Activities This Week

Let G be an algebraic group over a local field. We will show that the image of G under an arbitrary continuous homomorphism into any (Hausdorff) topological group is closed if and only if the center of G is compact. We will show how mixing properties for unitary representations follow from this topological property.

In 2003, a big bang occurred when the famous manuscript by Kechris, Pestov, and Todorcevic was uploaded to Arxiv, in which occurred a collusion between topological dynamics, Ramsey theory, and model theory, that left these fields not as they were before - the paper creates a strong link between properties of the automorphism group of a structure (as a topological group) and combinatorial properties of colourings of the same structure.

In this talk, I will survey the history leading to that bang, from Ramsey himself, through Ehrenfeucht and Mostowsky (and from the other side from Mitchell, through Herer, Christensen, Gromov, Milman, Frustenberg, Weiss, Glasner, up to Pestov) and finally Kechris, Pestov, and Todorcevic.

I will also survey applications in modern model theory, and, as time allows, a non-empty set of recent contributions to which I’m proud to be part of.

In a joint work with Tattwamasi Amrutam and Eli Glasner, we study intermediate $C^*$-algebras of the form $C^*_r(\Gamma) < \mathcal{A} < C(X) \rtimes \Gamma$, where $\Gamma \curvearrowright X$ is a given minimal action of a countable discrete group $\Gamma$ on a compact space $X$. Every $\Gamma$-factor of the given topological dynamical system $X \rightarrow Y$ gives rise to an intermediate algebra of the form $\mathcal{A} = C(Y) \rtimes \Gamma$, and by analogy we may think of more general factors as representing ‘‘non-abelian’’ factors. Let us call the dynamical system ``reflecting’’ if the only intermediate algebras come from dynamical factors.

We show that another source of intermediate algebras comes from ideals in $C^*_r(\Gamma)$. In particular, we show that if $\Gamma$ is not $C^*$-simple, $X$ admits a $\Gamma$-invariant probability measure, and the cardinality of $X$ is at least $3$, then the system is not reflecting.

In the talk, I will focus on the example highlighted in the title. In this case, we obtain a complete description of all intermediate algebras in terms of some combinatorial data described in terms of ideals in $C^*_r(\mathbb{Z})$. In particular there are uncountably many intermediate algebras, as compared to only countably many dynamical factors. I will show how our description can often be used in order to obtain structural information about the algebras, such as simplicity, the existence of a center, and a closed formula for the algebra generated by two given ones.

לעיתים קרובות במערכות בעולם ישנן כמה פתרונות אפשריים, אבל כל פתרון לבדו אינו אופטימלי. דוגמא שאנו נשתמש בה לצורך המחשה היא טיפול (תרופתי) מסוים למחלה. באופן קלאסי, חולה המגיעה לקבל סדרת טיפולים מקבלת סדרה מאפשרות אחת בלבד של התרופות האפשריות. בשנים האחרונות ישנן רופאות אשר נותנות סדרה של טיפולים מתרופה א, ואחר כך סדרה נוספת מתרופה ב.

אנו נשאל את השאלה כיצד אפשר לשלב שני סוגי טיפול באופן מיטבי (ורציונאלי)? נדגים בשיחה שלנו כיצד הופכים בעיה כזו לבעיה מתמטית, ונראה את הפתרון (היעיל) של הבעיה הזו.

השחקנית המתמטית המרכזית המופיעה בסיפור שלנו נקראת ״שרשרת מרקוב״. אין צורך בידע מוקדם, יש יתרון קטן למי שיודעת כפל מטריצות ו/או הסתברות בסיסית.