Periodic Reporting for period 1 - HOTHSPOH (Homotopy theory of spaces of homomorphisms)
Período documentado: 2019-09-01 hasta 2021-08-31
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.
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.
In the initial phase of the project the ER worked towards a stable decomposition of certain spaces of commuting transformations. The ER completed the first step by describing the “simpler pieces” into which these spaces are expected to split. The question of whether the spaces break up as expected motivated the study of certain low dimensional homology groups, leading to a collaboration with Alejandro Adem (University of British Columbia) and Jose Manuel Gomez (National University of Colombia Medellin). During this phase of the project the ER learned methods from equivariant topology to describe the second homology group of the space of commuting elements in a compact connected Lie group. The paper entitled “On the second homotopy group of spaces of commuting elements in Lie groups”, published in International Mathematics Research Notices, establishes furthermore some interesting connections with representation theory. The remaining steps in establishing the desired stable decomposition are part of an ongoing collaboration with Adem and Gomez.
In a related direction, the ER began a collaboration with Omar Antolin-Camarena and Bernardo Villarreal (both at UNAM, Mexico City). The joint work investigates a certain class of spaces built from spaces of commuting elements in Lie groups that have recently found applications in quantum information theory. The paper resulting from this collaboration is entitled “Higher generation by abelian subgroups in Lie groups” and was published in Transformation Groups.
In the second phase of the fellowship, the ER investigated the question of homology stability for certain spaces of commuting transformations. In an ongoing collaboration with Markus Hausmann (University of Bonn) a new approach to homology calculations is found by establishing a connection with a theory called “Hochschild homology”. Amongst other things, this allows for applying some recent results in homology stability to prove that certain spaces of commuting transformations exhibit this interesting stability behaviour.
The ER presented the findings from the fellowship in research seminars at the University of British Columbia, Bilkent University, University of Aberdeen, Ludwig-Maximilians-University Munich, UNAM Mexico City, and the University of Copenhagen, as well as at the workshop “Spaces of Homomorphisms and Classifying Spaces 2021”.
In a related direction, the ER began a collaboration with Omar Antolin-Camarena and Bernardo Villarreal (both at UNAM, Mexico City). The joint work investigates a certain class of spaces built from spaces of commuting elements in Lie groups that have recently found applications in quantum information theory. The paper resulting from this collaboration is entitled “Higher generation by abelian subgroups in Lie groups” and was published in Transformation Groups.
In the second phase of the fellowship, the ER investigated the question of homology stability for certain spaces of commuting transformations. In an ongoing collaboration with Markus Hausmann (University of Bonn) a new approach to homology calculations is found by establishing a connection with a theory called “Hochschild homology”. Amongst other things, this allows for applying some recent results in homology stability to prove that certain spaces of commuting transformations exhibit this interesting stability behaviour.
The ER presented the findings from the fellowship in research seminars at the University of British Columbia, Bilkent University, University of Aberdeen, Ludwig-Maximilians-University Munich, UNAM Mexico City, and the University of Copenhagen, as well as at the workshop “Spaces of Homomorphisms and Classifying Spaces 2021”.
During the fellowship new computational as well as new structural results about homology groups of spaces of commuting elements in Lie groups were obtained. Prior to this project the second homology group was unknown in most cases. The phenomenon of “homology stability” is of classical interest and has been observed for a great deal of examples. The project showed that spaces of commuting transformations define an interesting new class of examples that exhibit this behaviour. Furthermore, the results mark a major improvement on rational stability results that were known prior to this project. The project also established an unexpected connection with Hochschild homology which allows for a new approach to homology calculations. The findings from the fellowship are expected to be of interest to the mathematical physics community, as the spaces that were studied are examples of a more general class of moduli spaces arising in quantum field theory.