Abhyankar's conjectures in Galois theory: Current status and future directions

TL;DR

This paper surveys Abhyankar's influential conjectures in Galois theory, summarizing current progress, open problems, and future research directions in the study of fundamental groups and Galois covers in positive characteristic.

Contribution

It provides a comprehensive overview of the status of Abhyankar's conjectures and outlines promising future research directions inspired by his work.

Findings

Current status of four Abhyankar conjectures in positive characteristic

Identification of open problems and research directions

Summary of key contributions to fundamental groups and Galois covers

Abstract

In this paper, we survey the major contributions of Abhyankar to the development of the theory of fundamental groups and Galois covers in positive characteristic. We first discuss the current status of four conjectures of Abhyankar about Galois covers in positive characteristic. Then we discuss research directions inspired by Abhyankar's work, including many open problems.