Asymptotic Geometric Analysis: Achievements and Perspective

The reader will have noticed the non-standard appearance of this piece. Indeed, we are used to reading papers which are either survey papers or research ones (or a mixture of both). However, it seems to be beneficial for a given field to sometimes take a pause and to observe the picture of the field’s development in a broad sense, from “above”, and to examine the directions in which things are progressing, in various directions, at the same time. This collection of short essays on some particular subdirections of the theory is an attempt to present such an overview. In recent years Asymptotic Geometric Analysis has grown enormously in its areas of interest, directions and results. Trying to understand and digest the picture of this development, I asked several experts who are close to me and represent the centers of various directions, to write for me a very concise and short description of their present central interests. More precisely, that part of their interests that relates to Asymptotic Geometric Analysis. Many agreed, and I am posting below the short texts I received. After each of them, I will place my comments, as well as some problems that arise when reading these texts. Of course, I know that a few promising and interesting directions are missing. Some articles I expected, I did not receive, and some directions are not active at present around me. It is my hope that such a presentation will add curiosity to some questions for experts, but more important, it will have, hopefully, a positive influence on the young generation joining this field.