Por Carlos Florentino (Faculdade de Ciências da Universidade de Lisboa).

The idea of symmetry is present in ancient civilizations and became part of our mathematical tools with the introduction of groups and group actions by Galois and Lie. Understanding the orbits/invariants of these actions and the space they form turned out to be extremely useful both in algebraic and in geometric classification problems. These problems were greatly unified by the notion of moduli space, introduced by Riemann and developed by Mumford.

In this colloquium, we present some classification problems in algebra and geometry giving rise to nice moduli spaces, and explain some techniques used in their study, such as polynomial invariants (named after Euler, Poincaré, etc). We then compute some of these polynomials in simple cases (including a few new results), and show how they may allow us (following ideas of Grothendieck) to "add" and "multiply" algebraic varieties.