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Does one need a specific field in mind to apply to a PhD program in mathematics?
When applying for the PhD program in mathematics, usually, one is not required to specify in what field (e.g., PDE, dynamical system, etc.) he/she intends to do. However, I don't know whether it will be disadvantage that one does not have a specific field in mind at all.
I ask this question because some people suggest the students who are applying to the graduate school should talk to or connect with the professor who is in the school he/she intend to apply for. But if one does not even have a specific interest in mind, how can he/she talk to a professor about his/her application? (Even if the student is interested in analysis, say, there are lots of sub-field in analysis.)
So, here is my question:
Does one need a very specific field in mind to apply a PhD program in mathematics? Would this be thought as advantage or disadvantage of an application?

At least for most graduate programs in pure mathematics in the US, there's no need to have a specialization in mind when applying.  [This may be very different in other countries.]
It's valuable to demonstrate in your application that you have studied some serious mathematics, by discussing undergraduate research or advanced coursework.  However, there's no implication that you intend to focus on the same fields in graduate school.
It's common to indicate an interest in a few possible specialties, usually at a level of detail ranging from "algebra" to "analytic number theory".  If you are completely undecided, then that could come across negatively, by suggesting a general lack of enthusiasm.  However, being too specific is also problematic.  Matching a possible advisor too closely comes across as pandering, while being specific without matching anyone makes it look like you aren't a good fit for this particular department.
Overall, the general feeling is that incoming graduate students don't know enough to make well-informed decisions about specialization, and that anything they say is a little unreliable because their interests may shift as they learn more.  From that perspective, it's not worth worrying about this too much.
As for talking with professors, at least at the schools I'm familiar with this will not increase the chances of admission (the decisions are made by a committee).  After you've been admitted, it's important to talk with faculty and try to gauge the chances of finding a suitable advisor.  However, that can and should wait until February.

To add to what the Anonymous Mathematician said (so again, this has to do with United States)
If you have not already decided on a specialisation, you should certainly not try to shoehorn yourself into one just for the sake of applying for graduate school. If you have already decided on a specialisation, however, you should certainly specify it and try to talk to faculty members before and/or during the admission process. This is not so much to improve your chances at the school per se; this is mostly to find out whether that school would be a good fit for what you want to study.
When I was applying for graduate schools I had just one such experience. I was already quite sure about what I wanted to study, and so contacted some faculty members at places where I applied to who may be good advisors. One of them told me that (a) I have a pretty strong application and they will probably admit me (b) But he is personally busy with his existing students and won't want to take another one (c) There's no one in the same department that he sees as having similar enough interests to be a good advisor for me and (d) The department is running a bit low on money so unless I get a fellowship somehow I will have a hellish teaching load.
In the end I took an offer from another school.
For emphasis, however, what I mean by specialisation is very narrow: "partial differential equations" is still too broad. For my advice above to apply you need to be able to say which type of PDEs you are interested it (elliptic, parabolic, evolutionary, dispersive, transport, kinetic theory, fluids, and/or optimal control to list a few) and to be able to hold an informed (not necessarily expert) discussion on the subject and why you want to study it.

For other countries the system can vastly differ; this is especially so for degree programs which require a Masters or equivalent degree for admission. Many of those programs require submission of a research proposal and admission is contingent on the research being likely to be able to be conducted at said university and that a suitable advisor can be found for your research proposal. But for these kinds of degrees the opposite of the expectation described by Anonymous Mathematician holds: the incoming students are expected to be sufficiently educated (by the Masters degree) to make informed decisions and are also expected to have a good idea what is involved in pursuing PhD research and know what they want to work on. But you can general tell the expectations by reading up on the qualifications for admission and on what the departments/universities expect on the application form.

The experience you describe with the professor who disclosed the issues you might have at his school make this an excellent answer--those kinds of questions might not even have occurred to a person to think about when they are looking for a program.

In France the picture is very different from the US, since you start right away your PhD research (so you must have an adviser from the beginning, and an idea what you will be working on).
If one includes the M2 (second-year master) in graduate studies, then anyway most of them are specialized, so by picking one you do choose a specific field. You can switch for your PhD, but for some fields with many prerequisites it is usually a bad idea.


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Different CV for university position and research industry position?
Up to now, I've only applied to "public" academic positions (universities or public research centers). I have now an academic CV, that is roughly 7 pages long (2 pages of "classic" CV, and 5 pages of research experience + list of publications).
I would be interested in applying for research positions in industry. I know that if I wanted to apply for a regular position (for instance, as a software developer), only my classic CV is enough (and the long one might actually be counter-productive). However, the kind of position I'm interested in is very similar to a public research position, i.e. people there publish, applying for public funding, might even co-supervise PhD students.
So, what kind of CV should I send for such a position? Should I consider the application process as the standard public one, or rather as the standard industry one?
--
In case it's relevant, it's in Computer Science, and I'm thinking of positions such as those available at Microsoft Research, IBM Research, etc.

The general rule of thumb is that every resume should be custom tailored to the job you apply for. Granted CVs are a bit different but the same logic applies.

@bobthejoe Sure, but if there are no application guidelines for the CV, then it doesn't really help me to know what kind of CV should I send :)

So, for those in North America, the distinction would be between a CV and a resumé?

@HarrisonW.Inefuku Not sure about the difference, I only use the term CV :) But I've heard that CV has an academic meaning, while résumé is general (and short).

As a general rule of thumb, if the position you are applying for (1) has the term "researcher" or something similar in the position title and (2) requires a PhD (or research masters), definitely send the academic one. The fact they're looking for PhDs means they're interested in research experience, and your academic CV will highlight that much better than your industry one.
That being said, it is an industry position, so I would do my best to keep the resume two pages, and offer to send the full CV on request. Definitely include "Selected Publications" and "Selected Oral Presentations" sections, but keep it short.

