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".
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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]
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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]
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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]
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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]
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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]
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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]
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The endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture
L. Guth
arXiv:0811.2251.
[presenter: Kevin Hughes, Princeton]
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Maximal Theorems for the directional Hilbert transform on the plane.
M. Lacey, X. Li
Trans. AMS math/0310346
[presenter: Prabath Silva, IU Bloomington
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Maximal operators over arbitrary sets of directions.
by N. Katz
Duke Math. J. 97 (1999), no. 1, 67-79.
[presenter: James Scurry, Georgia Tech]
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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
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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]
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Lp estimates for the Hilbert transform along a one variable vector field
by M. Bateman, C. Thiele
preprint.
[presenter: Diogo Oliveira, Berkeley]
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An incidence theorem in higher dimensions
by
Jozsef Solymosi, Terence Tao
arxiv 1103.2926
[presenter: Mark Lewko, U Texas, Austin]
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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]