What is expected in a Thesis Abstract, on a math job application?
The few times I was asked for such a thing, I literally copy-pasted the abstract for my thesis into a document and submitted it. mathjobs doesn't want to tell me which jobs asked for it back when I applied so I'm not sure (I got two postdoc offers that round but I'm pretty sure those places didn't ask for a thesis abstract). No one emailed me to tell me that I'd done it wrong though. Usually there is a contact person listed in each mathjobs ad that you could contact to ask for details, but if say the deadline is tonight I would recommend just copy-pasting the thesis abstract, or making up a thesis abstract if your thesis doesn't exist yet.

In principle though, the introduction to your research statement and your thesis abstract can be (but isn't always) quite distinct (my thesis was a subset of my research statement). A research statement should have at least something to say about where you envision your research going, but a thesis abstract is mostly limited to the contents of your thesis.