Speaker: Nicolas Monod (Ecole Poly Fed. de Lausanne)

Title: Cutting and pasting: Frankenstein’s method in group theory

Abstract:
We have known for a century that a ball can be decomposed into five pieces and these pieces rearranged so as to produce two balls of the same size as the original.
This apparent paradox has led von Neumann to the notion of amenability which is now much studied in many areas of mathematics.
However, the initial paradox has remained tied down to an elementary property of free groups of rotations for most of the 20th century. I will describe recent progress leading to new paradoxical groups.

Speaker: Pekka Koskela (University of Jyväskylä)

Title: A geometric characterization for planar Sobolev extension domains

Speaker: Peter Binev (USC)

Title: Data Assimilation in Reduced Modeling

Abstract: We consider the problem of optimal recovery of an element $u$ of a Hilbert space $\mathcal{H}$ from measurements of the form $\ell_j(u)$, $j=1, \dots,m$, where the $\ell_j$ are known linear functionals on $\mathcal{H}$. Problems of this type are well studied and usually are carried out under an assumption that $u$ belongs to a prescribed model class, typically a known compact subset of $\mathcal{H}$.
Motivated by reduced modeling for solving parametric partial differential equations,
we consider another setting where the additional information about $u$ is in the form of how well $u$ can be approximated by a certain known subspace $V_n$ of $\mathcal{H}$ of dimension $n$,
or more generally, in the form of how well $u$ can be approximated by each of a sequence of nested subspaces $V_0\subset V_1 \cdots \subset V_n$ with each $V_k$ of dimension $k$. A recovery algorithm for the one-space formulation was proposed in
[Y. Maday, A.T. Patera, J.D. Penn and M. Yano (2015), {\em A parametrized-background data-weak approach to variational data assimilation: Formulation, analysis, and application to acoustics}, Int. J. Numer. Meth. Engng, 102: 933-965].
We prove that their algorithm is optimal and show how the recovery problem for the one-space problem, has a simple formulation, if certain favorable bases are chosen to represent $V_n$ and the measurements. Our major contribution is to analyze the multi-space case. It is shown that, in this multi-space case, the set of all $u$ that satisfy the given information can be described as the intersection of a family of known ellipsoids in $\mathcal{H}$. It follows that a near optimal recovery algorithm in the multi-space problem is provided by identifying any point in this intersection.
It is easy to see that the accuracy of recovery of $u$ in the multi-space setting can be much better than in the one-space problems. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem and one of them is analyzed in detail. This analysis includes an a posteriori estimate for the performance of the iterates. These a posteriori estimates can serve both as a stopping criteria in the algorithm and also as a method to derive convergence rates.
Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for $u$.

This is a joint work with Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk. The results are available at [arXiv:1506.04770].

A short meeting to provide an overview of the reorganized graduate program.

Speaker: Alexander Volberg (Michigan State)

Title: Non-homogeneous harmonic analysis, Geometric Measure Theory and fine structures of harmonic measure

One of the goals of harmonies analysis is to study singular integrals. Singular integrals are ubiquitous objects in PDE and in Mathematical Physics, and as it turned out recently, play an important part in Geometric Measure Theory. They have various degrees of singularity, and the simplest ones are called Calder’on–Zygmund operators. Their theory was completed in the 50′s by Zygmund and Calder’on. Or it seemed like that. The last 20 years saw the need to consider CZ operators in
very bad environment, so kernels are still very good, but the ambient set has no regularity whatsoever.
Initially such situations appeared from the wish to solve some outstanding problems in complex analysis: such as Painlev’e’s, Ahlfors’, Denjoy’s and Vitushkin’s problems.
But recently it turned out that the non-homogeneous harmonic analysis (=the analysis of CZ operators on very bad sets and measures) is also very fruitful in the part of Geometric Measure Theory that deals with rectifiability, and also helps a lot to understand the geometry of harmonic measure. The research on the geometric properties of harmonic measures was pioneered in the U. of M. in the 60-70′s by George Piranian, and came to fruition in the later works (80-90′s) by Lennart Carleson, Nikolai Makarov, Jean Bourgain, Peter Jones and Tom Wolff. But most of the results concerned the structure of harmonic measure of planar domains. As an example of the use of non-homogeneous harmonic analysis, we will show how it allows us to understand very fine property of harmonic measure of any domain in any dimension.