Activities This Week

One of the most interesting open questions in Galois theory these days is: Which profinite groups can be realized as absolute Galois groups of fields? Restricting our focus to the one-prime case, we begin with a simpler question: which pro-p groups can be realized as maximal pro-p Galois groups of fields?

For the finitely generated case over fields that contain a primitive root of unity of order p, we have a comprehensive conjecture, known as the Elementary Type Conjecture by Ido Efrat, which claims that every finitely generated pro-p group which can be realized as a maximal pro-p Galois group of a field containing a primitive root of unity of order p, can be constructed from free pro-p groups and finitely generated Demushkin groups, using free pro-p products and a certain semi-direct product.

The main objective of the presented work is to investigate the class of infinitely-ranked pro-p groups which can be realized as maximal pro-p Galois groups. Inspired by the Elementary Type Conjecture, we start our research with 2 main directions: 1. Generalizing the definition of Demushkin groups to arbitrary rank and studying their realization as absolute/ maximal pro-p Galois groups. 2. Investigating the possible realization of a free (pro-p) product of infinitely many absolute Galois groups.

In this talk we focus mainly on the second direction. In particular, we give a necessary and sufficient condition for a restricted free product of countably many Demushkin groups of infinite countable rank, to be realized as an absolute Galois group.

נניח שיש לנו סדרת אופרטורים חיוביים $T_n$ על $C([0,1])$. האם אפשר להבטיח שהתכנסות במידה שווה לכל פונקציה רציפה נובעת רק מבדיקת מספר קטן של פונקציות “מבחן”? בשנת 1953 הוכיח קורובקין תשובה מפתיעה—כן! מספיק לבדוק את ההתכנסות על שלוש פונקציות פשוטות כדי להשליך על כולן. בהמשך, ססקין הרחיב את הרעיון הזה למרחבים כלליים יותר, וחיבר אותו למבנה הגיאומטרי של שפת שוקה.

בהרצאה נצלול אל הרעיונות המרכזיים מאחורי המשפטים הללו, נראה כיצד הם מספקים הוכחות אלגנטיות למשפטי הקירוב של ויירשטראס ופייר, ונבין מדוע התוצאות האלה הרבה יותר חזקות ממה שנדמה במבט ראשון. אם הזמן יאפשר, נרחיב את הדיון להכללות מתקדמות ולכיווני מחקר עכשוויים, כולל הקשרים ל-hyperrigidity במצב הקומוטטיבי.

ההרצאה מתאימה לשנה ב’ ומעלה.

The Zilber-Pink conjecture is a far reaching and widely open conjecture in the area of “unlikely intersections” generalizing many previous results in the area, such as the recently established André-Oort conjecture. Recently the ``G-functions method’’ of Y. André has been able to consistently establish the missing arithmetic result needed to establish cases of this conjecture for Shimura varieties. I will discuss how, using properties of the p-adic values of G-functions, we can get new cases of this conjecture in $\mathcal{A}_2$.

In this talk, we will explore the structure of ends of Schreier graphs associated with stationary random subgroups (SRS). We begin with the classical Freudenthal-Hopf theorem on the possible number of ends of Cayley graphs and extend it to the setting of random subgroups. First, we establish the result for Schreier graphs of invariant random subgroups (IRS) before further generalizing it to stationary subgroups.

We will then introduce the concept of stationary actions, stationary random subgroups, and present the key result: For a group Γ with a symmetric generating set, the number of ends of the Schreier graph of an SRS is almost surely 0, 1, 2, or infinite.

Finally, we will outline the proof of this theorem, emphasizing the emergence of the “no-core” phenomenon in stationary actions, which is an interpretation of Kac’s lemma within the framework of stationary actions.