A (possibly non-Thom) ring spectrum of Lagrangian cobordisms, and the Fukaya category

If one allows for non-compact cobordisms, it turns out that the cobordism category of Lagrangians is a stable infinity-category. It's moreover symmetric monoidal, so its unit object is an E-infinity ring spectrum. I can't compute much about it. But it turns out that any (Calabi-Yau) symplectic manifold gives rise to a stable infinity-category of Lagrangian cobordisms, each of these is ``linear'' over the ring spectrum from above, and one can always construct an exact functor to that manifold's Fukaya category (which is linear over Z). The geometric result is that homotopy groups of cobordism spectra detect Floer cohomology groups--moreover, if two compact Lagrangians are related by a compact Lagrangian cobordism, the cobordism realizes an equivalence between them in the Fukaya category. I won't assume much prior knowledge about the Fukaya category or Floer cohomology, and will begin by reviewing these concepts.