Across
Single fellow who’s qualified in Arts or Science? (8)
Gave instruction in rigid pronouncement (6)
Writings on specific subjects soldier demolishing exam exploits (8)
I haven't cracked this one yet — but Fifteensquared have:
CORPORAL less ORAL (exam) + USES (exploits)
Highly trained person, not necessarily in medicine (6)
I haven't cracked this one yet — but Fifteensquared have:
Just an extended definition: a doctor could be highly trained in medicine, or in another subject for a PhD
Level of tension that rises by degrees (11)
Not advanced degree, as recorded (5)
Type that’s rejected Trinity coming to university in a train, oddly (9)
I haven't cracked this one yet — but Fifteensquared have:
U + (IN A TRAIN)*
Its teams are a shade lighter than opponents (9)
I haven't cracked this one yet — but Fifteensquared have:
Cambridge sports teams play in light blue, their Oxford opponents in dark
Part of education that could be spent in the past (5)
Innocence as misrepresented in small-scale theory and research (11)
Immediately in agreement about first in Classics (2,4)
I haven't cracked this one yet — but Fifteensquared have:
C[lassics] in AT ONE (in agreement)
Graduate, note, below me in musical club performances (8)
I haven't cracked this one yet — but Fifteensquared have:
BA (that degree again) + RE (the note below me) in CATS (musical)
Part of one’s old college, including second teacher (6)
I haven't cracked this one yet — but Fifteensquared have:
S[econd] in MATER (part of Alma Mater, one’s old college)
Learned university craft with editor after editor (8)
Down
Like term after term without vacation? Sponsor closed fund (4-2-4)
Rooms arranged in cold institution for term of biology (10)
I haven't cracked this one yet — but Fifteensquared have:
ROOMS in C HOME
Ignoring the odds, best buddies study abroad (5)
I haven't cracked this one yet — but Fifteensquared have:
The even letters of bEsT bUdDiEs – “study abroad” because it’s the French word for “study”
Excessively rich nut a brave don demolished (12)
As assistant, inserting warning on course already mentioned (9)
I haven't cracked this one yet — but Fifteensquared have:
FORE (warning on a golf course) in AS AID
Announced step to punish student (4)
A quarter of Americans, say, stored in computer memory (4)
Stiff paper, the passing of which facilitates working relationship (8,4)
Recording what’s in syllabus – overlapping, hence irritation (10)
I haven't cracked this one yet — but Fifteensquared have:
DISC (recording) + CONTENT (what’s in a syllabus), with the two Cs overlapping
Not subject to pressure, like bridge in 18 (10)
History professor finally working in funny cliche (9)
I haven't cracked this one yet — but Fifteensquared have:
[professo]R + ON (working) in CLICHE*
Patriarch who, to his friends, explained the gravity of the situation? (5)
Polite form of address reflected academic achievement (4)
Assumes they should be experts in their fields (4)
Type letters or click words in the clue