Open problems in deformations of Artinian algebras, Hilbert schemes and around

TL;DR

This paper reviews open problems in the deformation theory of zero-dimensional algebraic objects, highlighting both classical and new challenges, and discusses the potential for current methods to address them.

Contribution

It provides a comprehensive overview of open problems in deformations of Artinian algebras, modules, and tensors, including recent developments and emerging questions.

Findings

Many open problems remain in the deformation theory of Artinian algebras.

Recent advances suggest these problems may be approachable with current techniques.

The paper identifies both well-known and newly emerging challenges in the field.

Abstract

We review the open problems in the theory of deformations of zero-dimensional objects, such as algebras, modules or tensors. We list both the well-known ones and some new ones that emerge from applications. In view of many advances in recent years, we can hope that all of them are in the range of current methods.