 12.8. Skip sections 11, 13 and 14.
    [presenter: Palsson]

  8. The true complexity of a system of linear equations.
     by Gowers W. T. and Wolf J.
    http://arxiv.org/pdf/0711.0185


    Prove Theorem 3.13
    [presenter: Hart]

  9. Norm convergence of multiple ergodic averages for commuting transformations.
     by Tao T.
    http://arxiv.org/abs/0707.1117


    Prove Theorem 1.6. The main emphasis should be on sections 3-6.
    [presenter: Wang]

  10. On the Erdos-Volkmann and Katz-Tao ring conjectures.
     by Bourgain J.
    Geom. Funct. Anal. 13 (2003), no. 2, 334-365


    Part I: Cover sections O,I and II.
    [presenter: Azzam]

  11. On the Erdos-Volkmann and Katz-Tao ring conjectures.
     by Bourgain J.
    Geom. Funct. Anal. 13 (2003), no. 2, 334-365


    Part II: Cover sections III,IV and V.
    [presenter: Shen]

  12. A quantitative version of the idempotent theorem in harmonic analysis.
     by Green B. and Sanders T.
    http://arxiv.org/pdf/math/0611286


    Prove Theorem 1.3
    [presenter: Do]