HOTHSPOH · Homotopy theory of spaces of homomorphisms
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2019-09-01 → 2021-08-31
- EU contribution
- €207,312
- Participants
- 1
- Scheme
- MSCA-IF-EF-ST
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Results in brief
Homotopy theory of spaces of homomorphisms
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.
Data: CORDIS, © European Union
Project objective
In this project we propose to study homotopy theoretic properties of spaces of commuting elements in compact Lie groups. These spaces play an essential role in mathematical physics and geometry, but only in the last decade a systematic study by homotopy theoretic methods has been initiated. Important open questions in the field concern the homology as well as the (stable) homotopy type. In the first part of the project, we attempt to prove a conjectural stable splitting theorem, which would establish an intriguing relationship between spaces of commuting elements and commuting varieties in Lie algebras, an object of classical interest in algebraic geometry. In the second part, we propose to investigate the phenomenon of homology stability for spaces of commuting elements in the unitary and orthogonal groups. Building on recent work of the experienced researcher, an approach to calculate the stable homology is presented. This is expected to uncover a wealth of previously unknown homology groups of these interesting spaces.The research conducted to achieve the project goals, together with the training in teaching and management received during the fellowship, will have a major positive impact on the career development of the experienced researcher. On the research level, this impact is through the acquisition of knowledge in new research areas, in particular in homology stability and the homotopy theory of Lie group actions.The project will be carried out in an exceptionally active and successful scientific community at the University of Copenhagen, supervised by a world expert in the homotopy theory of Lie groups. Completion of the project will serve as a springboard to build new collaborations and to enter further advanced projects in a range of areas. It is thus a perfect preparation for a high-level research career in mathematics.
Original text from CORDIS.
Participants
- KOBENHAVNS UNIVERSITET · KOBENHAVNCoordinatorDenmark
Links
Data: CORDIS, © European Union