Activities This Week

$C^\ast$-covers of operator algebras, first studied by Arveson about 50 years ago, remain a vibrant area of research. Recently, Adam Humeniuk and Christopher Ramsey studied the lattice of $C^\ast$-covers of operator algebras, focusing on its structure and on whether this lattice uniquely determines an operator algebra up to completely isometric isomorphisms.

In my talk, I will give an introduction to operator algebras and their $C^\ast$-covers, and answer the question of whether there exists nontrivial operator algebras with a one point lattice. Additionally, I will characterize the possible cardinalities of the lattice of $C^\ast$-covers. This is a joint work with Adam Humeniuk and Christopher Ramsey.

Chromatic homotopy theory aims to study cohomology theories through a hierarchy of simpler layers, organized by a notion called height. In this talk I will introduce the basic ideas behind this viewpoint and explain two approaches to analyzing these monochromatic layers: the classical K(n)-local category, which is closely related to one-dimensional formal group laws, and the T(n)-local or telescopic category, which is more directly tied to periodic phenomena in the stable homotopy groups of spheres. I will then describe a framework for understanding periodicity inside the chromatic layers, and explain how this allows one to lift Picard elements from the K(n)-local setting to the telescopic setting. Finally, I will present an application to chromatic Galois theory, leading to the construction of a first example of a non-abelian Galois extension in the T(n)-local world.

A significant part of Mathematics boils down to “resolving systems of equations”, e.g. equations of implicit function type, F(x,y)=0. In many cases one has to resolve these just ``order-by-order”. The obtained power series, y(x), do not need to be analytic.

Artin approximation (A.P.) ensures: every formal solution is approximated by analytic solutions. This goes in contrast to various other (functional or differential) equations, for which the formal and analytic words are very different.

I will give a brief introduction to this topic, and then explain the recent results: A.P. for (quivers) of morphisms of scheme-germs. In simple words: A.P. holds for an additional class of functional equations (not of implicit function type).

מציאת פתרונות שלמים למשוואות פולינומיאליות, כמו למשל x² + y² = z³, הוא תחום רחב המעסיק מתמטיקאים מימי היוונים הקדומים ועד ימינו. בפרט, שאלות “תמימות” וקלות לניסוח עלולות להתברר כקשות ועמידות לפיתרון. בהרצאה נתמקד בהערכת מספר הפתרונות השלמים האפשריים למשוואה פולינומיאלית, ונציג רעיון אלגברי מפתיע, המאפשר לגשת לשאלה זו על ידי שימוש מחוכם באלגברה ליניארית (והערכת דטרמיננטה מתאימה). ההרצאה אינה דורשת ידע קודם בתורת המספרים.