Activities This Week

הממוצע החשבוני גיאומטרי $M(a,b)$ של שני מספרים $a,b$ מוגדר כגבול משותף של שתי סדרות מספרים ממשיים: אם $a,b$ הם שני מספרים ממשיים אז “מחליפים אותם” בממוצע הרגיל שלהם $(a+b)/2$ ובממוצע הגיאומטרי שלהם $\sqrt{ab}$ וממשיכים בתהליך ההחלפה. לא קשה להראות שהמספרים מתקרבים זה לזה והגבול המשותף הוא $M(a,b)$.

גאוס שם לב שהמספר הזה קשור לאינטגרלים שקשה מאוד לחשב אותם ובמיוחד להיקף של הלמינסקטה $r^2=\cos(2\phi)$ של ברנולי. נסביר את הקשר הזה וננסה להבין אותו מנקודת מבט מודרנית. בדרך ניפגש עם עקומים אליפטיים ותבניות מודולריות.

With each simple connected graph $G$ with $n$ vertices one can associate a generalized configuration space $Conf_{G}(n,X)$ consisting of $n$ points $(p_1,\ldots,p_n)$ on $X$, with $p_i\neq p_j$ whenever vertices $i$ and $j$ are connected by an edge. For $X=\mathbb{C}$ the generalized configuration space admits a compactification that coincides for a complete graph with Deligne-Mumford compactification of moduli spaces of rational curves with $n$ marked points. The latter is known under the name modular compactification. I will explain what kind of natural algebraic structure exists in the union of these spaces and how one can extract information about the Hilbert series of cohomology rings for different collections of graphs. Surprisingly, the same method can be used to obtain the generating series for different combinatorial data assigned with a graph: such as the number of Hamiltonian paths, Hamiltonian cycles, Acyclic orientations and Chromatic polynomials. The talk is based on the joint work with my student D.Lyskov: https://arxiv.org/abs/2406.05909

Ratio-limit boundaries were first studied for their applications to Toeplitz C-algebras of random walk, but are also interesting in their own right for measuring new types of behavior at infinity. For the purpose of describing Toeplitz C-algebras of random walks, new boundaries need to be identified in more precise terms. One such boundary is the so-called space-time Martin boundary, as studied by Lalley for random walks on the free group.

In this talk we will discuss ratio-limit boundaries and some work in progress on space-time Martin boundaries of random walks on discrete groups. The space-time Martin boundary is related to the notion of stability studied by Picardello and Woess, which elucidates potential descriptions of the space-time Martin boundaries for random walks on \mathbb{Z}^d and on hyperbolic groups.