The Nash problem from a geometric and topological perspective

TL;DR

This paper surveys the proof of the Nash conjecture for surfaces, discusses higher-dimensional cases, and reviews recent developments in the generalized Nash problem and holomorphic arcs, highlighting geometric and topological influences.

Contribution

It provides a comprehensive overview of the Nash conjecture's proof for surfaces, summarizes advances in higher dimensions, and discusses recent research on generalized Nash problems and holomorphic arcs.

Findings

Proof of Nash conjecture for surfaces surveyed

Summary of higher-dimensional Nash problem by de Fernex and Docampo

Discussion of recent developments in generalized Nash problem and holomorphic arcs

Abstract

We survey the proof of the Nash conjecture for surfaces and show how geometric and topological ideas developed in previous articles by the authors influenced it. Later we summarize the main ideas in the higher dimensional statement and proof by de Fernex and Docampo. We end the paper by explaining later developments on generalized Nash problem and on Koll\'ar and Nemethi's study about holomorphic arcs.