Non-product geometries for particle physics and cosmology

Introduction
	Noncommutative geometry
		The idea
		Fredholm modules and spectral triples
		Grading and reality
		The first-order condition and Hodge property
		Reconstruction theorem and beyond
		Twisted spectral triples
	The spectral action principle
		The general construction
		Spectral action in terms of Wodzicki residue
	Gauge theories and almost-commutative geometries
	Spectral Standard Model
		The fermion doubling problem
		Lorentzian formulation
New directions
	Non-product geometry for the Standard Model
		Formulation of the model and its basic predictions
		Bosonic spectral action
		Outlook
	Non-product geometry for modified gravity theories
		Doubled geometries - dynamical stability of solutions for FLRW models
		Analysis of the interaction term
		Towards generalized bimetric models
		Outlook
Conclusions and outlook