Schedule 
Sunday,
December 12  Jordan
Hall (450 Serra Mall, Bldg. 420), Room 040 
9:15
 9:45 
Coffee and Registration

9:45  10:00 
Opening
remarks 
10:00  10:45 
Jean
Bourgain  Harmonic analysis aspects of Ginzburg Landau minimizers 
11:00  11:45 
Wilhelm
Schlag  On ergodic matrix products and fine properties of
the eigenfunctions of the associated difference equations 
12:00  2:00 
Lunch
break 
2:00  2:45 
Ben
Green  Progressions of length four in finite fields 
3:00  3:30 
Coffee
and refreshments 
3:30  4:15 
Terence
Tao  Quantitative ergodic theory 
4:30  5:20 
Open
problems session 

 Izzy Katznelson  Topological
Recurrence and Bohr Recurrence
 Mate Wierdl  Multiple convergence
and recurrence along random sequences
 James Campbell  Oscillation
and variation estimates for the Carleson operator, and rotated
ergodic averages.
 Daniel Rudolph  Another Ergodic
theorem, maybe
 Idris Assani  On the pointwise
convergence of ergodic averages along cubes for not necessarily
commuting measure preservng transformations
 Yuval Peres  Projections of
planar Cantor sets and a related Kakeya set

Monday,
December 13  Jordan
Hall (450 Serra Mall, Bldg. 420), Room 040 
9:30  10:15 
Mikhail
Sodin  Zeroes of Gaussian Analytic Functions 
10:30  11:00 
Coffee
and refreshments 
11:00  11:45 
Yuval
Peres  Determinantal
Processes And The IID Gaussian Power Series 
12:00  2:00 
Lunch
break 
2:00  2:45 
Assaf
Naor  The Lipschitz Extension Problem 
3:00  3:30 
Coffee
and refreshments 
3:30  4:15 
Elon
Lindenstrauss  Invariant measures for partially isometric
maps 
4:30  5:20 
Open problems
session 

 Scott Sheffield  Infinity harmonic
functions
 Boris Solomyak  Interior points
of selfsimilar sets
 Emmanuel Lesigne  Two questions
around ergodic disintegration
 Don Ornstein
 Sinan Gunturk  Invariant sets
of a class of piecewise affine maps on Euclidean space and/or
other problems
 JeanPierre Kahane  genericity
and prevalence

6:30  9:30
.
. 
Banquet
at the Faculty
Club in honor of Y. Katznelson,
with Bob Osserman
as emcee, a historical talk by JeanPierre Kahane and a toast
by Yonatan
Katznelson.

Tuesday,
December 14  Jordan
Hall (450 Serra Mall, Bldg. 420), Room 040 
9:30  10:15 
Benjamin Weiss  On entropy of stochastic processes

10:30  11:00 
Coffee
and refreshments 
11:00  11:45

Bryna
Kra  Lower bounds for
multiple ergodic averages 
12:00  1:30 
Lunch
break 
1:30  2:15 
Vitaly
Bergelson  IP versus Cesaro 
2:30  3:15 
Hillel
Furstenberg  Hausdorff Dimension of Orbit Closures and Transversality
of Fractals 
3:30  4:00

Conference conclusion (Coffee and refreshments)

Abstracts 
Vitaly
Bergelson  Ohio State University
Title: IP versus Cesaro
Abstract: While traditional ergodic theory concerns itself with
the study of the limiting behavior of various Cesaro averages,
IP ergodic theory utilizes the notion of IPconvergence which
is based on Hindman's finite sums theorem. This usually allows
one to refine and enhance the results obtained via the Cesaro
averages. An example of such an enhancement is provided by the
FurstenbergKatznelson IP Szemeredi theorem. We shall review
some of recent developments in IP ergodic theory and formulate
new interesting problems and conjectures.

Jean
Bourgain  Institute
for Advanced Study
Title: Harmonic analysis aspects of Ginzburg Landau minimizers

Hillel
Furstenberg  Hebrew
University
Title: Hausdorff Dimension of Orbit Closures and Transversality
of Fractals
Abstract: It sometimes happens that the dynamics of a commuting
pair of transformations is more easily described than that of
the individual transformations. We describe a situation of this
type and the possibility that "large" orbits under
the two transformations together and a "tiny" orbit
under one of the transformations may imply a "large"
orbit under the other. Here size is measured by Hausdorff dimension,
and we are naturally led to problems regarding fractals.

Ben
Green  University of British Columbia
Title: Progressions of length four in finite fields
Abstract: If G is a finite abelian group with size N, define
r_4(G) to be the size of the largest subset of G which does
not contain four distinct elements in arithmetic progression.
Gowers showed that r_4(Z/NZ) is bounded above by N/(log log
N)^c for some c > 0. I would like to discuss some joint work
with Terry Tao in which we show that r_4(G) = N/(log N)^c for
the particular group G = (Z/5Z)^n. The argument involves `quadratic
fourier analysis' on `quadratic submanifolds': I will attempt
to explain what that
means. I will also discuss the prospects of generalising out
result to arbitrary G.

Bryna
Kra  Penn State University & Northwestern
Title: Lower bounds for multiple ergodic averages
Abstract: Recent developments in multiple ergodic averages have
lead to new combinatorial consequences. I will discuss what
happens to a set of integers with positive upper density when
it is translated along certain sequences of integers and one
takes the intersection of these sets. It turns out that substantially
different behavior occurs if the sequence is formed using linear
integer polynomials or formed using rationally independent integer
polynomials. The first case corresponds to Szemeredi's Theorem
for arithmetic progressions, and we have tight lower bounds
on the size of the intersection for progressions of length 3
and 4, but no such bounds for longer progressions. For independent
polynomials, such as polynomials of differing degrees, we have
tight lower bounds on the size of every intersection.

