Activities This Week

What is the relationship between a Souslin tree and its reduced powers? and is there any difference between, say, the reduced w-power and the reduced w1-power of the same tree? In this talk, we shall present tools recently developed to answer these sort of questions. For instance, these tools allow to construct an w6-Souslin tree whose reduced w_n-power is Aronszajn iff n is not a prime number.

This is joint work with Ari Brodsky.

We describe a striking conjectured relation between ``dimensions’’ in topological dynamics and C-algebras. (No previous knowledge of C-algebras or dimension theory will be assumed.) Let $X$ be a compact metric space, and let $h \colon X \to X$ be a minimal homeomorphism (no nontrivial invariant closed subsets). The mean dimension ${\mathit{mdim}} (h)$ of $h$ is a dynamical invariant, which I will describe in the talk, and which was invented for purposes having nothing to do with C-algebras. One can also form a crossed product C-algebra $C^* ({\mathbb{Z}}, X, h)$. It is simple and unital, and there is an explicit description in terms of operators on Hilbert space, which I will give in the talk. The radius of comparison ${\mathit{rc}} (A)$ of a simple unital C-algebra $A$ is an invariant introduced for reasons having nothing to do with dynamics; I will give the motivation for its definition in the talk (but not the definition itself). It has been conjectured, originally on very thin evidence, that the radius of comparison of $C^({\mathbb{Z}},X,h)$ is equal to half the mean dimension of $h$ for any minimal homeomorphism $h$.

In this talk, I will give elementary introductions to mean dimension, the crossed product construction, and the ideas behind the radius of comparison. I will then describe the motivation for the conjecture and some partial results towards it.

נעיין בשורה של נמלים, כך שכל אחת רצה אחרי הנמלה שלפניה. אם מספר הנמלים הוא סופי אזי יש נמלה ראשונה. שלכאורה אין לה אחרי מי לרוץ, וכולן יגיעו ליעד שהיא קובעת. יותר מעניין אם נניח שהיא רצה אחרי האחרונה בטור. מצב כזה נקרא “ריצה ציקלית” ועל פי כללי מהירות טבעיים הן יתכנסו לממוצע של נקודות ההתחלה שלהן. ברם, אם מספר הנמלים הוא אינסופי בן מניה, ולטור אין סוף בשני הכיוונים המצב הרבה יותר מורכב. זה נושא הדיון שלנו.