May 8
23 p.m.
library

David Damanik (Rice University)
The repetition property and applications to spectral
theory
We present joint work with Michael Boshernitzan on a fairly
weak
repetition property of a topological dynamical system, which
implies for generic sampling functions a property for their
orbits, which was introduced by Alexander Gordon in 1976
to exclude eigenvalues of the associated Schr"odinger
operators. A particular consequence is that a typical skewshift
model with a
generic sampling function has no eigenvalues for almost
every element in the
hull.

Apr.
4
23 p.m.
library 
Vitaly Bergelson (Ohio State
University)
Van der Corput trick and its ramifications.
Let (x_n) be a sequence of real numbers. Van der Corput's
difference theorem states that if for every k in N the sequence
x_(n+k)x_n, n=1,2,... is uniformly distributed mod1, then
the sequence (x_n) also is uniformly distributed mod1. For
example, the uniform distribution of the sequence (n^2a mod1),
where a is an irrational number, is an immediate consequence
of this theorem. We will discuss various refinements of this
result which lead to interesting applications in ergodic theory
and combinatorics.

Apr.
3
23 p.m.
(Rm 230 or library) 
Vitaly Bergelson (Ohio State
University)
Applications of Poincare recurrence theorem to combinatorics
and number theory.
Poincare recurrence theorem, one of the earliest results
in ergodic theory, was introduced in 1890 by Henri Poincare
in his famous Acta Mathematica paper on the stability of solar
system. The goal of this lecture is to discuss some modern
applications of this classical theorem to problems in combinatorics,
algebra and number theory. 
Mar.
27
1:00 p.m.
Library 
Ilya Shkredov
3D corners 
Mar.
25
1:00 p.m.
Library 
Ilya Shkredov
Theorem on 2D corners

Mar.
20
1:00 p.m.
Library 
Michael Lacey
Progressions of Length 3 in finite fields. RothMeschulum
argument 
Feb.
7 
Serban Costea
Sobolev capacity and Hausdorff measures in metric measure
spaces 
Feb.
5 
1 pm Tuesday:
Michael Lacey
Lipschitz Kakeya Maximal Function

2 pm Tuesday:
Daryl Geller
Heat Trace Asymptotics, Spherical Wavelets and Dark
Energy
This is joint work with Azita Mayeli. The heat trace asymptotics
on the sphere $S^n$ are known explicitly. (That is, if $\Delta$
is the LaplaceBeltrami operator on the sphere, one knows
the numbers $a_m$ for which $\mbox{tr}(e^{t\Delta}) \sim
\sum_{m=0}^{\infty} t^{(mn)/2}a_m$. I. Polterovich recently
gave a very explicit expression for them.) On the sphere,
the heat kernel $K_t(x,y) := g_t(x \cdot y)$ is a function
of $x \cdot y$. Using the heat trace asymptotics we show how
one can compute the Maclaurin series for $4\pi g_t(\cos \theta)$.
For small $t$ this gives rise to what appears to be an excellent
approximation to the heat kernel, \[ 4\pi g_t(\cos \theta)
\sim \frac{e^{\theta^2/4t}}{s}[(1+\frac{t}{3}+\frac{t^2}{15}+\frac{4t^3}{315}+\frac{t^4}{315})
+ \frac{\theta^2}{4}(\frac{1}{3}+\frac{2t}{15}+\frac{4t^2}{105}+\frac{4t^3}{315})],
\] on $S^2$. Using this approximation we are able to approximately
evaluate new wavelets for the sphere, which we have devised,
and which have significant advantages over those currently
in use. Such wavelets are being used by teams of statisticians
and astrophysicists to analyze cosmic microwave background
radiation, in particular to estimate the percentage of dark
energy in the universe and its properties. (We have been invited
to visit one of these teams in March, in Rome.) 
Jan.
24 
Brett Wick (University of
South Carolina)
Bounded analytic projections 
Jan.
29 
1 pm Tuesday:
Alex Iosevich (University of Missouri)
Sums and products in finite fields via higher dimensional
geometry and Fourier analysis
2 pm Tuesday
Hrant Hakobyan (University of Toronto)
Conformal dimension: Cantor sets and curve families.
The infimal Hausdorff dimension of all quasysimmetric
(qs) images of a metric space X is called Conformal dimension
of X. A subset X of a line is called qs thick if every qs
self map of the line maps X to a set of positive measure.
Bishop and Tyson asked if there are sets which are not qs
thick but still have conformal dimension 1. We will answer
this affirmatively by proving that many Cantor sets of Hausdorff
dimension 1 have also Conformal dimension 1.
Very often the obstruction for minimizing the dimension
of a space X is the existence of "sufficiently many"
curves in X. If the time permits we will define a dimension
type quasiconformal invariant for a curve family in the
plane. We will discuss some examples and will illustrate
some connections of this invariant with the Conformal dimension
of X.

Jan.
24 
Thomas Hytonen
Thoughts about tent spaces 
Jan.
22 
Ignacio UriarteTuero
Removability problems for bounded, BMO and H\"{o}lder
quasiregular mappings
A classical problem in complex analysis is to characterize
the removable sets for various classes of analytic functions:
H\"{o}lder, Lipschitz, BMO, bounded (this last case gives
rise to the analytic capacity and the Painlev\'{e} problem
which has been recently solved by Tolsa.) One can ask the
same questions in the setting of Kquasiregular maps (since
they are a Kquasiconformal map followed by an analytic map.)
Most of the bounded case was dealt with in a joint paper with
K. Astala, A. Clop, J.Mateu and J.Orobitg, [ACMOUT]. The BMO
case was dealt with in [ACMOUT] except for a gap at the critical
dimension (Question 4.2 in [ACMOUT].) I answered the question
filling the gap in [UT]. The Lipschitz case was dealt with
by A. Clop, as well as most of the H\"{o}lder case, where
again a gap at the critical dimension was left. In a joint
paper with A. Clop [CUT] we closed the gap. I will summarize
the results and give some ideas of the proofs in the above
papers. The talk will be selfcontained.
References:
[ACMOUT] Kari Astala, Albert Clop,
Joan Mateu, Joan Orobitg and Ignacio UriarteTuero. Distortion
of Hausdorff measures and improved Painlev\'{e} removability
for bounded quasiregular mappings. Duke Math J., to appear.
[CUT] Albert Clop and Ignacio UriarteTuero.
Sharp Nonremovability Examples for H\"{o}lder continuous
quasiregular mappings in the plane. Preprint.
[UT] Ignacio UriarteTuero. Sharp
Examples for Planar Quasiconformal Distortion of Hausdorff
Measures and Removability. Submitted.
