| Time/Day | Monday | Tuesday | Wednesday | Thursday | Friday |
| 10:00-11:00 | C. Thiele | C. Thiele | C. Thiele | C. Thiele | C. Thiele |
| 11:30-12:30 | A. Iosevich | C. Muscalu | A. Magyar | X. Li | M. Lacey |
| lunch break | C. Thiele, begins at 14:00
or earlier. |
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| 14:30-15:30 | C. Thiele | C. Thiele | C. Thiele | ||
| 16:00-17:00 | A. Seeger | I. Laba | K. Oskolkov | ||
| 7 pm | light dinner at Lacey's | ||||
| Alex Iosevich: | "Convex sequences and weighted incidence theorems." |
| Abstract: |
We show that the equation bi1+bi2+
.... +bid=bid+1+ ..... +bi2d
has O( N^{2d-2 + 2^{-d+1} } ) solutions for any strictly convex sequence {bi}i=1..N without any additional arithmetic assumptions. The proof is based on weighted incidence theory and an inductive procedure which allows us to effectively deal with higher dimensional interactions. We also explain a connection between this problem and the Falconer distance problem in geometric measure theory. |
| Andreas Seeger: | "Some maximal inequalities." |
| Abstract: | I will talk about some recent results concerning
maximal functions associated to Fourier multipliers of Mikhlin-Hörmander
type (joint with M. Christ,L. Grafakos and P. Honzik and about some
vector valued inequalities for maximal operators (joint with M. Christ). |
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| Camil Muscalu: | "Paraproducts on the bi-disc." |
| Abstract: |
We will describe a theorem which generalizes the
classical Coifman-Meyer multilinear theorem to the bi-parameter setting
of the bi-dics. This is recent joint work with Jill Pipher, Terry
Tao and Christoph Thiele. |
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| Izabella Laba: | "Distance sets corresponding to polygonal norms." |
| Abstract: |
Let X be the 2-dimensional plane equipped with a
non-Euclidean norm in which the unit ball is a polygon K, and let S be a well-distributed
subset of X. We address the question of how small the distance set of S
in X can be, depending on properties of K. In particular, we prove that
there is a well-distributed S with a distance set of bounded density if and
only if there is a coordinate system in which all sides of K have algebraic
slopes. The proofs display a connection to additive number theory, in particular
to Balog-Szemeredi type results. |
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| Akos Magyar: | "On distance sets of large subsets of integer points." |
| Abstract: |
A result of Y. Katznelson and B.Weiss states that
if A is a measurable subset of R2 of positive upper density, its
distance set: d(A)={ |x-y| : xεA , yεA } contains all large numbers. We prove a similar result for subsets A of Zn of positive density, namely that: d2(A )={ |m-l|2 : mεA , lεA } contains all large multiples of a fixed number Q, where Q depends only on the density of the set A . If time permits we discuss some related results, such as the fact that the difference set A-A intersect homogenous algebraic sets of the form $P(m)=0$. |
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| Xiaochun Li: | "Hilbert transforms on smooth families of lines." |
| Abstract: | Let v be a vector field from R^2 to the unit circle
S^1. We study the operator H_vf(x)= p.v. \int_{-1}^{1} f(x-tv(x)) dt/t We prove that if the vector field $v$ has $1+\epsilon$ derivatives, then $H_v$ extends to a bounded map from $L^2$ onto itself. Our method is based on a crucial maximal function estimate. This is joint work with Michael Lacey. |
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| Konstantin Oskolkov: | "Talbot's effect for Schroedinger equation with integrable potential." |
| Abstract: | In 1836, W.H.F. Talbot, the English inventor of photography,
observed a phenomenon of self-reproduction (revival), with variable scaling
factors, of the original optical image on the grate. This phenomenon is especially
"visible" in the solutions of the time-dependent Schroedinger equation
of a free particle with the periodic initial data. The classical Gauss' sums
play the role of scaling factors. In the talk, I will describe an approach
to Schroedinger equations with rather non-smooth potentials, and show that
Talbot's phenomenon is still an inherent feature of the solutions of such
equations. |
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| Michael Lacey: | "Hankel Operators, Commutators, Product BMO." |
| Abstract: |
We discuss the boundedness of Hankel operators in
two and more complex variables. The natural extension of the classcial Nehari
theorem has several equivalent formuations. These are in terms of weak factorization
of H^1, boundedness of commutators, and the product BMO theory of Chang and
Fefferman. We will explain the theorem, its equivalences, and the proof of
the theorem, which sheds new light on the nature of product BMO. This is
joint work with Sarah Ferguson and Erin Terwilleger. |