sci.math.research archive? Does there exist an archive somewhere of posts to the USENET newsgroup sci.math.research?
The best approximation I'm aware of is Google Groups.  However, despite the Google brand name, the search capability of Google Groups is abysmal, and it is hard to tell how much content is really archived there.
I have an old bookmark to a Math Forum URL that currently seems to be dead. An old website called the The Mathematical Atlas indicates that the Math Forum moved to forum.swarthmore.edu but that site currently seems not to be working either.
The Usenet Archives website, as of this writing, seems to have archived a grand total of two (2) posts from sci.math.research. EDIT August 2022: The Usenet Archives has now archived much more of the content, but there is no easy way to search it.
This question was prompted in part by another MO question, Looking for source: “How not to be a graduate student”, which hints that the desired transcript may have been posted to a USENET newsgroup such as sci.math.research or sci.math. In my opinion there was some valuable content posted to sci.math.research and it would be a shame if it were lost forever.
 A: This is probably better suited as a comment to Gro-Tsen's answer, but I do not have an account on this site.
A relatively complete archive (including messages spanning 1991 through 2013) appears to be available on the Archive.org site at:
https://archive.org/details/usenet-sci
and in particular at the link:
https://archive.org/download/usenet-sci/sci.math.research.mbox.zip
A: There's a bit of content on Archive.org: the link says sci.math, but among the files in the collection there is a sci.math.research.20140626.mbox.gz which contains ~11k messages apparently posted between 2003 and 2014.  Sadly, this does not seem to contain the specific content you were looking for.
You could try contacting Archive.org: even though they don't have the data, they might be in position to persuade Google to give them a full dump of a low-volume high-quality newsgroup like sci.math.research since it's part of their service mission to archive this sort of data.
Another option is to ask Daniel Grayson, who was moderator for sci.math.research (he wasn't the only one, but there seems to be a connection between the newsgroup and UIUC so maybe the chances are higher): it isn't unlikely that the moderation process kept a complete archive at some level.
