פעילויות השבוע

דקארט מצא קשר עמוק בין אלגברה לגאומטריה: המישור הקרטזי, שבו מתאימים נקודה לכל זוג של מספרים ממשיים. בחצי הראשון של המאה ה-20, מתמטיקאים מצאו קשר יותר עמוק: לכל צורה גאומטרית מוגדרת ע“י משוואות פולינומיאליות, התאימו חוג (במובן של מבנים אלגבריים), שבו תכונות גאומטריות של הצורה משתקפות בתכונות אלגבריות של החוג. באמצע המאה ה-20, גאון בשם אלכסנדר גרוטנדיק הוביל מהפכה שבה הבין שאפשר להתאים פירוש גאומטרי למגוון מבנים אלגבריים מופשטים. בפרט, אפשר להתאים לחוג המספרים השלמים צורה או ”מרחב“ שהנקודות שלו הן בדיוק המספרים הראשוניים. מהפכה זו השפיעה לא רק על הגאומטריה האלגברית אלא על מגוון תחומים במתמטיקה, מהם תורת המספרים ופיזיקה מתמטית. אם יישאר זמן, ננסה לתת מושג של הקשר שהוא פתח בין טופולוגיה לתורת המספרים בעזרת תורת גלואה.

This talk is based on my ongoing work with Steffen Muller and Padma Srinivasan. I will explain the idea of the Quadratic Chabauty method for finding rational points on curves and how one reinterprets previous work by Balakrishnan and Dogra, on their own and with collaborators, using p-adic adelic metrics on line bundles. Time permitting I will explain how one can use this to compute the ”local contributions away from p“ for Quadratic Chabauty, which are crucial for computations.

During the 60,s and the 70,s Furstenberg developed two parallel theories regarding boundaries of groups of different flavours. One is topological, and the other is measurable and relates to random walks. The research of these two theories and their connections with rigidity theory and operator algebra theory is still very active, yet many questions are open. In an attempt to understand better the connections between them, I‘ll show that they share the same driving force. We סwill develop one machinery to produce them both at the same time. Two Boundary Theories for the price of one.

The talk will be aimed at an audience which is not necessarily familiar with the concepts below, and I’ll aim to explain them at the expense of providing proofs. The Elliott classification program for simple nuclear C-algebras reached an important milestone in the past decade. This program aims to classify simple separable nuclear C-algebras in terms of the Elliott invariant, consisting of the K-theory groups and tracial data. It is now established that such C-algebras can be classified provided they are Z-stable (a key regularlity condition) and satisfy the Rosenberg-Schochet Universal Coefficient Theorem (where it is a major open problem to determine whether this theorem holds for all simple nuclear separable C-algebras). The classification theorem was shown not to extend beyond the Z-stable case. Specifically, Toms constructed examples of simple separable nuclear unital C*-algebras with the same Elliott invariant, but which can be distinguished by another invariant he called the radius of comparison, which measures to what extent positive elements fail to be compared via tracial data. While there were good reasons to think that simply adding the radius of comparison to the invariant would not be sufficient to extend the classification theorem further, intriguingly, recently, Elliott, Li and Niu proved a classification theorem for so-called Villadsen-type algebras using the combination of the Elliott invariant and the radius of comparison, an invariant which was introduced by Toms in order to distinguish between certain non-isomorphic AH algebras with the same Elliott invariant. This might have raised the prospect that the Elliott classification program can be extended beyond the Z-stable case by adding the radius of comparison to the invariant. I will discuss a recent preprint in which we show that this is not the case: we construct an uncountable family of non-isomorphic AH algebras with the same Elliott and same radius of comparison. We can distinguish between them using a finer invariant, which we call the local radius of comparison. This is joint work with N. Christopher Phillips.