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The problem of reduction of multisymplectic manifolds by the action of Lie groups is stated and discussed, as a previous step to give a fully covariant scheme of reduction for classical field theories with symmetries., 9 pages. The definition of bracket of Hamiltonian forms has been corrected (and an acknowledgment to Prof. C. Blacker has been added in the "Acknowledgments")

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2015, Director: Vicenç Navarro Aznar, Even though the title of this composition you are reading is Gauge fields, this is just one of the many concepts that appear throughout the work. Moreover, despite the fact that this is rather a physical term (the actual mathematical name would be connections on principal G-bundles), the contents in this exposition are mathematical. As a student of both mathematics and physics, I wanted to choose a topic that could be covered in both aspects, a subject which could be interpreted physically but which had a mathematical background. This composition intends to contain the mathematical background to develop the so-called gauge theories used in modern physics such as quantum physics, particularly quantum field theory (QFT), for example quantum electrodynamics (QED) or quantum chromodynamics (QCD). However, as stated, this contains no physical conclusions, but only mathematical, with maybe some references to their physical applications. Thus, the motivation was a combination of my interest on physics (which may explain the outline and selection of topics) but also my fascination for differential geometry and topology.

It is natural to extend the Grothendieck theorem on completeness, valid for locally convex topological vector spaces, to Abelian topological groups. The adequate framework to do it seems to be the class of locally quasi-convex groups. However, in this paper we present examples of metrizable locally quasi-convex groups for which the analogue to the Grothendieck theorem does not hold. By means of the continuous convergence structure on the dual of a topological group, we also state some weaker forms of the Grothendieck theorem valid for the class of locally quasi-convex groups. Finally, we prove that for the smaller class of nuclear groups, BB-reflexivity is equivalent to completeness. (C) 2001 Elsevier Science B.V.

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups G determined by three requirements. They must have the Baire property, satisfy the open refinement condition, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group G∧ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group G of weight ≥ 2!, we find proper dense subgroups H1 and H2 of G such that H1 is reflexive and pseudocompact, while H2 is non-reflexive and almost metrizable.

We look at finitely generated Bohr groups G# (groups G equipped with the topology inherited from their Bohr compactification bG). Among others, the following results are proved: every finitely generated group without free non abelian subgropus either contains non trivial convergent sequiences in G# or is a finite extension of an abelian group; containing the free non abelian group with two generators does not have the extension property for finite dimensional representations, therefore, it does not belong to the class D introduced by Poguntke in [27]: if a countable FC group, then the topology that the commutator subgroup [G,G] inherits from G# is profinite and metrizable