Noncommutative geometry encodes and extends classical geometric spaces through the framework of spectral triples. A natural question arises regarding how to study the convergence of these geometries and what it rigorously means to take the limit of spectral triples, such as when using matrix models to approximate quantum spaces or finite graphs to approximate fractals. To address this, it is necessary to establish a natural geometry on the space of metric spectral triples.

This talk introduces the Spectral Propinquity, an analogue of the Gromov-Hausdorff distance for metric spectral triples, which provides a distance function on the class of metric spectral triples up to unitary equivalence. A central theorem demonstrates that convergence under the spectral propinquity guarantees the continuity and convergence of the spectra of the corresponding Dirac operators.

The presentation will conclude by exploring several concrete examples of geometric convergence within this space. Key applications discussed will include:

• The convergence of Riemannian spin geometries and almost commutative models.

• The recovery of the Sierpiński gasket's entire spectral geometry as a limit of finite graph approximations.

• The introduction of metric twisted spectral triples to resolve failures of the Leibniz identity in quantized calculus for fuzzy and quantum tori.

• The behavior of inductive limits, featuring examples like noncommutative solenoids and Bunce-Deddens algebras.