One extra idea for consideration: I sent my (academic) resume to an industry position, and they sent it back asking for a short (one to three) sentence blurb describing each of my projects (at the time, my resume included work as a research assistant, graduate student, and post-doc). I ended up using that format for all the jobs I applied to, since most people have no idea what "Temporal Dynamics of the Cortico-limbic System" actually means. It's a good way to help them actually get a feel for what you accomplished and what your skills are.

Thanks, right now, I have a "short" CV, of two pages, with major points of each positions (project involved, number/kind of publications) and some kind of long appendix (3/4 pages), that describes in detail my research experience. But I like the idea of 2/3 sentences blurb for each project!

I didn't really change my CV for industrial research positions versus academia, but I did tailor my research statement a fair bit. As I recently explained to a student, industrial research labs are often more interested in your skills, and academic positions are more interested in your contributions to an area (all these statements are "on average"). That might indicate ways to modify your CV ? maybe have a blurb highlighting your interests/skills up front for the industry position ?

Thanks, indeed, I might not focus enough on my skills right now, I'm going to expand this section.

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Do senior PhD students form a part of the admission commitee?
Recently, I was having a chat with a few senior PhD students (almost done with their thesis) and they pointed out that the professors would, at times, solicit their opinion regarding the admission of a particular student. This was especially true when the prospective student belongs to the same country as that of the senior PhD student.
Does this happen in all institutes?
If it does happen, is it official or discussed-over-coffee unofficial?

Sounds rather unofficial.

Just so you realize, you're asking two questions; (1) are they part of the committee, and (2) is their input solicited/considered during the admission process.

Not a particularity helpful answer, but ...
No, it does not happen at all institutes, but it does happen at many. In some cases it is official and in other cases it is unofficial.
As an applicant you should assume that the entire time you are on campus is part of the interview including any drinks/meals.

+1 for "your entire visit is part of the interview". Anyone you meet can give feedback that may affect your acceptance.

I'd add, from personal experience, in case of large (on-going) research projects "unofficially" meeting with senior PhD students can happen and is "part of the interview."

My department's graduate admissions committee officially includes senior graduate students.  And yes, we specifically ask student committee members for opinions about applicants from their home countries.  We sometimes also solicit unofficial "over-coffee" opinions if a student knows an applicant (or letter writer!) personally but isn't on the committee.

In our department, we employ the help of grad students (through the grad student committee) to assist with rating institutions (especially in countries that have seen a proliferation of colleges in recent years). We don't usually let students read things like reference letters, since that gets into confidentiality issues.
If the applicant knows people at the department, then we often get informal feedback.

In my university international grad students help with rating institutions, too, which is very informal (and subjective). But I am unaware of any other form of help from grad students.

Graduate students will almost certainly not be part of the official committee. However, as Daniel stated, they will may be asked about their feelings about a given applicant to the graduate program.
When applying to join a lab, it is very common for individual lab professors to consult the postdocs/graduate students in the lab regarding candidates.

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What are the most important factors when considering graduate school in chemistry?
I'm considering applying to chemistry graduate schools within the next six months and am hoping for some guidance in prioritizing aspects.  For instance, among my list of considerations there exists: renown of school, programs of interest, appeal of location, amount of financial aid, duration of program, and success of past alumni.
Do others who have already gone through this process have suggestions about how to prioritize these considerations?  Were there things in your experience that you thought would be important and ended up not being so.  Or visa-versa, are there aspects that you neglected and wish you hadn't?

[chemistry.se] is more for conceptual questions relating to chemistry, so I've migrated this to [academia.se], where it seems to be on-topic.

As a second year grad student in chemistry, I would say the two most important factors are if there are multiple professors you could see yourself working for and if you feel like you could spend the next 5+ years of your life at the school/in the area. You are going to be stressed out with work, so you want to enjoy where you're living.
I haven't heard of any US chemistry PhD grad programs that don't pay tuition + insurance + stipend, so I don't think that's much of a worry (I would, however, make sure you get all that).
The renown of the school and success of past alumni is actually less important than you'd think. Grad school is all about your research advisor and your research (i.e. a better professor at a lower tier school is better than the other way around).

scientifics is right. Your first priority should be figuring out what flavor(s) of chemistry (analytical, organic, physical, etc.) interests you. Even better would be identifying a specific research area, like photovoltaics, total synthesis, spectroscopy, medicinal chemistry, structural biology, etc. Then, your second priority is finding research advisers working in your field of interest. Pick schools that have more than one person that you could work for.

Prestige.
Prestige is basically the currency of the academic world. And in the current, grimmer-than-death job market for physical sciences, the prestige of the institution you got your PhD from is the main factor in your post-PhD success, or absence thereof.
It might sound stupid, unfair and cynical. Because it is. Your success should be proportional to your own value, hard-work and scientific output. But with 200 to 300 graduates applying for every opening position, even in third-class colleges and small firms, the prestige of your last-attended institution is the actual cut-off. I have personally met people sitting in hiring committees who openly admitted that any application from someone who didn't graduate from a top-10 US programme or EU equivalent (Oxford, Cambridge, ETH) will go straight to the bin without further reading. Only after this first filter has been applied will they actually start going through your publications, letters of recommendation and actual personal achievements.
So yes, all in all, working for an awesome group in a second-class university might trump working in a mediocre group from an Ivy League. But nothing trumps working in an awesome group in a top-tier university, like dozens, not to say hundreds of graduate students do every year. Those same students you will be faced against when applying for jobs and post-docs. So in today's gigantic battle royale of a job market, where less than 10% science graduates actually land a tenure-track position, if you don't have the full package, you don't stand a chance. That's just how it is.
Personally, I simply wouldn't do a PhD in physical sciences again. And if I did, I certainly wouldn't even consider doing one out of the top-10 programmes of your field. It simply isn't worth your time in terms of employability.

