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My understanding is that postdocs, research assistants (which I understand are master's level analogues of postdocs) and PhD students are all paid positions except for self-funded PhD which I hear is rare, but regardless, PhD students like postdocs and research assistants are expected to assist with the specific research that the professor is doing.

But, considering that a PhD studentship is the pursuit of a degree rather than the application of acquired knowledge from a degree, how much is expected of background knowledge for a PhD applicant?

There was this quora post that I can't find anymore that asked why professors took on PhD students to assist with research instead of hiring a research team or something. I was supposed to link it as an introduction.

Anyway, my question is different from the following because I'm asking specifically for PhD students relative to postdocs or research assistants.

• How much is expected to know before starting a PhD

• What kind of knowledge is required to do a PhD?

• How can an undergraduate student conducting mathematical research with a professor expect to contribute?

I'm going to use math for examples.

• If I know elementary topics of complex analysis, abstract algebra, algebraic topology, algebraic geometry and differential geometry, can I apply as a PhD student of a professor who specializes in complex analysis? Complex geometry? Functional analysis? Differential geometry?

• For location: I don't mind if people post an answer for US or Europe, but in case I really need to say: I live in Country A, which is somewhere in the middle.

My understanding is that postdocs, research assistants (which I understand are master's level analogues of postdocs) and PhD students are all paid positions except for self-funded PhD which I hear is rare, but regardless, PhD students like postdocs and research assistants are expected to assist with the specific research that the professor is doing.

But, considering that a PhD studentship is the pursuit of a degree rather than the application of acquired knowledge from a degree, how much is expected of background knowledge for a PhD applicant?

There was this quora post that I can't find anymore that asked why professors took on PhD students to assist with research instead of hiring a research team or something. I was supposed to link it as an introduction.

• Update: I found the question! Why would a professor want a Ph.D student instead of a research staff? What are the advantages and disadvantages?

Anyway, my question is different from the following because I'm asking specifically for PhD students relative to postdocs or research assistants.

• How much is expected to know before starting a PhD

• What kind of knowledge is required to do a PhD?

• How can an undergraduate student conducting mathematical research with a professor expect to contribute?

I'm going to use math for examples.

• If I know elementary topics of complex analysis, abstract algebra, algebraic topology, algebraic geometry and differential geometry, can I apply as a PhD student of a professor who specializes in complex analysis? Complex geometry? Functional analysis? Differential geometry?

• For location: I don't mind if people post an answer for US or Europe, but in case I really need to say: I live in Country A, which is somewhere in the middle.

My understanding is that postdocs, research assistants (which I understand are master's level analogues of postdocs) and PhD students are all paid positions except for self-funded PhD which I hear is rare, but regardless, PhD students like postdocs and research assistants are expected to assist with the specific research that the professor is doing.

But, considering that a PhD studentship is the pursuit of a degree rather than the application of acquired knowledge from a degree, how much is expected of background knowledge for a PhD applicant?

There was this quora post that I can't find anymore that asked why professors took on PhD students to assist with research instead of hiring a research team or something. I was supposed to link it as an introduction.

Anyway, my question is different from the following because I'm asking specifically for PhD students relative to postdocs or research assistants.

• How much is expected to know before starting a PhD

• What kind of knowledge is required to do a PhD?

• How can an undergraduate student conducting mathematical research with a professor expect to contribute?

I'm going to use math for examples.

• If I have 1 class in complex analysis but otherwise come from an applied math background where I haven't taken geometry, topology, algebra, number theory, can I apply as a PhD student of a professor who specializes in complex analysis? I'd think that those other courses would be relevant because how much pure math intersects.

• How about same as above but 1 of each of the courses?

• If I have 1 class in stochastic calculus (I actually have 2) or operations research and come from an applied math background, can I apply as a PhD student of a professor who specializes in stochastic calculus or operations research?

I'd think the lack of pure math background would not be relevant even if, for instance, probability uses complex analysis as early as in characteristic functions because the student can just take complex analysis in the program. I don't think the lack of any 1 of the pure math courses above would be a deal breaker for an applied math professor.

