when I applied to PhD programs (not in math) it was basically CV + personal statement + recommendation letters + short chats with interested faculty :shrug: Maybe it was because my CV was "strong" but the chats were more see if interests were aligned, rather than actually interviewing me.
I also know that another major difference is that American universities tend to hire without a professors involvement, into a generic "program", then the PhD student seeks an advisor after being accepted, so applicants have to woo some unconnected committee pursuing various goals misaligned from the PIs instead of convincing the PI. In much of Europe a professor basically "owns" a chair and really is boss and decides hiring pretty much alone.
Anyways, if they don't have a masters' yet, they could present their bachelor thesis. But I already think it's a bad model to combine the masters (courses) and the phd (research) into this American hybrid that's the direct-bachelor-to-PhD jump, but that is somewhat unrelated.
I'm not a big fan of the US application setup. IDK how it is in Europe, but in the US, it feels like there's a lot of not-very-meritocratic "secret" stuff you need to know to up your chances.
In Germany (countries differ quite a bit on these details) a significant chunk (though not necessarily majority) is directly hiring a master's student who worked at the same chair on a part-time student research assistant job and/or a master's thesis and did a good job and typically turned the thesis into a paper submission. In other cases it's still somewhat inside baseball like getting recommendations from other scientists that the PI knows, or coming from a university that the professor knows well. It can also be from an impressive email, from a chat at a conference poster, etc. There is much less of this global casting that committees do, like "we need this and this type of character now" to have a vibrant student body composition, since each PI makes a decision just about their lab, without having a global view, and their goals are basically to get research out of that person, papers, results useful for funding applications etc. So they want to see aptitude in that. Through talk quality, communication efficiency when talking with existing team member one on one, recommendations, sometimes even phone calls with the PI with whom the student worked (these are small worlds). I think it's more organic and natural, though still prone to exclusionary gatekeeping, than the HR-style narrative essay-and-extracurriculars-based admissions though maybe that's more prominent for undergrad admissions even in the US.