As a post-doc applicant, should I address a professor by their first name in email after they addressed me that way?

Continue to use the surname until explicitly given permission to do otherwise. One way for her to give that permission, actually, would be to sign a mail with only her first name.

There is a power imbalance between you. She is probably not being entirely correct in assuming she can use your first name, but it is common for people in authority to take some liberties.

But I'd recommend keeping it a bit formal until that obviously no longer works. And my full name is "Bobby Buffy" then I'd never be Professor Bobby. It would either be Professor Buffy or just Bobby. (For the record, though, I'm not Bobby.)

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Note that this answer may not apply everywhere. It has a US perspective.

If she "signed" the Email, see how she signed it. People usually sign letters and emails (if at all), the way they'd like to be addressed by the recipient.

• If she signed with only (one of) her given name(s) (Y1), feel free to address her using that/those name(s) (the same she used), without any title, i.e.

  Dear Y1

  • The same applies in the unlikely case that she signed with a nickname (Z): In your reply, address her as

    Dear Z

• If she signed with anything including her last name / family name (X), be it

  • only her last name (X)
  • her given name(s) and her last name (Y1 X)
  • (some of) her academic title(s) and her last name (Prof. X)
  • all of the above (Prof. Y1 X)

  , then continue to address her with title(s) and last name / family name, i.e.

  Dear Prof. X

• If she didn't sign the email, or just with a greeting but without any part of her name, also stick to

  Dear Prof. X

  unless there is another compelling reason to believe she wants to be addressed differently. (E.g., she explicitly mentioned it in the body of the email.)

As buffy mentioned, academic titles and given names without the family name don't go together. So if she were to sign her first email to me with

Cheers,

Prof. Y1

I'd be rather baffled and wouldn't be sure how to address her in a reply.

This rule of thumb applies outside of academia, too. (In absence of academic or other titles, use "Mr." or "Ms." as appropriate with last names.)

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While my experience stems from Switzerland, I believe it can (regarding this) be applied to all of the "western world", thus to the US as well. While even western countries and cultures (and sometimes, individuals) can be very different in when what person allows what other person to be on (uni- or bilateral) first-name basis with them, the way of signalling this in signatures is, to my understanding, much more universal.

Stick with Professor X until you get a blindingly obvious cue. It will not cause offence even if you missed a hidden cue that just Y1 would be appropriate.

As a male, white former academic married to a (woman) professor, there is another unfortunate dynamic to be aware of. Unless Professor X has been extraordinarily lucky, she will have doubtless experienced at some point in her career being taken less seriously by someone due to her gender, and this may well have translated into that person addressing her informally by first name, uninvited, while continuing to address her male peers formally. While this has nothing to do with your situation, a too-early uninvited transition to informality may -- not necessarily, but more likely -- set off niggles of concern than in someone else who is less likely to have had to ever deal with this unfortunate dynamic.