"As planned, work on the first component begun with an attempt to detect nontrivial classes in the homology of automorphism groups of free groups. In the course of this work, the researcher unexpectedly discovered a novel perspective on a very classical topic in algebraic topology, namely the mod p homology of E-infinity spaces. Upon an evaluation of the mathematical significance of the material, the researcher decided to refocus the research effort in the first part of the project into the development of this perspective. The researcher then proceeded to successfully develop the foundations of the mod p homology of E-infinity spaces from the new point of view. The major research results obtained are presented in the preprint ""Homology operations revisited"" (arXiv:2001.02781 47 pages) posted on the arXiv preprint repository.
The research work in the second part, joint with Grodal, centered on the strengthening and extension of string topology of finite groups of Lie type, with a particular focus on understanding when the Tezuka conjecture holds. Using a new ""untwisting theorem,"" we showed how to dispense with the customary congruence condition in the conjecture at the expense of relating the twisted finite group of Lie type tG(q) not to the free loop space of the classifying space of the corresponding compact Lie group, but rather to the free loop space of a different l-compact group depending on G, the twisting t, and the congruence class of q modulo l. With this setup, we proved that a string topological variant of the Tezuka conjecture holds in the simply connected case for any q and any twisting t whose order is not divisible by l, except possibly in a finite number of cases requiring special attention. The results are incorporated into the preprint ""String topology of finite groups of Lie type"" (arXiv:2003.07852 58 pages) posted on the arXiv preprint repository. The researcher also mapped out a proof that the string product constructed in the paper is commutative and that it agrees with a product previously constructed by Chataur and Menichi, providing a key link between the string topology of finite groups of Lie type and previous work on string topology of classifying spaces using Chataur and Menichi's product. These enhancements will be incorporated into a future revision of the preprint.
In addition to the research work, during the project the researcher organized a workshop together with Grodal (""Workshop on classifying spaces of finite groups of Lie type,"" University of Copenhagen, July 8–12, 2019); supervised a master's thesis project concerning invertible two-dimensional topological quantum field theories; and organized mathematical gong shows showcasing junior researchers as part of the host department's outreach efforts during Copenhagen's annual Culture Night festival in October 2018 and 2019. In addition to the preprints mentioned above, the research results have been disseminated through talks the researcher has given on the work at various conferences, workshops, and seminars."
