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Background: Due to COVID-19 pandemic I'm doing my summer internship on Mathematics in online mode.

I'm not sure whether this will make my resume stronger. Is it possible to get good recommendation letter form this online internship or will it be frowned upon during the PhD admission as an international applicant?

Also how to make most of these online internships because interaction with the professor is virtual along with professors busy schedule?

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will it be frowned upon during the PhD admission as an international applicant?

During a pandemic, reasonable people will not penalize you for doing your research online.

Before the pandemic, people might have been confused by the idea of an online research internship.

It's not possible to predict attitudes towards online internships performed after the pandemic.

how to make most of these online internships

Communicate. It's always good to communicate with your colleagues, but when working online it is especially important.

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(I am assuming you're an undergrad. Most of what I say below might apply to a masters student as well. I am an undergrad myself).

To first note, the following things aren't affected online -

  1. Your results. Whether you get them online or in a lab, if you have verified results that means your work has been worthwhile.
  2. Communication with your advisor. From what I have noticed. Your advisor will be as busy as he was if the session was offline. If the internship has promised your advisor will dedicate time to you, he or she will even though it's on a call. How comfortable you are in an online format is separate issue.
  3. Resume. Going back to the first point, if you get your desired results, it will look great on your resume. Honestly there would be no difference if you got the results online or offline. I don't think anyone has that much time in hand to look into that perspective of the results.

What Might get affected -

  1. The depth of your results. If your research demands a lab environment, you might not be able to explore few aspects of your problem. I don't think that is something you will personally face given your domain.
  2. Pay. All online internships pay less and provide lesser benefits.
  3. Networking. This is the biggest drawback in my opinion. I don't think you will have a lot of opportunities interacting with people who aren't directly involved in your work. (If the lab has a slack group, it might be easier to network.)

The only positive I see in an online environment is flexibility. I don't think people pay much attention "when" you get the work done. (Though this is in my and a couple of friend's experience).

What I am unsure about is the LOR and PhD Admit.

A. If you get your desired results, I don't think there is any issue what so ever. Would be the same LOR you would've gotten offline with minor differences.

B. If you do not get the desired results, it depends how much effort is put in and how your advisor is willing to interpret that.

The PhD panel most likely will sway depending on the above as well. (please correct me if I'm wrong here.)

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