Por Carlos Florentino (Departamento de Matemática, Faculdade de Ciências Universidade de Lisboa, CMAFcIO).

The idea of symmetry, present in ancient civilizations, became part of our mathematical tools with the introduction of groups and group actions by Galois and Lie. Understanding the geometry of the spaces of orbits, and the algebra of the invariant functions on them is extremely useful both in algebraic and in geometric classification problems. These problems were greatly unified by the notion of a moduli space, introduced by Riemann and developed by Mumford.

In this colloquium, we present some classification problems in algebra and geometry which give rise to interesting moduli spaces - polygon spaces, character varieties, Higgs bundles -, and show some of the tools used in their study, such as polynomial invariants (named after Euler, Poincaré, Hodge, etc.). In some simple cases, there are explicit formulas for these polynomials (some just recently computed), and we end up announcing an interesting form of Langlands duality for character varieties of free groups.

The Mathematics Colloquium is a series of monthly talks organized by the Department of Mathematics of IST, aiming to be a forum for the presentation of mathematical ideas or ideas about Mathematics. The Colloquium welcomes the participation of faculty, researchers and undergraduate or graduate students, of IST or other institutions, and is seen as an opportunity of bringing together and fostering the building up of ideas in an informal atmosphere.