Thom polynomials for singularities of maps

TL;DR

This paper introduces Thom polynomials, a universal tool in intersection theory for classifying singularities of maps, with broad implications in enumerative geometry and related fields.

Contribution

It provides an accessible overview of the theory of Thom polynomials and their role in classifying map-germ singularities, highlighting their historical development and applications.

Findings

Thom polynomials serve as universal intersection invariants for singularities.

The theory connects classical and modern enumerative geometry.

Potential applications span various geometric and topological problems.

Abstract

This is a gentle introduction to a general theory of universal polynomials associated to classification of map-germs, called Thom polynomials. The theory was originated by Ren\'e Thom in the 1950s and has since been evolved in various aspects by many authors. In a nutshell, this is about intersection theory on certain moduli spaces, say `classifying spaces of mono/multi-singularities of maps', which provides consistent and deep insights into both classical and modern enumerative geometry with many potential applications.