These problem true for the whole academia, nothing specific to physical sciences. Actually, the situation is somewhat better in physical sciences.


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On citing books as references in papers

I am an undergraduate student with no experience and no guidance from any teacher whatsoever.

I have done research on Recreational Mathematics (Maths tricks). I have a lot of new tricks to introduce, in a completely new format.
But the problem is that there doesn't seems to been any research paper on this topic before, but there are many books. Can I quote books as a source of reference instead of research papers published? (I am quoting research papers related to the theorems and concepts I have used - but there is no reference to the trick on which I have worked upon.) Also the trick seems to be age-old and extremely simple, and I think that the inventor of it is not known to the world at all.
Can I cite the book? (I know that the author of the book didn't invent the concept behind the trick.)

Yes! If you cite a book, I recommend you to provide a chapter/section number (you can do it in LaTeX with \cite[Section~5.7]{PolyaErdos54}), so that your readers know where to look without having to go through 300+ pages.

You can and you should.
Sure, peer-review papers are preferred over regular (not peer-reviewed books) but if the information you want to cite only is written down in a book (peer reviewed or not) rather than in a research paper, of course you should reference to the former.
Technically, you can refer to whatever you want (a painting, a tv show, a private discussion with a friend of yours or The Journal of Ghosts and Humbug). However, if you want to write a believable paper, you should strive towards citing as trusted and/or high quality sources as possible.


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Why is Paper Publishing and getting a patent completely unrelated?
I know that not all, but many papers (that are accepted and published) are based on ideas that can be patented as well. So, why is that there is no system in which a person can apply for a Publishing a Paper and also getting the Idea patented at the same time?
Is it possible that I publish a paper, do not get it patented (going to patent it at a later point of time - because of not having sufficient money to pay for the patent application or whatever) and during that time someone else patents my work in his name? Or Someone else picks up my theory, develops it further and gets a patent? How would the situation be then? What should I do under such a situation?

not all, but many papers (that are accepted and published) are based on ideas that can be patented as well

This is a misconception.  Even in engineering, only a small minority of papers contain ideas worth patenting.  The further you get from engineering, the smaller this fraction gets.

So, why is that there is no system in which a person can apply for a publishing a paper and also getting the idea patented at the same time?

You can indeed do both things in parallel, but not via the same process.  Publishing a paper is an academic process, whereas filing a patent application is a legal process, and the two things just don't overlap very much.
It's kind of like asking why you can't write your dissertation and apply for jobs via the same process.  You can certainly work on both at the same time, and there is a relationship between them, but fundamentally they aren't the same thing.

Is it possible that I publish a paper, do not get it patented and during that time someone else patents my work in his name?

No, it's not possible.  Well, technically someone could try, but they would be wasting their time, since your paper would serve as "prior art" that would invalidate the patent.

Or someone else picks up my theory, develops it further and gets a patent?

That could happen, whether or not you file for a patent.  They would own the intellectual property rights to the extension of your work, but not to your work itself.

At least in the US, you have one year from the first date of publication to file for a patent.  (Posting on the internet apparently counts as "publication".)  And since when did prior art actually matter?

The one-year grace period applies only to the inventor (the disclosure is prior art for anyone else).  The stupid patent you link to is great, but it's got the opposite difficulty.  The problem is that it's hard to find prior art, because there aren't many recorded descriptions of swinging techniques.  That patent is pretty frivolous and would presumably be thrown out if it made it to court, on the grounds that it was obvious to anyone skilled in the art of swinging.  :-)  I imagine it got approved because it really didn't matter, and approving it was easier than trying to dig up prior art.

Patents are about owning and protecting an idea to make money from it in the future.
Papers are about advancing science. A paper lays claim to an idea, but anyone else is allowed to build on that idea without having to pay royalties.

I voted up for the first two paragraphs, but the third paragraph has some mistakes: http://en.wikipedia.org/wiki/Method_(patent)

I removed the last paragraph.

To add to Dave Clarke's answer: Even in the rare cases when a publishable result is also patentable, filing for a patent is a serious amount of work.  Filing a patent application costs thousands of dollars; submitting a paper is free.  Filling a successful patent requires very specific and formal language, which is not the same as the specific and formal language used to communicate with other researchers.  Why should we spend all that extra effort for so little reward?
Or Someone else picks up my theory, develops it further and gets a patent? How would the situation be then? What should I do under such a situation?
You should cite the patent in your tenure case as iron-clad evidence that your research has real-world impact.  Congratulations!


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How do grad schools design courses for potential students?
For graduate schools in STEM which take students with a Bachelor's degree (BS/BE/BTech etc.) and graduate them with an MS and/or PhD, what do they assume about the student's prerequisite knowledge when designing courses, their difficulty and overall dynamics?
My question is pointed towards knowledge and not what is on the student's transcripts. For instance, a student might have "Linear Algebra" on his transcripts but might not remember Singular Value Decomposition very well.
Do they:

assume that student knows all the courses he has taken very well and build from there? (What happens to people who had a BS in allied fields? For instance a student may have a BS in Pure Math but enrol for a MS/PhD in Computational Math with little idea about a "compiler".)
assume he knows nothing and take him to proficiency?
anything in between?

This really varies among different universities, even in the same field.  The assumptions for computer science PhD students at MIT or Berkeley are very different from the assumptions for computer science PhD students at the University of Southern North Dakota at Hoople.

@JeffE, in that case, I'll direct my question to the Top 3 of the field. I assume there won't be much deviation within them?

