Lorentzian geometry and physics in Kasparov's theory. (Abstract of thesis) (Q2800104)

This is the summary of the Ph.D thesis of the author. The main results of the thesis were announced in 4 scientific articles below -- [\textit{K. van den Dungen} et al., J. Geom. Phys. 73, 37--55 (2013; Zbl 1285.53060); \textit{J. Boeijink} and \textit{K. van den Dungen}, J. Math. Phys. 55, No. 10, 103508, 33 p. (2014; Zbl 1366.58003); ``Indefinite Kasparov modules and pseudo-Riemannian manifolds'', Preprint, \url{arXiv:1503.06916}; ``Krein spectral triples and the fermionic action'', Preprint, \url{arXiv: 1505.01939}].NEWLINENEWLINEFirst, the author has introduced a preliminary of the main geometric objects of Lorentzian geometry and gauge theory from the Conne's perspective in non-commutative geometry. After that, using the internal unbounded Kasparov product, the author has proved that the framework can be adapted to allow for globally nontrivial as well as trivial gauge theories. Next, the author has combines two geometric themes of Lorentzian geometry and gauge theory to introduce the concept of Krein spectral triples by generalizing spectral triples from Hilbert spaces to Krein spaces. Using Krein spectral triples, the author has constructed the so-called almost-commutative Lorentzian manifolds and proposed a Lorentzian alternative for the fermionic action. Finally, the author has derived the fermionic part of the Lagrangian of gauge theory by using the fermionic action and proved that alternative fermionic action recovers exactly the correct physical Lagrangian.