Symmetries are ubiquitous in nature and in our mathematical formulation of the laws of nature. In mathematics symmetries are encoded by the notion of a “group”. A sphere, for instance, is symmetric under spatial rotations, and the collection of all spatial rotations form a group. Two elements in a group may be “composed”, just like two spatial rotations can be performed one after another. The result of such a transformation may depend on the order in which the two rotations are carried out. If the order of composition does not matter, then we say that the two transformations “commute”. The central objective of the EU-funded project “Homotopy Theory of Spaces of Homomorphisms” (HOTHSPOH) was to understand collections of commuting transformations for a special class of groups called Lie groups. Lie groups are groups of “continuous” symmetries, the group of rotations being one amongst many examples. The collection of commuting transformations constitutes a “topological space”, a geometric object so to say whose shape one wishes to understand. This is important, as the spaces arising this way are examples of “moduli spaces”, that is, spaces that parametrize other mathematical objects. The moduli spaces considered in this project appear in mathematical physics, where they parametrize ground states of quantum field theories. Their understanding is therefore important not only from the viewpoint of mathematics, but also from the viewpoint of physics.
The shape of a space can be studied by means of algebraic invariants. An invariant could be a number that we assign to a space which reflects a small part but not all of the intricate geometric structure of the space. There are more powerful invariants called “homology groups”. Both objectives of the project HOTHSPOH aimed at facilitating the computation of homology groups. The first objective aimed at achieving this by breaking the space into simpler pieces, that is, by establishing a so-called “stable decomposition”. The second objective aimed at showing that the homology groups of certain spaces of commuting transformation “stabilize”, a structural result which too facilitates the computation of these invariants.
As to the first objective, the project concludes with a description of the “simpler pieces” that appear in a conjectural stable decomposition while the decomposition remains to be proved. Further results obtained during the fellowship are concrete computations of homology groups. As to the second objective, the project concludes with a proof of homology stability, and with a new approach to homology calculations.
