Scilla siberica, Siberian Squill and Puschkinia libanotica, Striped Squill.

Do you have a picture of the flowers? Or is this what you think are the flowers? These are the leaves, not the flowers. it could be something like Balotta pseudodictamnus. (Bear in mind that all the plants you have shown are shrivelled with the summer heat and not at their best.)

Wow interesting, Southern Chervil . Looks like chervil but they say it is not a substitute for chervil in the kitchen. To bad. I would love a native good tasting chervil Southern Chervil

Casia, I'm just north of Concord NH and have this same weed in my gravel drive, so I am glad that you asked what it is. I don't know what it is, but I don't think that you have the ID quite right. The flowers look right for something closely related to the Cerastium that you found, but the overall plant growth style doesn't. If you look in the second photo, there are green leaves which certainly look like Cerastium/chickweed of some kind, but the leaves for the plant you want ID'd are the deep purple ones in a tiny rosette. The only leaves are in this rosette, and the flowering stem is leafless, wiry and straight upright with the flowers in a small cluster (cyme) at the top. If anyone wants to take another shot at ID on this, the seeds are oval in shape, flat and papery thin, reminding me of elm seeds on a tiny scale. It seeds quite prolifically!