I agree with @aeismail's and @dmckee's answers, but let me add a different spin:
Students are admitted to strong PhD programs not on the basis of how much or what they know, but rather on their potential for successful research.  Every program admits students from all over the world, who may or may not have undergraduate degrees in exactly the same field.  Beyond a few fundamental concepts, it is not reasonable to assume that incoming graduate students have any specific prior knowledge.
That said, most courses for PhD students are generally taught as if the students have a strong undergraduate background in the same field.  The definition of "strong undergraduate background" depends strongly on the graduate program; the expectations at the top PhD programs are generally extremely high.  More importantly, PhD students in top departments are expected to have the intellectual maturity to recognize and correct weaknesses in their background, even if the missing material is not normally covered in a strong undergraduate program.
Also: Courses are arguably the least important part of any PhD program.

+1 Students are expected in my department to bone up on what they need - including reading up on or retaking classes that they've technically "done" but will be needing to use heavily.

In my experience in chemistry, a lot of PhD programs give first year students exams to check their knowledge and see where they might need remedial classes.

They are coming to grad school, not some hand-holding, jolly the kids along, summer program.
It is assumed that if students arrive with a deficiency they will take the necessary remedial classes (often that means getting in with the upper-division undergrads) and if you have forgotten something they will do the necessary boning up.
They will be boning up on things from time to time for the rest of their lives, after all. Might as well get in some practice in school.

In my experience it is not uncommon for first year grad classes to share a syllabus (and lectures) with an undergraduate class. The grad class often has some additional assignments (e.g., an extra or longer paper or additional problems in homework assignments).

This is a very difficult question to answer, since there are so many disciplines and so many programs at so many universities, all of which have their own rules.
But in general, the standard master's program (or coursework phase of a doctoral program) is designed to take someone who has a bachelor's degree in the same field and bring them up to a level of competence sufficient to pursue graduate-level research in that field. Very little allowances, if any, are made for people coming from other disciplines and programs, and they're certainly not designed for someone starting with no knowledge whatsoever.


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Important qualities for Research Assistanship applicants
1.What qualities do professors (assume STEM if necessary) look for in students (in their MS) when recruiting them for an RA (Research Assistantship)?

2.Consider the follow scenario:

Person 1 : Has excellent fundamentals, thirst for knowledge and good grasping power but no exposure to the academic research per se.
Person 2 : Above average fundamentals (few random holes here and there. Not too minor, not too major either) and prior exposure to academic research.

Who would a professor rather pick?

The motivation of the question is to find out how a potential grad student can prepare himself to increase the likelihood of getting an RA with a professor he/she likes.

Here is a similar question from the recruiters perspective, although aimed at advanced undergraduates, they don't seem that different from the MS students you describe.

From your question, I'm guessing that you're coming from the American system (or similar) in which it is common for students intending to pursue the Ph.D. to start in the MS program at the same school.  While they are completing MS requirements, they try to find an advisor and a research project for the Ph.D.
I think that often the advisor will expect the student to start a project and demonstrate some competence and progress while the student is still TA'ing and taking classes.  That way the advisor can measure the student's capability before committing to funding.  I typically give a student some material to read and ask them to implement and test some existing algorithm, in order to help the student get started and to test the student's capability for research.  The most important thing is the ability to rapidly understand new ideas well enough to implement them from scratch.
Coursework is not sufficient to get you a research position, but it can certainly disqualify you.  Usually prospective RAs have taken or are taking my (graduate-level) class, and I won't consider taking them on unless they get at least an A-.

When you give the interested student material to read, is it necessarily your work or just some basic/intermediate work in the same field?

Usually it's not mine, but sometimes it is (now that I've co-authored the only monograph in the main subfield I work on).  Depends on what I plan to have the student do, and how prepared he/she is.

There are two issues here: what do advisors want, and what can they observe?  In principle, I think most would choose Person 1 over Person 2, if they knew all the facts.  However, this choice will not typically arise in practice, because they won't be able to verify the "thirst for knowledge and good grasping power".
Classroom performance, no matter how excellent, is generally not sufficient for admission to a strong graduate school.  The problem is that lots of students get excellent grades, and these grades are only loosely correlated with research ability.  The only way to stand out is to do something that's fundamentally more impressive than doing well in courses; this could be a research project, or a substantial exposition, or coding, or any number of other things.  Research has some advantages, since that's what you're aiming to do in grad school, but it's by no means necessary.  However, if you don't do any research, then you'd better have some other way of demonstrating your talent.
For example, if Person 1 has done no research, but wrote a beautiful, 80-page undergraduate thesis giving an exposition of forcing and the continuum hypothesis, then that might count for more than most undergraduate research projects.  On the other hand, if Person 1 can point to nothing concrete except course grades, then that will likely be a problem.

In what way is "a beautiful, 80-page undergraduate thesis giving an exposition of forcing and the continuum hypothesis" not research?

I'd reserve the term research for original work that might be publishable in a research journal.  That's not to say it wouldn't be impressive, though.


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Publishing paper using an open-source software
Can someone publish a paper on the mathematical model of the open source software i.e. parameter analysis of the model (i.e relevant to specific area) which is the backbone of the software.

Is it possible to publish paper on model analysis?
Apart from citing the software, what kind of credit I should include in the paper?

This seems to depend a lot on the particular area of study, and isn't a generic 'academia' question

Yes, it is possible to publish a paper on model analysis. I would treat this like any other situation in which I wanted to use someone else's "data". Generally, it would be the owner of the data (in this case the developer of the model) that would do the analysis. In the absence of an analysis that I need (or want, or think would be interesting), I would contact the owner(s)/developer(s) and see if they have plans for a similar analysis or would like to collaborate.

Yes, https://conference.scipy.org/proceedings/ annually publishes papers related to scientific softwares.

This seems like (Sci) python specific, correct me if I am wrong. I think the question aims at a more general model, possibly language independent.


