Across
Stern child sorry about tense picture (15)
I haven't cracked this one yet — but Fifteensquared have:
Back(=stern) + tot(=child) + eh<(=sorry backwards) + future(=tense of a verb)
Navy tracking wayward marine who made unproven claims (7)
I haven't cracked this one yet — but Fifteensquared have:
N{avy} after(=tracking) after marine*. I think the def refers to the Riemann hypothesis, which I won't attempt to explain as I haven't the foggiest what it's about, but it is one of the most important unproven conjectures in mathematics.
Do turn into Headless Man? (7)
I haven't cracked this one yet — but Fifteensquared have:
Wind(=turn) in [i]sle(Isle of Man being an example thereof)
Spare bank (4)
See inside Curie's second home (10)
Riotous pastoral events? (14)
Dan etc developed alarmist traits (14)
Walk through temporary clearing (10)
King once called joint (4)
Some proclaim it a terrible ape (7)
School literature of elite becoming tender to Shylock (7)
I haven't cracked this one yet — but Fifteensquared have:
Elite with the lit{erature} being replaced by ducat. Ducats are the currency used in the Merchant Of Venice by Shakespeare.
Mysterious character behind golfer to pitch film (15)
I haven't cracked this one yet — but Fifteensquared have:
Rump(=behind) + Els(=Ernie Els, golfer) + tilt(=pitch) + skin(=film)
Down
Just having nothing to wear? (6)
End another board game perhaps after onset of Chinese burn (7)
I haven't cracked this one yet — but Fifteensquared have:
Re-mate (which might be a way of ending another chess game) after C[hinese]
Duck in lift bumping head (4)
I haven't cracked this one yet — but Fifteensquared have:
[S]teal, lift here being a slang term for steal with the first letter "bumped" i.e. removed.
Pulling 3 around new Jerusalem (9)
I haven't cracked this one yet — but Fifteensquared have:
Teal(=answer to 3) around n{ew} + Sion. Sion is one of the names of Jerusalem used in the bible.
Relieved and happy but out of place (5)
I haven't cracked this one yet — but Fifteensquared have:
[Pl]eased
Where pilots go to be confirmed (10)
Keep rare pair of diamonds including queen in card game (7)
I haven't cracked this one yet — but Fifteensquared have:
(Two D{iamonds} around ER) in uno(=card game). The def refers to undercooking something like a steak.
Contents of table's bread and wine (8)
A ship with one crew member wrapping article from Italy to take on board (10)
I haven't cracked this one yet — but Fifteensquared have:
A SS(=ship) + ((I mate) around il(=masculine form of definite article in Italian))
National park replaced green site (9)
Filthier S&M previously shunned by liberal (8)
I haven't cracked this one yet — but Fifteensquared have:
S & M + ear[l]ier (minus l{iberal})
Rock band covering mostly soft sad music (7)
I haven't cracked this one yet — but Fifteensquared have:
REM(=US rock band) around quie[t](=soft).
PIN number much appreciated being in credit (7)
River always going in opposite directions (6)
I haven't cracked this one yet — but Fifteensquared have:
Ever(=always) in S & N(=opposite directions)
Brigitte Bardot's very special hair (5)
I haven't cracked this one yet — but Fifteensquared have:
Tres(=very in French, the language of Bardot) + s{pecial}
Stony sculpture (4)
Type letters or click words in the clue