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I'd be a bit wary of working with this person if this is really them and is really their communication style.

20 minutes is plenty of time to determine a student is not a fit and to politely deny them. 20 minutes is not really enough time to commit to supervising a student.

This level of communication might be typical from some professors, but I think those are the worst to have as advisors:

"What do you think of my research ideas?"

"Good"

"Where should we try to submit this paper?"

"Journal"

"I'm worried I'm not making progress towards my thesis"

"Ok"

I would look at the admissions process for this university. If the process is such that you need to have some professor's sign off to apply, then maybe that's all they are trying to convey: that you get their sign off, and they aren't putting more effort in because the application process tends to winnow students.

In that case, clarify they are willing to sponsor your application, and go ahead and apply, but make sure you get a chance to meet further with this person before you take them as an advisor, and continue to apply elsewhere: you should not take this as a sure thing by any means, nor should you be sure yourself. If there are other people there you are interested in working with, by all means contact them.

Good luck.

It means there was likely human error on the professor's part. The single word doesn't convey any message. Most likely explanations could be the professor replied to the wrong email, or accidentally pressed send before finishing his reply.

I'd suggest writing back and asking for clarification.

Edit: to answer your edit - you've asked for clarification and he comes back with another ambiguous answer. This isn't a good sign. If you're really, really keen to work with him I'd suggest writing back again asking if you should do anything next; otherwise I'd just assume he's not really interested and look elsewhere.

I wouldn't read it as a "no". But it might have been an automatic reply, given the timing. It probably has the same meaning as "Thanks for your interest". Your next step should probably be to ask how you can formally apply for a position. You can also explore, separately, the school's admission requirements.

But I would take it as a positive signal, not a negative one.

All you can conclude from this is that this professor does not give reasonable responses to e-mail.

How to proceed: not by e-mail. If it is feasible to visit him in person, that is ideal. If not, you could request a phone call to discuss further. Either way, your response should be very concise. "Great! Could we have a phone call this week to discuss? When would be a good time for you?"

If you succeed in getting a meeting, you can assess whether he would be a reasonable colleague from there. Conversely, if you cannot get a meeting after a reasonable amount of effort, it's probably better to walk away.

It seems, especially from your comment that "I was hoping that they would advocate my admission", that you were hoping that based on a cold email the professor would go out of his way to help you. You mentioned two research areas and said that "I would like to know more about them." What were you hoping for here? That he would reply with a detailed explanation?

You asked if you could look forward to a position in his group, and he replied with "yes, you can". Here's one possible thing it might mean: "If you apply to my university, are accepted, enroll, pass your qualifying exams, and make a positive impression, then I would be happy to take you on as a student."

It is difficult to guess exactly what the professor meant. My best guess is that he wants to be encouraging, but is trying to steer you to "normal" channels: to apply to his university for admission using whatever standard procedure is in place, to read his papers if you find them interesting, and if you have any questions to ask them more directly.

I think "welcome" means he is happy to welcome you into his group. This is supported by the statement "yes, you can." Perhaps the professor is under the impression that you already have admission into the university and are simply looking for a guide.

If you want to work with the professor, you should email and ask how to proceed further. How should you apply, where do you get funding etc.

I would say that's a "No", and I would move on because this professor hadn't shown the signals which are sent from a typical "eager" professor to an applicant in whom she is interested. An interested professor often gets into details fast. Namely, she may: ask about the details of your research; show her curiosity about your background (which would be critical for your research) by asking about the related courses you have passed; ask about your potential funding resources to make up her mind about your needs to financial support. This list can be easily continued...

To my experience, if you look like the "right person" to a professor, she would not give you such kinds of short telegraphic responses encouraging you to pursue the way through official channels (The exception would be the professors affiliated with super famous schools in which the admissions are essentially committee-based). Professors are often competing with each other in attracting good students. So, if a professor finds someone who looks like a great fit to her requirements, she would try to show her interest. In fact some PIs often return to candidates by these kind of short not-so-expressive answers in favor of their institutions! In particular, if a professor is not in need of recruiting someone, she may not explicitly express her point. Instead, those kinds of short answers keep the applicant hopeful for basically nothing. Thus, he applies to that program thinking that those "welcome"s or "yes"s were positive signs toward success. But the whole point would be the application fee paid by the applicant; the money that would not be steered to the university's pocket if that PI had been clear to the applicant about the lack of any interest in his case.

As others have noted, this is ambiguous and somewhat confidence-lowering.

In your next interaction, be sure to ask an open-ended question, not a polar yes-no one, and get a reply that is satisfying to you. Specifically you should really ask:

What are the next steps I should take?

1: received gladly into one's presence or companionship
was always welcome in their home
2: giving pleasure : received with gladness or delight especially in response to a need
a welcome relief
3: willingly permitted or admitted
he was welcome to come and go
— W. M. Thackeray
4—used in the phrase "You're welcome" as a reply to an expression of thanks

https://www.merriam-webster.com/dictionary/welcome

The fourth definition doesn't fit, as you didn't do anything for them. This leaves the other ones, which all are variations on "received gladly". Given just that response, the professor appears to be is admitting you into the position. It is rather ambiguous, especially since apparently there is more than one position, and it also indicates a low degree of selectivity. You certainly should get more clarification, but it appears to be a positive response.

Back in 2011, my roommate received similar email: 'yes plz aply' (with single p). He was disappointed, but still applied (was not admitted to that university though). I decided not to send any email to professors. We both applied through regular admission processes and got accepted.

Many years later, after several years in academia, I understand that this response does not mean anything. Well-known professors receive dozens of such emails, and have to save their time.

Just apply.

On the basis of your edit, you should double check that you sent the emails to the correct address - confirm this by looking for the contact details of the professor on the department website.

If the email address on the professor's website and on the departmental website are different, try the other address and let the professor know that there may be an error on their website. If these are the same, there may be a phone number for the professor on the departmental website, which could be worth giving a ring.

Failing that, try emailing the department secretary (or local equivalent), say that you've been having trouble contacting the professor, and ask for their help, since the other methods of contact haven't worked satisfactorily.

"What's the next step?"

Remember that question when interviewing for a "real" job. Don't wait for the professor to make the next move, take the initiative to push the process forward. At some point, you need his offer in writing in some form, I've had people verbally offer me jobs that were more about their wanting to work with me than their authority to hire me.

In grad school, I was offered 3 assistantships; the prof who gave me a payroll sign up form got me, "This education brought to you by a grant from Exxon Corp." Offers can come in strange forms!