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What are "getcited.org" and "getcited.com" for?
What is the purpose of getcited.org and getcited.com?
I am completely new to researching and paper publishing and I frequently get results from this site in Google Scholar Search.
I am absolutely not able to make out what this site does, it just gives the name of the paper, the author and the publication. It does not give a link to download them, neither does it give any abstract.
There are these link buttons which ask you to enter a username and password each time. Even after creating an account and logging in, I am asked to log in each time I click on a link button and the link button just opens my profile on the website.

I'm not going to make this a formal answer, since I know nothing about these sites except for what I've found by spending a few minutes looking.  However, I think they serve no useful purpose: poorly designed web pages, slow databases, not that much content.  GetCITED started in 2001 and may not have been maintained in recent years.  It was probably a good idea in 2001, but it didn't catch on and is no longer useful.  I'm surprised you frequently get search results from there, since I never do, but it probably depends on the topics.

(See http://en.wikipedia.org/wiki/Talk:GetCITED, http://www.archives.expressnews.ualberta.ca/article/2001/07/222.html.)

From the wikipedia page:

GetCITED is a website database that lists publication and citation information on academic articles whose information is entered by members. It aims to include not only journal articles but also book chapters and other publications, both peer-reviewed and non-reviewed. The objective is to make this information publicly available, as such information is presently held in restricted databases. It indexes over 3,000,000 publications from over 300,000 authors.

However, on the actual getcited.org webpage, we see the copyright dates 2000-2006, which makes me think the page has not been maintained in 5 or 6 years.  So basically, it seems to be a dead website, that is still available mainly because domain registration is so cheap.


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Etiquette for establishing contact with multiple professors for PhD guidance
Assuming that one has begun a PhD program at a university which allows a student to choose an advisor after his/her first year in the program and the student has narrowed down to a couple of potential advisors, what is the etiquette for approaching them for advisor commitment? Specifically,

How do you get across that you aren't sure yet about him being your advisor  and are fishing around before you decide without sounding rude?

Do ask multiple questions, @Nunoxic... We are short on numbers here: http://area51.stackexchange.com/proposals/16617/academia :)

related (but exactly the same): http://academia.stackexchange.com/q/252/79

@shan23. I beg to differ (In fact, my answer is the accepted one :|). That question is regarding how to get in touch for the first time. The assumption is that the 2 of you don't know each other at all. This question is more to do with you knowing the professor, he knowing you and now you are about to make the first step to go meet him in his Office to discuss the possibility of he advising you for your thesis.

@Bravo. Are you sure? How do I split?

@Nunoxic they seem like related, but different question. What happens if I answer the questions and someone likes my answer to 1, but hates my answer to 2.

@Nunoxic - That was a typo on my part - I meant not exactly the same. I thought that anyone looking at this question for the first time might also benefit from the answer you gave to that question, as that would be applicable before this question came into the picture!

@Nunoxic There are five good questions here with related but not intertwined answers. To split the question, edit this one so that you are only asking the first part, which matches the question title. Then create new questions for the second, third, fourth, and fifth parts of your question.

@Bravo, Daniel, Ben. I've made 5 different questions.

Advisors understand that students are fishing around - that's normal. What's less common (but possible) is students fishing around in wildly different areas. Just keep in mind that either an advisor is going to see something in you that will motivate them to try and convince you to work with them, or they'll view you as someone who needs to make up their mind first, and won't spend too much time thinking about you.
In that respect, matching up with an advisor is more like dating than an job interview, at least in the US :)

Ben Norris is basically correct—as advisors, we know that students have multiple options for choosing advisors, particularly in setups where the students pick advisors after their arrival. Therefore, we won't (or at least shouldn't) take it personally when someone tells us they're considering other advisors.
In general, however, you should also remember that these processes are often double-sided: in case of competition, the advisors also have a choice in who they want as their top choice. If you are overly hesitant, the advisor may choose another candidate who is more certain as a top choice. So, you should be honest if you're not immediately "sold" on working for a particular advisor, but don't be so negative as to make the advisor look elsewhere.

The adviser already knows that you are not 100% committed to his or her research group. If the system lets students choose, then the advisers know the rules. The faculty in that department very likely made those rules. They know that you are considering other advisers.
It is never rude to consider other advisers. Some programs even require it. My graduate program required that I interview at least three potential advisers and rank them. The faculty would then be informed of the list of students who chose each faculty member at the first choice only. Faculty had to fill open positions from this first.
This system was implemented by the faculty a few years previous, when they became dissatisfied with an approach that left matching entirely up to students and faculty meeting and getting along. Faculty would then choose their favorites from all interested students. Usually, this meant that a small number of students were not chosen at all, which was not acceptable.

+1 for "interview at least three potential advisers and rank them".  -1 for "informed ... first choice only".  Gale-Shapley, anyone?


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Etiquette for discussing authorships, expectations, etc. with potential advisors
Assuming that one has begun a PhD program at a university which allows a student to choose an advisor after his/her first year in the program and the student has narrowed down on a couple of potential advisors, once a student has established contact with his potential advisor,
At what point do you discuss things like authorship, expectations and other such details? First meet, second meet or otherwise?

Think of your first meeting as a mutual interview. The adviser wants to know if you are worth spending time and money on. You need to ask questions to determine if the adviser is worth spending years of your life on.
Good topics for the first meeting are:

The adviser's expectations of you (including hours worked per week, number of papers to publish before graduating, etc.)
Your expectations of your adviser (Do you get vacation time? How long does it take your students to graduate? How much grant funding do you have? What earns me authorship on papers? etc.)
Your potential project (What are the long-term goals? What are the short-term objectives? Have other students worked on this project before?)
What preparations do you need to make? (Should you take a particular course? Should you read certain papers or books?)

You should also try to get candid answers to these questions from the students who already work for your potential adviser.

Also: Do you mind if I talk with some of your students?  Do any of your students also work with other faculty?  What do your students do over the summer?  Where do your former students work now?  What earns you authorship on papers?