As for the applied math analogues to pure math courses above, which I guess would be numerical analysis/scientific computing, PDE, operations research, advanced statistics and advanced probability (in my university, BS and MS math students aren't required to have these but are free to take any as electives), I guess an applied math professor may have an analogous problem for an applicant with a pure math background with, for instance, only 1 class in stochastic calculus or operations research. If not, then good for pure math background applicants!

• What if the professor researches on something that isn't taught in the average bachelor's/master's in math/applied math? For instance, mathematical biology. I think the closest thing my university had to that was the brief mention of Lotka–Volterra in ODE classes. I mean, it's not like there are a bunch of people lying around who major in both mathematics and biology. Another instance would be image reconstruction. I think image reconstruction was in some exercises in our numerical analysis book, but they were never discussed in class, and we were never required to do them.

• For location: I don't mind if people post an answer for US or Europe, but in case I really need to say: I live in Country A, which is somewhere in the middle.

My understanding is that postdocs, research assistants (which I understand are master's level analogues of postdocs) and PhD students are all paid positions except for self-funded PhD which I hear is rare, but regardless, PhD students like postdocs and research assistants are expected to assist with the specific research that the professor is doing.

But, considering that a PhD studentship is the pursuit of a degree rather than the application of acquired knowledge from a degree, how much is expected of background knowledge for a PhD applicant?

There was this quora post that I can't find anymore that asked why professors took on PhD students to assist with research instead of hiring a research team or something. I was supposed to link it as an introduction.

Anyway, my question is different from the following because I'm asking specifically for PhD students relative to postdocs or research assistants.

• How much is expected to know before starting a PhD

• What kind of knowledge is required to do a PhD?

• How can an undergraduate student conducting mathematical research with a professor expect to contribute?

I'm going to use math for examples.

• If I know elementary topics of complex analysis, abstract algebra, algebraic topology, algebraic geometry and differential geometry, can I apply as a PhD student of a professor who specializes in complex analysis? Complex geometry? Functional analysis? Differential geometry?

• For location: I don't mind if people post an answer for US or Europe, but in case I really need to say: I live in Country A, which is somewhere in the middle.

• What if the professor researches on something that isn't taught in the average bachelor's/master's in math/applied math? For instance, mathematical biology. I think the closest thing my university had to that was the brief mention of Lotka–Volterra in ODE classes. I mean, it's not like there are a bunch of people lying around who major in both mathematics and biology. Another instance would be image reconstruction. I think image reconstruction was in some exercises in our numerical analysis book, but they were never discussed in class, and we were never required to do them.

• For location: I don't mind if people post an answer for US or Europe, but in case I really need to say: I live in Country A, which is somewhere in the middle.

There was this quora post that I can't find anymore that asked why professors took on PhD students to assist with research instead of hiring a research team or something. I was supposed to link it as an introduction. Anyway, my question is different from these because I'm asking specifically for PhD students relative to postdocs or research assistants.

• How much is expected to know before starting a PhD

• What kind of knowledge is required to do a PhD?

• How can an undergraduate student conducting mathematical research with a professor expect to contribute?

There was this quora post that I can't find anymore that asked why professors took on PhD students to assist with research instead of hiring a research team or something. I was supposed to link it as an introduction.

Anyway, my question is different from the following because I'm asking specifically for PhD students relative to postdocs or research assistants.

• How much is expected to know before starting a PhD

• What kind of knowledge is required to do a PhD?

• How can an undergraduate student conducting mathematical research with a professor expect to contribute?

• What if the professor researches on something that isn't taught in the average bachelor's/master's in math/applied math? For instance, mathematical biology. I think the closest thing my university had to that was the brief mention of Lotka–Volterra in ODE classes. I mean, it's not like there are a bunch of people lying around who major in both mathematics and biology. Another instance would be image reconstruction. I think image reconstruction was in some exercises in our numerical analysis book, but they were never discussed in class, and we were never required to do them.

• For location: I don't mind if people post an answer for US or Europe, but in case I really need to say: I live in Country A, which is somewhere in the middle.