SUMMER SCHOOL
Geometric and Fourier analytic questions in Euclidean space

Note: this list of topics does not aim to be a representative cross-section of current research in "Geometric and Fourier analytic methods".

  1. Decay of circular means of Fourier transforms of measures.
    by T. Wolff
    Internat. Math. Res. Notices 1999, no. 10, 547 - 567.

    [presenter: Taryn Flock, Berkeley]

  2. Algebraic methods in discrete analogs of the Kakeya problem.
    by L. Guth and N. Katz
    Adv. Math. 225 (2010), no. 5, 2828-2839.
    If time allows, presenter may also discuss
    On the size of Kakeya sets in finite fields.
    by Z. Dvir
    J. Amer. Math. Soc. 22 (2009), no. 4, 1093-1097.

    [presenter: Faruk Temur, UIUC]

  3. On the Erdos distinct distance problem in the plane
    by L. Guth and N. Katz
    arXiv:1011.4105.
    (22 pages)

    [presenter: Matt Bond, Michigan State/UBC]

  4. Bounds on oscillatory integral operators based on multilinear estimates.
    J. Bourgain and L. Guth
    arXiv:1012.3760.

    Presenter should focus on Sections 2 and 4 of this paper. Section 6 can be covered by presenter of next topic.
    [presenter: Josh Zahl, UCLA]

  5. The Bochner-Riesz conjecture implies the restriction conjecture.
    T. Tao
    Duke Math. J. 96 (1999), no. 2, 363--375.
    Presenter should focus on Sections 1-3, we do not emphasize the additional remarks of Section 4. However, if time allows, it would be very nice to discuss Section 6 of the previous topic by Bourgain and Guth.
    [presenter: Yi Hu, UIUC]

  6. A bilinear Fourier extension theorem and applications to the distance set problem.
    B. Erdogan
    Int. Math Res. Notices 2005:23 from Erdogans webpage


    [presenter: Francesco DiPlinio, IU Bloomington]

  7. The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture
    L. Guth
    arXiv:0811.2251.


    [presenter: Kevin Hughes, Princeton]

  8. Maximal Theorems for the directional Hilbert transform on the plane.
    M. Lacey, X. Li
    Trans. AMS math/0310346

    [presenter: Prabath Silva, IU Bloomington

  9. Maximal operators over arbitrary sets of directions.
    by N. Katz
    Duke Math. J. 97 (1999), no. 1, 67-79.

    [presenter: James Scurry, Georgia Tech]

  10. A remark on the maximal function associated to an analytic vector field.
    by J. Bourgain
    Analysis at Urbana, Vol. I (Urbana, IL, 1986-1987), 111-132, London Math. Soc. Lecture Note Ser., 137, Cambridge Univ. Press, Cambridge, 1989.
    The emphasis is on Introduction and Sections 3 and following, Section 2 can be discussed briefly if time allows.
    [presenter: Stefan Steinerberger, Bonn

  11. Single annulus Lp estimates for Hilbert transform along vector fields.
    by M. Bateman
    preprint (available shortly) This paper has a fair amount of overlap with the paper by Lacey and Li above. Presenters may want to coordinate for efficiency and to avoid duplication.

    [presenter: Michael Bateman, UCLA]

  12. Lp estimates for the Hilbert transform along a one variable vector field
    by M. Bateman, C. Thiele
    preprint.



    [presenter: Diogo Oliveira, Berkeley]

  13. An incidence theorem in higher dimensions
    by Jozsef Solymosi, Terence Tao
    arxiv 1103.2926



    [presenter: Mark Lewko, U Texas, Austin]

  14. The power law for the Buffon needle probability of the four-corner Cantor set
    by Fedor Nazarov, Yuval Peres, Alexander Volberg
    arXiv:0801.2942

    If time allows, it would be nice to do in the generality of the following generalization:
    The Favard length of product Cantor sets.
    by Izabella Laba, Kelan Zhai
    arxiv 0902.0964


    [presenter: Krystal Taylor, U. Rochester]