Before the first meeting it is worthwhile to review the PI's publication history. You can get an idea of how often they publish with students.

This question should be asked during your interview, in a polite but directed manner. The question is simply, "How do you assign authorship on papers co-authored with graduate students?" The answer should address who is on the paper, the ordering of the authorship, and what criterion is used to determine who is a co-author.
You should also definitely ask his current graduate students the exact same question, preferably students who have gone through the process of writing a paper with the advisor at least once, just to make sure that what the advisor says is mostly mirrored in what the students report.

+1 for "talk to the students". I routinely tell new students to chat with my students to get the scoop on me :)

I assume the main issue for you is the question of authorship.
In academia, there are rules about authorship, and they differ between the disciplines slightly. You can expect that supervisor will request to be last author of every paper. I very much advise against trying to bargain around that, this is the only scientifically relevant reward your supervisor will get for supervising you. And since professors talk to each other, you may unknowingly ruin your reputation by trying to negotiate on this.
There are details about authorship of other people who might give you research ideas (like other grad students in the lab, outside collaborators) but I don't believe the general answer can be provided to you by the professor in a way that would protect you in the future if he will be trying to coax you into accepting another author that you don't see as having significant contribution to the paper. Usually can people agree on everything in general, but then it is always down to the question, what amount of work does qualify someone to demand co-authorship.
For example, a person who does some simple physical/electrical measurements is certainly not justified to be a co-author, but if someone performs experiment for you that requires trial of toxicity of the chemical substance including 50 live mice, and you depend on his knowledge and experiences to actually design the experiment properly, then such person will probably be expected to be co-author, especially if you only took the results and ran them through SPSS.


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Etiquette for an initial meeting with a prospective advisor: What is expected of the student?
Assuming that one has begun a PhD program at a university which allows a student to choose an advisor after his/her first year in the program and the student has narrowed down on a couple of potential advisors and has taken AT LEAST one course with each of his potential advisors:
What should the student take to the first meet? A rough research idea, summary
of the professor's important papers or otherwise?
Note: I have made 5 separate questions as suggested (As opposed to the single one which I started off with)

For an initial meeting, I would take an general idea of the kinds of things you'd like to do in a particular advisor's group, a sense of what the advisor works on—you don't necessarily have to have read papers in advance, and an open mind. That said, you'll make a better impression if you have read a few recent papers, and have a few intelligent questions to ask. It will pique the advisor's interest, and show your sincerity.
As for coming in with ideas for a research project, this depends a lot on how the project will be funded. If you have your own fellowship and can design your own project, then you should definitely have an idea or two ready. On the other hand, if you will be funded by an existing research grant within the group, your choice of projects will be more constrained.
One final note—I would say that it's definitely not required that you take a course with an advisor before considering them as a research advisor. If that were the case, I would have been on a completely different career path than the one I'm on now, as I'd be working for my fourth-choice advisor (who was the only one I took a class with before choosing a research group!).

Basically you bring the same things to an interview with a potential Ph.D. adviser as you would to a job interview: 1) evidence you've done your homework about the group, 2) knowledge about the project(s) you might be working on, 3) an open mind and positive attitude, and 4) a description of how you will fit in with that adviser's current group and projects.

@BenNorris: (1) and (3) are far more important. You might not know what (2) and (4) are before the meeting! (That was partially the case for me.)


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Appropriate time for contacting professors after PhD enrolment
Assume that one has begun a PhD program at a university which allows a student to choose an advisor after his/her first year in the program and the student has narrowed down to a couple of potential advisors and has taken at least one course with each of them.

How early should one contact potential advisors? Just at the beginning of the program (early bird?) or after 6 months (after taking courses with each one) or right when you need to choose (don't make decisions till you have to?)?

Related: http://academia.stackexchange.com/questions/924/contacting-professors-for-phd-vacancies

As soon as is reasonable.
Think of it this way. There are possibly other students competing to work with this advisor, and they might have funds only for one new student. There might be other considerations - maybe the advisor is going on sabbatical, or isn't taking new students, and so on.
So as soon as possible, set up a meeting with the potential advisor, indicate your potential interest (and why!), and ask them point blank if they're taking students, and how they prefer to screen students (some do it through course work, some might like to work with you on an independent study project, and so on).
Depending on what they say, you can take further followup action - maybe they want you to take their class, and maybe they have a research group that has open meetings, and so on.
Bottom line: don't wait till your year is over, because it is quite possible that someone has made a decision already.
Caveat: if this year-delay is institutionalized, it might be that all advisors wait till the end of the year to even think about new students. I suspect this is unlikely, but in any case that first meeting will help clarify it.

It is never too early to contact potential advisers. If you have your heart set on a particular individual, you may even want to initiate contact before you start (and perhaps as early as right after you have applied). Early contacts usually come from motivated students.
Waiting a while gives you the ability to get to know the adviser and helps them get to know you. This could be beneficial, but you should not wait too long. Last minute contacts suggest a variety of behaviors that are unappealing in a graduate student, like laziness, forgetfulness, lack of dedication, etc.
If the program lets graduate students choose, then the advisers are well aware of how the system works. A short email requesting more information and an in person interview would be appropriate. If you know the adviser better (you have already taken a class with this person), then an informal stop by their office would be fine, too. Most advisers expect this kind of contact in this situation. An initial contact by email does not have to be anything more than:

Dear Professor {X}, I am (or will be soon) a graduate student in your department. I am interested in joining your research group. I have read about your work in the field of {Y} and found it interesting, and I want to learn more. Are you available to meet with me at one of the following times {suggest three specific times}? Thanks.


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Procedure for having multiple advisors for a PhD thesis
Assume that one has begun a PhD program at a university which allows a student to choose an advisor after his/her first year in the program and the student has narrowed down to a couple of potential advisors and that he/she wishes to do his thesis under multiple advisors.
What is the procedure for suggesting multiple advisors? Whom do you speak to first? What if the advisors don't go so well together?

