The first call for applications for the new edition of the PhD Program in Mathematics UC|UP (starting September 2019) will be open during January 02, 2019 - February 04, 2019
The prize Eng. António de Almeida for the best doctoral thesis defended at FCUP in the years 2015 to 2017 in the areas of Mathematics, Statistics and Computer Science was awarded to Célia Borlido for her thesis "The word problem and some reducibility properties for pseudo- varieties of the form DRH", supervised by Prof. Jorge Almeida within the scope of PIUDM.
Representation theory is the study of algebraic objects, such as groups and algebras, from the point of view of symmetry and invariants in linear spaces. It is a mathematical theory with a wide range of applications in group theory, combinatorics, number theory, probability, geometry and physics.
The syllabus will vary from year to year. The following is a list of optional suggested topics which together cover many basic aspects of the representation theory of finite groups, quivers, Lie groups and Lie algebras:
1. Associative algebras, group algebras, quivers and path algebras, irreducible and indecomposable representations, Schur's lemma, semisimple algebras, Jordan-Holder and Krull-Schmidt theorems, representations of finite-dimensional algebras.
2. Representations of quivers, indecomposable representations of quivers of type A1, A2, A3, D4. The triple subspace problem, simply laced root systems, reflection functors, Gabriel's theorem.
3. Lie groups, Lie algebras and enveloping algebras. Classification of semisimple Lie algebras and their representations. Quantized enveloping algebras.
4. Representations of finite groups: Maschke's theorem, duals and tensor products of representations, characters, orthogonality of characters, character tables, Burnside's theorem, induced representations and their characters (Mackey formula), Frobenius reciprocity.
5. Representations of the symmetric group and the general linear group, Schur-Weyl duality, first fundamental theorem of invariant theory.