Harmonic Analysis, Ergodic Theory and Probability
a conference in honor of Izzy Katznelson's 70th Birthday

Partially supported by Stanford University, the Clay Mathematics Institute
and the National Science Foundation

Stanford University, December 12-14, 2004


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 self-similar 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
  • Jean-Pierre Kahane - genericity and prevalence

6:30 - 9:30

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Banquet at the Faculty Club in honor of Y. Katznelson, with Bob Osserman
as emcee, a historical talk by Jean-Pierre 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 IP-convergence 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 Furstenberg-Katznelson 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 non-hyperbolic 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 Y-valued 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, zero-one 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 permanent-determinant 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 one-dimensional lattice. They are typically studied by means of transfer matrices, which define the associated co-cycles. 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 - Tel-Aviv 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 Fourier-analytic 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 measure-preserving 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.

 


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