TITLE: Algebraic P vs. NP
QUESTION [30 upvotes]: I recently attended a lecture where the speaker mentioned that what he was talking about was connected to the algebraic version of the $P$ vs. $NP$ problem. Could someone explain what that means in a simple way or point me to a source suitable for a non-expert mathematician? Thanks.

REPLY [25 votes]: I suspect that the question under consideration is whether or not $VP=VNP$; this is the problem directly studied by geometric complexity theorists, as I understand their work.  This project is described in some detail here (this paper is by Burgisser, Landsberg, Manivel, and Weyman, describing work of Mulmuley and related people)--it is aimed at algebraic geometers; so you will likely be comfortable with it.  The description of the complexity problem under consideration is in section 9.
The algebraic $VP$ vs. $VNP$ conjeture is due to Valiant, in this paper and his paper "Reducibility by algebraic projections" which I can't find online at the moment, unfortunately; these are references [63] and [64] in the paper I link to above.  Valiant is, as I recall, a very clear writer, so hopefully you will find these papers readable as well.
Essentially, Valiant argues that some algebraic properties of the permanent and related varieties should have complexity-theoretic implications; a reasonable heuristic for this might be the many combinatorial interpretations of the permanent.  Unfortunately, as far as I know there are few implications between these algebraic versions of P vs. NP and the problem itself; there are some results assuming GRH.  See e.g. this paper by Burgisser.
Hopefully, this is the algebraic analogue of P vs. NP you were looking for.