Singularities arise naturally in a huge number of different areas of mathematics and science. As a consequence, Singularity Theory lies at the crossroads of the paths connecting applications of mathematics with its most abstract parts. For example, it connects the investigation of optical caustics with simple Lie algebras and regular polyhedra theory, while also relating hyperbolic PDE wavefronts to knot theory and the theory of the shape of solids to commutative algebra.

The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis and geometry, or physics, or from some other science on parameters. For generic points in the parameter space their exact values influence only the quantitative aspects of the phenomena, their qualitative, topological features remaining stable under small changes of parameter values.

However, for certain exceptional values of the parameters these qualitative features may suddenly change under a small variation of the parameter. This change is called a perestroika, bifurcation or catastrophe in different branches of the sciences. A typical example is that of Morse surgery, describing the perestroika of the level variety of a function as the function crosses through a critical value. (This has an important complex counterpart the Picard-Lefschetz theory concerning the branching of integrals.) Other familiar examples include caustics and outlines or profiles of surfaces obtained from viewing or projecting from a point, or in a given direction.

Algebraic geometry classically studies solutions of systems of polynomial equations in several variables multivariate polynomials. It is even more important to understand the intrinsic and geometric properties of the totality of solutions of a system of equations, than to find a specific solution.

It is based on using and combining abstract algebraic techniques, mainly from commutative algebra, and analytic techniques, mainly from complex variables theory and complex differential geometry. Recently deep interactions with gauge theory are also at the core of current research.

The fundamental objects of study in algebraic geometry are algebraic varieties. Basic questions involve the study of geometry and topology, both globally and locally near the points of special interest like the singular points, the inflection points...

Algebraic geometry has deep connections with other parts of pure mathematics, like topology, number theory or differential geometry. Despite being a part of pure mathematics it has found important applications in cryptography, coding theory and physics (it is at the core of string theory, for example).