Unfortunately, I think the answer is going to be - "it depends"! I have advised several graduate students, including 2 who are currently "co-advised" with other faculty. In one case, the student approached me first, but I already had a working relationship (co-teaching, serving on supervisory committees together etc.) with the other faculty member. In the other case, I was approached 2nd, and again, I had previously worked with the other faculty and knew them well. When talking with potential students, if it is clear that a different faculty member would have better or equally relevant expertise, I always suggest talking to them. However, that's my opinion. I also have colleagues who would flat out refuse to consider co-advising a student.
I think the key "test" is to imagine your dissertation work as a 3 way collaboration - what does each faculty member bring to the table? What are they going to get out of it? How you approach them is completely dependent on the individual personalities involved. You should also find out if there are any policies on co-advising at your institution or department - there are none at mine, but that might not be universal.


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On citing "common knowledge" statements
I am an undergraduate student with no experience at all.
If you are writing a research paper, that you are planning to submit for publication, how do you state that a particular statement/fact is commonly known to all and it's dicoverer/inventor is not known, and that there exists no research paper that introduced the concept but there do exist various books/websites that discuss the very concepts and a few derivatives of the concept?
I am not able to cite a particular statement/bunch of statements as I am not able to find any research paper related to that at all.
The particular thing in question is: https://math.stackexchange.com/questions/161661/source-of-probably-the-most-simplest-math-trick
What should I do?

You don't need to cite common knowledge. If you think it is common knowledge, wait for the reviewer to ask for a reference, before going to all the trouble of sorting out the history. Often if a reviewer doesn't think the statement is common knowledge he/she will suggest a reference or provide a conflicting reference.

If you found a book that discusses the statement, and you really can't track down where it comes from (does the book cite anything?), then citing the book (with chapter/page numbers) is acceptable.

Track down as far as you can, until you get to the first study in the field ever.

If the statement/concept/idea that you think you want to cite is covered (without citation) in the introductory undergraduate textbook(s) on the subject, then it is likely common knowledge in the field. This is particularly true in science and social science. For example, you should not need to cite things like the following:

Fluorine atoms are more electronegative than carbon atoms.
Asexual reproduction of eukaryotic cells usually occurs by mitosis.
Force is a vector quantity, while power is a scalar quantity.
The free market equilibrium price for a commodity is found at the intersection of the supply and demand curves for that commodity.
Sigmund Freud is commonly considered the founder of psychoanalysis.

Thanks for that but it's in the topic of recreational mathematics - and there is no undergraduate course for that at all.

Check out the references and links at the bottom of the wikipedia page on recreational mathematics. Look for the good references that those references reference.

I will answer your question in a somewhat sideways manner: I think anyone who is trying to write a math research paper who does not already have a PhD in mathematics or a closely related field should do so in close consultation with an adviser who does have such a PhD.  It is not practical to learn all the standards of journals and publishing on your own.  Virtually every undergraduate is also in the following situation: their own perspective on mathematics is not yet mature, and thus if they submit a paper to a research journal they will be submitting their work to an audience with far more experience and background knowledge.  (A few truly brilliant undergraduates do work up to the standards of other mature, professional mathematicians, but even for them the work is probably much less than what they will be capable of later on.  I can't think of a single instance of a professional mathematician whose undergraduate work was in the same league as their later research.)
In fact -- and I don't mean to be discouraging by saying this -- most research done by undergraduates is not of a publishable quality.  This does not mean that undergraduates should not engage in research -- I think it is very valuable and enjoyable for them to do so (if anyone cares: I did research as an undergraduate, had a blast doing it, and did not try to publish it) -- but only that formal publication should probably be a goal for later in one's career than that, especially nowadays when it is so easy to put your work on the web.
In particular, you write

a particular statement/fact is commonly known to all and its discoverer/inventor is not known, and that there exists no research paper that introduced the concept but there do exist various books/websites that discuss the very concepts and a few derivatives of the concept?

Honestly, this sounds unlikely to me.  The vast majority of mathematical topics that are discussed in books are also discussed in research papers (maybe it is somewhat different in recreational mathematics, but even there I imagine it's still mostly true).  How do you know that no research paper treats the concept in question?  Searching the mathematical literature is itself a skill that takes both general experience and specific expertise in the subfield you're searching in: as a research mathematician it is common enough for me to come across a mathematical concept, try to find it in the literature, and only find it several weeks or months later when I have become more familiar with the local terminology and standard results.  Similarly, most mathematical concepts are traceable to a specific discoverer/inventor, although admittedly the generic mathematician does not feel as honorbound to track down primary sources as academics in most other fields (in my opinion this is a rare "character flaw" of the generic mathematician!).
Even if you have done all the mathematical research yourself, consulting an experienced adviser on how to write up and submit your work could save you a lot of time and headache.  Some journals/editors/referees are relatively supportive of authors writing their first paper (everyone who has published a paper was in that situation at one time!), but if your paper is, or looks to be, below the level of papers they want to publish, it will probably get bounced back to you with little constructive criticism.  An adviser can be much more kind...

I have always felt the need of an advisor, I was in school when the idea first stuck me, I contacted my teachers in school but none of them showed any interest in helping me. Now I am in a university with Computer as my major - taking your advise I will contact math teachers in my university soon. The particular fact that I want to site is this extremely simple math Trick: http://math.stackexchange.com/questions/161661/source-of-probably-the-most-simplest-math-trick Do you have an Idea of it's source?

