פעילויות השבוע

Using the Bruhat decomposition, a general linear group over a p-adic field may be thought of as a ”quantum affine“ version of a finite group of permutations. I would like to discuss some analogies and explore the implications of this point view on the spectral properties of the two groups. For one, restriction of an irreducible smooth representation to its finite counterpart gives the correct notion of the wavefront set - an invariant of arithmetic significance which is often approached using microlocal analysis. From another perspective, the class of cyclotomic Hecke algebras is a natural interpolation between the finite and p-adic groups. I will show how the class of RSK representations (developed with Erez Lapid) serves as a bridge between the Langlands classification for the p-adic group and the classical Specht construction of the finite domain.

נאמר שלתת-קבוצה סופית $A$ של $Z^d$ יש ריצוף אם קיימת קבוצה $B$ כך ש-$Z^d$ ניתן להצגה כאיחוד זר של הזזות $A$ על ידי איברי $B$.
נאמר של-$A$ יש ריצוף מחזורי אם קיימת $B$ כנ“ל שהיא יתר על כן מחזורית ביחס ל-$d$ הזזות בלתי תלויות לינארית.

האם לכל קבוצה סופית בעלת ריצוף יש ריצוף מחזורי?

השאלה הזו עדיין פתוחה במקרה הכללי, אבל תשובה חיובית ידועה במקרים פרטיים, למשל כאשר גודל הקבוצה $A$ הוא מספר ראשוני או כאשר המימד $d$ הוא $2$ לכל היותר.

בהרצאות נציג מושגים ותיאוריה, משפטים והוכחות לגבי ריצופים והקשר שלהם לדינמיקה (וגם אלגברה ואנליזה).

In the last twenty years algebraic geometry has evolved rapidly, from the geometry of schemes and stacks, to the derived algebraic geometry (DAG) of today. The flavor of contemporary DAG is very homotopical, in the sense that is largely based on simplicial sets and Quillen model structures.

This talk is on another approach to DAG, of a very algebraic flavor, which avoids simplicial methods and model structures altogether. Instead, the fundamental concept is that of DG rings, traditionally called unital associative cochain DG algebras. DG rings are of two distinct kinds: noncommutative and commutative. These two kinds of DG rings interact, primarily through central DG ring homomorphisms; and this interaction is quite fruitful. The main tool for studying DG rings, DG modules over them, and the associated derived categories, is the construction and manipulation of resolutions. Hence ”constructive approach“.

I will define the notions mentioned above, and state several results, among them: (1) The squaring operation and Van den Bergh‘s rigid dualizing complexes in the commutative arithmetic setting; (2) Theorems on derived Morita theory; (3) Duality and tilting for commutative DG rings. I will try to demonstrate that this constructive approach is very amenable to calculation. I will also mention work of Shaul, within this framework, on derived completion of DG rings and on the derived CM property.

The talk will conclude with a couple of conjectural ideas: (a) A structural description of the derived category of DG categories; (b) A construction of the cotangent DG module within the framework of commutative DG rings, without any arithmetic restrictions.

Some of this work is joint with J. Zhang, L. Shaul, M. Ornaghi and S. Singh.

Slides for the talk are available here:

https://sites.google.com/view/amyekut-math-bgu/home/lectures/constr-der-algebra

(updated 15 March 2022)

For an action of a countable amenable group $G$ on a compact metric space $X$, the mean dimension $mdim (G, X)$ was introduced by Lindenstrauss and Weiss, for reasons unrelated to $C^*$-algebras. The radius of comparison $rc (A)$ of a $C^*$-algebra $A$ was introduced by Toms, for use on $C^*$-algebras having nothing to do with dynamics.

A construction called the crossed product $C^* (G, X)$ associates a $C^*$-algebra to a dynamical system. There is significant evidence for the conjecture that $rc ( C^* (G, X) ) = (1/2) mdim (G, X)$ when the action is free and minimal. We give the first general partial results towards the direction $rc ( C^* (G, X) ) \geq (1/2) mdim (G, X)$. We don‘t get the exact conjectured bound, but we get nontrivial results for many of the known examples of free minimal systems with $mdim (G, X) > 0$. The proof depends, among other things, on Cech cohomology, and uses something we call the mean cohomological independence dimension. Unlike the currently known results in the other direction, it works for all choices of $G$.

The talk will include something about the crossed product construction; no previous knowledge of it will be assumed.

This is joint work with Ilan Hirshberg.