I want to try to analyze the magic schools, making wild guesses off of the little information we know.
White Necromancy is largely characterized through the summoning of willing spirits and souls (which is still illegal by the way) and far more importantly its use in medicine. White Necromancy spells allow the removal of dead tissue, cleaning of wounds and other things beyond that. In conjunction with Life Magic it allows the use of magic on rotting or decaying wounds, removal of parasites and similiar finnicky operations where the use of Life Magic would only make the problem worse. As such it is under strict surveillance within the Kingdom of Wenst but still allowed.
Black Necromancy. The reanimation of corpses with lesser demons and the damned souls gained through bargain with demons from beyond the Pale. Black Necromancy is forbidden almost everywhere, since it cleaves very, very close to actual demonology. Beyond its reanimation spells, Black Necromancy also contains harmful spells used for combat aswell as several crossover disciplines from Demonology.
I'm not sure how exploitable necromancy really is, aside from the obvious "hordes of the dead" approach. You can build a factory or have tireless unskilled workers, but they probably don't learn normally, and so would only be better than peasants for industrial purposes in that they're logistically easier to maintain. You can probably use it to make logic gates and a low-yield Turing Machine, but it'll be huge.
You can use it on a path to immortality, but that upsets the Inquisition even more than necromancy already does, and I'm not keen on granting immortality to any of our broken professors, or the students who will gain all sorts of interesting traumas from them. You can sell it to nobles, but this leads to a distinctly horrifying society.
Asavius practices a barely known form of magic. Sympathetic Magic deals with the linking of objects or sentients and hwo they can be affected by each other. Voodoo dolls are one child of Sympathetic Magic.
OP, best school of any of us.
I expect it to have low-hanging fruits of two-way comms mirrors (with a secret third for evil surveillance, if you want to go that route), which nobles and merchants would absolutely love. You could sell dolls to military wives so they know when their husbands die, or sell chastity rings so nobles know when their spouse cheats. If it can affect emotions one-way, you can keep soldiers from routing, or just nearly mind-control people. And of course there's the basic service of voodoo dolls for agents on sensitive work or high value courier runs, to ensure loyalty.
At the higher end, you have immortality by Portrait of Dorian Grey. Make it so it needs regular maintenance by high-level Sympathetic Magic users, and you have half the nobles and merchants in your pocket. If you can get around the Inquisition. It still has most of the problems of necromantic immortality, but at least you'll have the keys.
Best if all is that you have a near monopoly on this branch of magic.
Ibin Al-Lodril is trained in the ancient elven art of crystal-singing. Mostly used to craft the infamous (and extremely illegal) soul crystals which elven necromancers and blood sorcerers use, this magic is not only restricted but also rare.
I think it can make crystal magi-computers, which should be far more efficient than zombie-based computers. This is good. That or they're just magic batteries, which you can still make good use of.
Sinell...is special. Yes lets leave it at that. she seems to do a lot of sticking pencils through folded papers...so theres that.
Sinell seems to think it's about dimensions and bridging between them. I can probably use it as a game-spanning megaproject to get rich through a portal network.
I expect this field to be about exactly how standard magic works on a fundamental level. I'm not sure what good it is, but I might be able to optimize spells or get some exotic effects out of it.
He is leading in the field of Elder Lore, Magic Knowledge dealing with the beings beyond the Pale
This being a blue sky thaumaturgy field, my first thought is that magic is inherently part of outer gods. If "fireball" is a function call, and normal spellcrafters write new functions, this could eventually be used to write new programs or programming languages.
But after re-reading Black Necromancy, demons are also "from beyond the Pale." They might use a different magical system I can tap into, if the world/universe is basically floating in a +dimensional reality bubble.
I don't think a field that's just "demon lore" would be given under Thaumaturgy, and direct demonology would have dire warnings since the Inquisition is right behind me with a sword, daring me to step wrong.
TC is less of a magical field and more how to apply various spells to reduce and contain magic and its effects. You could call it Anti-Magic studies if you wanted to...
This is a black sheep that barely fits in with thaumaturgy, other than being needed for thaumaturgy.
I think I can use knowledge about the basics of magic to get better at countering all forms of magic, and this is my field of practical magic. It's limited, but the Inquisition likes it, merchants might like a room that's hard to scry on, and secure facilities would be pleased with hiring anti-mages. If I can figure out a way to enchant things with magic resistance, I can use it on walls or on infantry shields for warfare.
It's definitely something I need to keep up with for my other fields, and since the Inquisition will be recruiting them, I need to make training TC mages a primary focus.