Elon Lindenstrauss  NYU & Clay Mathematics
Institute
Title: Invariant measures for partially isometric maps
Abstract: Consider an irreducible nonhyperbolic toral automorhpism
on the four turus (which is the minimal dimension possible).
This map contracts in one (one dimensional) direction, expands
in another, and acts isometrically by rotations on the remaining
two dimensions. This is prototypical to a larger class of maps
which are partially isometric in the same sense. One invariant
measure for this map is Lebesgue measure, which as an abstract
measure preserving system has been shown by Katznelson to be
Bernoulli, just like for the hyperbolic case. However if one
consider the class of all invariant measures, there are nontrivial
restrictions, and thare are many intriguing open questions.
As I will explain in my talk,
classifying these invariant measures is closely related to Furstenberg's
famous conjecture about x2 x3 invariant measures on the circle
R/Z.
Part of my talk will be based
on joint work with Klaus Schmidt

Assaf
Naor  Microsoft Research
Title: The Lipschitz Extension Problem
Abstract: The Lipschitz extension problem asks for conditions
on a pair of metric spaces X,Y such that every Yvalued Lipschitz
map on a subset of X can be extended to all of X with only a
bounded multiplicative loss in the Lipschitz constant. This
problem dates back to the work of Kirszbraun and Whitney in
the 1930s, and has been extensively investigated in the past
two decades. The methods used in this direction are based on
geometric, analytic and probabilistic arguments. In particular,
the methods involve stable processes, random projections, random
partitions of unity and the analysis of Markov chains in metric
spaces. In this talk we will present the main known results
on the Lipschitz extension problem, as well as several recent
breakthroughs.

Yuval
Peres  University of California, Berkeley
Title: Determinantal Processes And The IID Gaussian Power
Series
Abstract: Discrete and continuous point processes where the
joint intensities are determinants arise in Combinatorics (noncolliding
paths, random spanning trees) and Physics (Fermions, eigenvalues
of Random matrices). For these processes the number of points
in a region can be represented as a sum of independent, zeroone
valued variables, one for each eigenvalue of the relevant operator.
In recent work with B. Virag, we found that for the Gaussian
power series with i.i.d. coefficients, the zeros form a determinantal
process, governed by the Bergman Kernel. A partition identity
of Euler, and a permanentdeterminant identity of Borchardt
(1855) appear in the proof. The determinantal description yields
the exact distribution of the number of zeros in a disk. The
process of zeros is invariant for a natural dynamics (we'll
see a movie).

Wilhelm
Schlag  Caltech
Title: On ergodic matrix products and fine properties of
the eigenfunctions of the associated difference equations
Abstract: We will present some recent work on difference equations
on the onedimensional lattice. They are typically studied by
means of transfer matrices, which define the associated cocycles.
Assuming positive Lyapunov exponents, we will discuss properties
of the distribution of eigenvalues in the stochastic limit.
This is joint work with Michael Goldstein.

Misha
Sodin  TelAviv University
Title: Zeroes of Gaussian Analytic Functions
Abstract: Geometrically, zeroes of a Gaussian analytic function
are intersection points of an analytic curve in a Hilbert space
with a randomly chosen hyperplane. Mathematical physics provides
another interpretation as a gas of interacting particles. In
the last decade, these interpretations influenced progress in
understanding statistical patterns in the zeroes of Gaussian
analytic functions, and led to the discovery of remarkable canonical
models with invariant zero distribution. We shall discuss some
of recent results in this area. The talk is based on joint works
with Boris Tsirelson.

Terence Tao
 UCLA
Title: Quantitative ergodic theory
Abstract: There are many techniques used in the study of multiple
recurrence (or equivalently in detecting arithmetic progressions
and similar objects). The combinatorial and Fourieranalytic
approaches tend to work in "finitary" settings such
as the cyclic group Z/NZ, whereas the ergodic theory approach
works instead in the setting of an infinite measurepreserving
system, with the two settings being linked via a transference
principle which requires the axiom of choice. The ergodic theory
methods are technically simpler (modulo standard machinery such
as measure theory and conditional expectation) and are more
easily applied to a wide range of problems, but the combinatorial
and Fourier methods give more concrete bounds and can apply
to certain settings (notably to subsets of sparse pseudorandom
sets, of which the primes are a good example) for which there
does not yet appear to be an infinitary analogue.
In this talk we discuss a compromise
approach, which we dub "quantitative ergodic theory",
in which we work in the finitary setting
of Z/NZ but still exploit the philosophy and ideas from ergodic
theory (e.g. sigma algebra factors, almost periodic functions,
conditional expectation). This for instance allows one to give
an elementary proof of Szemeredi's theorem based on ergodic
methods (requiring no Fourier analysis, and no sophisticated
combinatorial tools otherthan van der Waerden's theorem) which
also provides a quantitative (but rather poor) bound. This theory
was also a crucial component of the recent result of Ben Green
and the author that the primes contain arbitrarily long arithmetic
progressions.

Benjamin Weiss
 Hebrew Univ. of Jerusalem
Title: On entropy of stochastic processes
Abstract: I will discuss some new observations on the relationship
between the entropies of two stochastic processes, one of
which is a linear factor of the other. The notion of "Finitely
Observable" functions of processes will be defined and
a remarkable new characterization of the entropy will be given.
