The aim of these lecture notes is to provide the students with tools and techniques used in the theory of real singularities and to apply them in order to get interesting results on the topology and geometry of real singularities.

These notes provide an introduction to the theory of singularities of mappings and right-left equivalence. They cover a part of Mather theory, concerned with stability, classification and deformations, and go on to study the vanishing homology of images and discriminants of families of mappings. A number of open questions are discussed in the last three sections.

In these lectures we shall give results about non-isolated singularities of complex algebraic sets. We shall indicate many references to the reader.

We shall try to enunciate the results as clearly as possible. Detailed proofs of the results are not always given, but to complete them will be an excellent exercise for the reader. We hope that these notes will be a lead to students in the maze of the theory of singularity.

These notes follows essentially the lectures given at São Carlos in the school of 2012.

Gröbner bases are the main computational tool available for algebraic geometry. Building on top of Gröbner bases algorithms for ideal theoretical operations (intersection, quotient, saturation, free resolution,...) will be presented. Combining these algorithms with (multivariate) factorization leads to several algorithms for primary decomposition of ideals.

We introduce a new branch of the Thom polynomial theory for local and multi-singularities of maps.

A polynomial system of differential equations on the plane with a singularity at which the eigenvalues of the linear part are complex can be placed, by means of an affine transformation and a rescaling of time, in the form $\dot x = \lambda x - y + P(x, y)$, $\dot y = x + \lambda y + Q(x, y)$. The problem of determining, when $\lambda = 0$, whether the origin is a spiral focus or a center dates back to Poincaré. This is the center problem. We discuss an approach to this problem that uses methods of computational commutative algebra. We treat generalizations of the center problem to the complex setting and to higher dimensions. The theory developed also has bearing on the cyclicity problem at the origin, the problem of determining bounds on the number of isolated periodic orbits that can bifurcate from the origin under small perturbation of the coefficients of the original system. We also treat this application of the theory. Some attention is also devoted to periodic solutions on center manifolds and their bifurcations.