Many major advances in geometry have involved producing invariants from auxiliary spaces. Finite-dimensional approximation spaces shed light on the geometry of geodesics (in Bott's work and Viterbo's work) and Seiberg-Witten moduli spaces over families are powerful tools in studying 4-manifolds (see, e.g., Konno's work). Perhaps surprisingly, various communities are in the midst of discovering a class of geometric problems where the auxiliary spaces are so rich they behave like abelian groups -- that is, where the spaces arise as spectra. Many results, including in the two threads of ideas above, can now be recovered as corollaries of statements regarding these spectra! In this colloquium talk, I will speak about an unexpected continuation of this trend: A newly discovered spectrum-level invariant associated to Legendrian submanifolds and generating families, conjecturally giving a spectral lift of infinitesimal Fukaya categories. This is joint work with Lisa Traynor.