Activities This Week

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.

I will give an overview of some recent and some older results on rational dynamics and on rational difference equations. The main model theoretic tool here is the theory ACFA, the model companion of the theory of difference fields.

Locally analytic representations of p-adic Lie groups with Q_p coefficients are powerful tools in p-adic Hodge theory and the p-adic Langlands program. This perspective reveals important differential structures, such as the Sen and Casimir operators. A few years ago, Rodrigues Jacinto and Rodriguez Camargo developed a “solid” version of this theory using the language of condensed mathematics, which provides more robust homological tools (comparison theorems, spectral sequences…) for studying these representations. This talk will present work that extends this solid theory to a much broader class of mixed characteristic coefficients, such as F_p((X)) or Z_p[[X]]<p/x>, as well as semilinear representations. I will conclude by exploring how these ideas relate to mixed characteristic phenomena in p-adic Hodge theory and the Langlands program.