It's buried on like page 1,000 by now but we had just a really brief exchange about this question last year:

Originally Posted by
Rongway
Based on their Encyclopaedia Eorzea entries, their fields of expertise go something like this:
- Fourchenault Levailleur - something magical, probably, but he could possibly also be an expert on "learning"
- G'raha Tia - specifically interested in the Allagan Royal Eye, which would probably require studies in Allagan history
- Louisoix - Ancient prophecies
- Krile - Healing magicks, probably?
- Matoya - Aetherology
- Moenbryda - Aetherology
- Papalymo - Arcane studies
- Rammbroes - Anthropogeography
- Thancred - Survival and intelligence gathering (spycraft)
- Urianger - Prophecy
- Yda - Hand-to-hand combat
- Y'shtola - Arcane arts
Kesey mentions Mikoto below. The Encyclopaedia Eorzea II indicates her extensive achievements in aetherology.
Personally I'm hoping that Alisaie can start building an Archon thesis out of
what she's learned about tempering and the novel treatments she/we have been rolling out