Title: Some Explicit Results on Exotic 7-spheres

Abstract: In this talk, I will present some differential-geometric properties of exotic 7-spheres through the lens of Kaluza-Klein formalism, as well as some differential-topological ones through the construction of an explicit homeomorphism.  Firstly, I will review both established facts and recent developments, beginning with the natural geometry of Milnor’s bundles and proceeding with the results from arXiv:2309.01703 (J. High Energ. Phys. 2023, 100) and arXiv:2410.01909 (J. Geom. Phys. 216 2025, 105590); comments about the role of Kaluza-Klein theory and (possible) appearance of exotic spheres in supergravity will also be made.  Then, I will focus more on the differential topology side, presenting an explicit homeomorphic map which might provide some insights into one realisation of the Gromoll-Meyer sphere (work in progress, to appear soon).