Schedule for: 17w5068 - Mean Dimension and Sofic Entropy Meet Dynamical Systems, Geometric Analysis and Information Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this work, we are concerned with metric mean dimension in the case of actions of a group G by automorphisms of a compact, metrizable abelian group X. We relate metric mean dimension of this action the von Neumann rank of the dual of X as a Z(G)-module, provided this module is finitely generated (this assumption was later removed and the theorem proved in full generality by Li-Liang). This may be viewed as part of the recent phenomenon of relating the L2-invariants of a Z(G) module A to dynamical properties of the action of G on the Pontryagin dual of A. No knowledge of sofic groups, mean dimension, or von Neumann algebras will be assumed.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will show that if a free ergodic action of a countable group has positive Rokhlin entropy (or, less generally, positive sofic entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countable groups the well-known factor theorem of Sinai.

For fixed A, N, L > 0, consider the set M of all L2 functions (signals) on the real line whose spectrum (i.e. support of the Fourier transform) is contained in the interval [-A, A] and simultanuosly in a sum of N intervals, each of length at most L (which can be positioned anywhere inside the interval [-A, A]). Blind multiband sampling deals with the problem of existence of a (stable) sampling set for M, i.e. a discrete subset of the real line, such that the signal from M is uniquely determined by its values on this set. During the talk I would present effective (yet, in general, not providing perfect reconstruction) algorithms for blind multiband sampling proposed by Y. Eldar and C. Mishali, which are based on the idea of compressed sensing.

I would like to give a talk about a new Ornstein-Weiss type subadditive convergence theorem along hyperfinite exhaustions of pmp Borel equivalence relations. In collaboration with Amos Nevo (Techion), we used this result to define a new notion of entropy (cocycle entropy) for pmp actions of abritrary countable groups. It has turned out that for free actions, cocycle entropy coincides with Rokhlin entropy which is studied by Brandon Seward et al. However, using subadditive convergence techniques in order to assign entropy values to measured partitions, our definition is a priori quite different and more in line with the classical Kolmogorov-Sinai approach. Moreover, extending Elon Lindenstrauss' techniques to hyperfinite equivalence relations, we were able to settle the underlying Shannon-McMillan-Breiman theorem for a vast collection of pmp actions of general countable groups. Being of very general nature, we expect that our subadditive convergence theorem will have further important applications for non-amenable entropy theory and beyond. In the framework of future reserach activities, it is planned to define topological cocycle entropy and investigate the validity of a variational principle in terms of relation invariant measures. Further projects might concern (cocycle) mean dimension and determining the entropy for algebraic actions.

In this talk I discuss higher dimensional versions of topological Rokhlin lemmas in a noncommutative context. I also describe the use of such lemmas in the structure and classification theory of simple amenable C*-algebras. Finally, I describe how to use Cartan MASAs to extract dynamical information from crossed product C*-algebras, both in the case of classical topological dynamical systems and coarse metric spaces.

Slow dimension growth condition is one of essentials in the classification of simple amenable C*-algebras, and it implies that the algebra is regular for the purpose of the classification, i.e., the algebra absorbs the Jiang-Su algebra Z tensorially. Consider a minimal homeomorphism with mean dimension zero, then the corresponding C*-algebra is shown to have slow dimension growth, and hence is covered by the recent progress of the classification program. In particular, this includes the C*-algebra of any uniquely ergodic system. The talk is based on a joint work with George A. Elliott.

The classic coding theorem of Shannon for zero error channels has as its ergodic theoretic counterpart the Krieger generator theorem. Dobrushin, Gray, Ornstein and Kieffer gave in a similar fashion an ergodic theoretic version of Shannon's coding theorem for noisy channels. I will discuss similar ergodic theoretic versions of of the Slepian-Wolf coding theorem for multiple correlated sources which are constrained to use distinct independent channels to a common end without communicating between themselves.

Let h be a minimal homeomorphism of an infinite compact metric space. Then $rc (C* (Z, X, h)) \leq 1 + 2 mdim (h)$. If $X$ has infinitekly many connected components, then $rc (C* (Z, X, h)) \leq (1/2) mdim (h)$.

For free actions of amenable groups, we discuss an equivalent characterization of the small boundary property in terms of the existence of disjoint open towers with Folner shapes, small diameters, and the remainder having small orbit capacity. An ingredient behind this, which might be of independent interest, is a characterization of the small boundary property as having a special kind of extension on a totally disconnected space. We pose the question of whether this result can be strengthened for the remainder of the towers to be small in a stronger sense, and relate this to a desirable outcome for transformation group C*-algebras. If time permits, we may discuss actions of groups with the Tempelman condition, or possible applications for embeddings into cubical shifts. This is joint work in progress with David Kerr.

Generalizing classical work of Day and Følner for discrete groups, I will present characterizations of amenability of (not necessarily locally compact) topological groups in terms of the existence of almost invariant vectors and almost invariant finite subsets, and discuss some applications of these results, e.g., concerning the coarse geometry of Polish groups. This is joint work with Andreas Thom. Furthermore, linked with concentration of measure, the mentioned amenability criteria provide sufficient conditions for (a strong form of) extreme amenability, which can be used to prove the extreme amenability of topological groups of measurable maps. This is joint work with Vladimir Pestov.

Given a module M of the integral group ring ZG of a discrete group G, one has the von Neumann-Lueck rank of M defined in the L2-invariants theory, closely related to the l2-Betti numbers. One also has the algebraic action of G on the Pontryagin dual of M defined, which makes the dynamical invariants available. I will discuss that the von Neumann-Lueck rank corresponds to the mean dimension in the case G is sofic. Two ingredients are needed. The first is that we have to introduce the mean dimension of one action relative to an extension, which is the same as the absolute mean dimension for amenable group actions but new for sofic group actions. The second is a way to define mean length for modules of the group ring for sofic group G so that an addition formula holds, which also has application to the direct finiteness problem for group rings of G. This is joint work with Bingbing Liang.

We improve Ornstein-Weiss' quasitiling of a countable amenable group to a "perfect" tiling. The price is that the number of shapes increases drastically, nonetheless we manage to keep the entropy equal to zero. It remains a challenge to encode the system of tilings in a 0-1 subshift. So far we can do that only under the additional Tempelman's condition. Joint work with Dawid Huczek and Guohua Zhang. Last part with Maxence Phalempin.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.