Importance of domain selection for undergrad applying to a PhD program (USA CS)

Question

Background: CS undergrad applying to CS PhD programs in the USA right after graduation.

As an undergrad, I research on (and will be till applications are submitted) broadly on 3 applications of a single "Technique/Algorithm".

Reasons for the domain of choice: 2 of the 3 really interest me a lot and the third I took up because the lab is well renowned in the country and I learn a lot in the process.

While this has helped me develop good skills on a technical basis, I really haven't dived deep into a particular "problem" with a fixed domain. Which is the construct of a PhD (right?).

Question 1: Will this look bad on my profile? Will it show that I'm confused?

All 3 domains, I have good research work done. I invested additional time and made sure I am not at a superficial level for any of the domains.

I actually am not confused. There is one broad problem I have in mind which I want to solve during my PhD and that involves the whole "specialisation" aspect and deep diving into a single domain. I will be mentioning this in my SOP.

I am not able to proceed with that now as I won't be able to produce anything great by my own and it demands computational resources and proper guidance which I tried but failed to obtain.

(If the answer to question 1 is yes) Question 2: Will mentioning my plan to specialise benefit me in any way in the eyes of the admission committee?

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Note: If it adds any value to the question. All 3 my letter writers will be backing me for the 3 different applications I work on. 1 letter each for 1 separate broad problem that I work on. (The 3 different problems are done at 3 different labs headed by 3 different profs).

Answer 1

Actually, in the US, any research experience at entrance with a bachelors is a plus, possibly a big plus. Having an idea of where you want to go in research is a big plus, even if not entirely settled. So, yes, it is good to mention the specific areas that most interest you and any experience you have with those.

But acceptance depends on many things, from GPA to letters of recommendation.

I'd suggest that, other things being equal, having an idea of what you want to do in grad school will be an advantage and will help you choose more precisely when it comes time to do so.

But doctoral studies in the US normally start with a lot of advanced coursework leading up to the comprehensive exams so that candidates have a fairly broad exposure to mathematical concepts even before they specialize (and specialize...) on a dissertation topic.

It might be a mistake, however, to refuse to consider other options. Keep an open mind. You will have plenty of time to develop a formal plan.

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My own experience was that, at entry, I would have been pretty happy with anything in analysis or topology, but nothing specific beyond that. And pretty unhappy with algebra. My only "research" experience was in the philosophy of math and it was pretty embarrassing. Long time ago, however.