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Thanks Wintonian. I think this one will take me all week!

It will be a pity if this puzzle puts people off having a go because it has a reputation as "mathematical". A maths background will make people at ease with the theme, but is wholly unnecessary to solution - though maybe a peep or two at the Internet will help the antimath feel more at home.
My problem is more mundane: 24 across. I can't make any of the few words that would fit match up with any misprint-corrected version of the clue. Could someone explain, please?
Looks like it will be a grand day - enjoy it!

Hi, Durban

24ac was a word completely new to me. The definition in TCD is “a creature produced by artificially combining DNA from a goat and a sheep”.

The misprint is in the word “alleges”, and the correct word is a form of a gene. I got this only because I’d already worked out what the correct letter had to be.

Thanks, Wintonian, but that's 24 DOWN and - yes - there is no 24 across! What this old fool meant was 26 across - still can't see it .

No problem, Durban!

I got 26ac as a byproduct of checking the corrected definition for 11ac. The answer is a past participle of a two-letter abbreviation that can be used as a verb (like “to MC”). The misprint is in the first word of the clue.

Durban, aimed becomes aired

My maths A level waits in the wings as my brain goes noooooooooooo. This is very tough. I will plug away with the hints here and the pleasure that I actually had Rizla as one of my few achievements. Perhaps I need some of that college inspiration. As I think I have 25 and 27, can it be confirmed that they begin rows 7 and 8? I cant seem to fit others around them so maybe I have that wrong....although I'm not really sure how else they could fit.

PAYE for 1a has not been helping me!

yes, rrrobbo, they do go there

Woodlouse gives he answers to this equation as -3 1/2 +/- 6.089i

but I've tried a couple of quadratic solvers which give -3 1/2 +/- 6.383572667401852i

Does the fact that it is an argand diagram somehow affect this answer? I've no real idea what any of this is about to be honest. I can see a way of obeying the instruction so that no cell is entirely blank, with either of these solutions, but I'm confused as to which is the correct solution.

Thanks, Wintonian and Keepatit. I arrived at that answer long ago, and discarded it because C did not have it per se. More imagination needed! All clear now.(To quibble, the relevant usage of "air" means broadcast using radio waves; DJ's don't do that, I think - though I'm no authority here)