One helpful approach can be to ask a few people that you think of as familiar with the field (in your case, this could be professors or certain other students).  Even though most schools offer no class in recreational mathematics, you can ask 3 or 4 people who you see as at least "intermediate" level (whatever that means) in recreational math.  If all of them immediately recall the information in question, then it's a good bet that you don't have to cite it.  Alternatively, if most of them don't know it, then perhaps you should cite it.
Specifically for recreational math, I recommend that you ask around on the website Art of Problem Solving: http://www.artofproblemsolving.com/ (from the front page, click on "online community").  This site is focused on contest mathematics, which is distinct from recreational math, but a close cousin (and many people interested in one are also interested in the other).

I did post it as a question: http://math.stackexchange.com/questions/161661/source-of-probably-the-most-simplest-math-trick before asking this question.

@myselfpoddar Yes, that post of yours looks fine.  For this particular fact, I agree with the answers you received: you really don't need to cite anything.  I say that because your trick is an immediate consequence of algebra (by that I mean "college algebra" that is taught in grade 9 or earlier).


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What category does a collection of mathematics tricks come under?
I am currently researching on Recreational Mathematics - Mathematical Tricks and I have come across this article:
Ten Amazing Mathematical Tricks
.
This article is of free style and does not follows the pattern/format I have seen in all the other articles/research paper so far.
My question is: Is this a research paper? Is it a paper? If not, what is it? Can such a paper be cited as a reference? I want to know more about these freestyle articles.

A number of people have correctly told you (I can see the article) that this is a collection of mathematical puzzles, rather than a research paper.  So let me tell you a little more about this type of article.  Martin Gardner is considered by many to be the biggest popularizer of recreational mathematics in the 20th century (perhaps ever).  For 25 years (1956-1981), he wrote a column for Scientific American called Mathematical Games.  Many of his columns have been collected into books (such as Martin Gardner's Mathematical Games).  Some other well-known examples of similar writing include that of Douglas Hofstadter, Ian Stewart, and Keith Devlin.

I am so glad this question was asked, because I hadn't heard of Ian Stewart and Keith Devlin. I'm looking forward to reading their works.

I can't see the actual article, but if it's an MAA article by Martin Gardner, it's probably a list of interesting puzzles. Peter Winkler also writes a column of this kind.
These might not be "research papers" in the sense of proving new theorems on a topic, but they're interesting communications. You can of course cite such a paper as a reference if you use some material from it (and you should!)

This is the description of the Journal:

Math Horizons is intended primarily for undergraduates interested in
   mathematics. Thus, while we especially value and desire to publish
   high-quality exposition of beautiful mathematics, we also wish to
   publish lively articles about the culture of mathematics. We interpret
   this quite broadly—we welcome stories of mathematical people, the
   history of an idea or circle of ideas, applications, fiction, folklore,
   traditions, institutions, humor, puzzles, games, book reviews,
   student math club activities, and career opportunities and advice.

So, it don't contain research papers.

As the others have stated, what you refer to is an article in a magazine, and not a research paper in a journal.  If your goal is to show that you came upon an idea while reading this article, then you should cite the article.  (Research papers do sometimes include magazine articles in their bibliographies.)
However, research papers usually cite sources where ideas were first published, and it is rare for original ideas to be first published in magazines as articles.  So there is a good chance that the ideas you found in the article were published earlier.  I recommend that you search for earlier sources and cite them also.
I also recommend that you read a few issues of the three publications I mentioned in this answer to see how they cite work in recreational mathematics.  (In my opinion, the most prestigious place to publish work on recreational mathematics is the American Mathematical Monthly.)

And just in case you didn't know, Martin Gardner (http://en.wikipedia.org/wiki/Martin_gardner) is considered a giant in the field of popularizing recreational mathematics.

And if you really didn't know about Martin Gardner, I envy you.  You get to read all his columns for the first time!!

@JeffE I really didn't knew, I have learned it only recently.


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How does one maintain academic contacts?
How do you stay in touch with researchers whom you meet at conferences? My intention is to stay in touch for possible collaborations and job opportunities.
Usually I meet people, we talk and I send them a "thank you" mail but later I get confused how to take this forward and maintain this contact.
I do send links, my preprints and interesting website questions and stuff once in a while but more often than not, I don't get a reply.

Related question: http://academia.stackexchange.com/q/271/102

It's worth noting that this skill is not specific to Academia. Any discussion on how to network successfully will contain tips and ideas that will be directly applicable to your question.

@eykanal I don't agree. Networking in other areas is following LinkedIn, the concept of following the other persons work is nonexistent.

@Inquest - I strongly disagree, but this is slightly off-topic... lets take this to chat for any further discussion.

I don't think my rep permits chat. I agree with you about it being a skill but it's different on a lot of levels as well. Notably, the best I can ask a buddy working at google is what he works on and some more. With a friend at utexas I can read his paper and discuss his work for hours. With the google friend, I was run into a wall sooner.

academia is a little different in that you have to almost hide your networking efforts otherwise it starts seeming pushy. Not saying that this is fair or unfair, but it is the culture

To build and maintain a network of professional relationships, one should develop, maintain, and express genuine interest in other people and their work. Introducing oneself at conferences is a good start, as is recording and updating contact information, and setting up a system to remind yourself to reach out to people regularly. (There are various tools for this, ranging from a spreadsheet, to social network services, to CRM software. If you use gmail, you may want to try "contactually" for the mechanical part. )
Sending your contacts updates on your work is useful. And these should be targeted... it's ok to use a broadcast medium like twitter to announce new work to the world, but avoid sending out mass-email -- instead target those people who are likely to be most interested, and include a brief personal note that puts the new work in the context of their interests.
Most important, keep track of what others in your network are doing -- new publications, new projects, working papers, promotions, etc. (Social networks can be useful for this in some fields for tracking people directly, tracking working papers in your discipline, and disciplinary news can be quite useful, google scholar can also help alert you to new publications from people in your network. ) And when someone in your network is starting something of interest, re-connect -- with congratulations, a useful comment, and possibly some information on your own work (if it really is of interest with respect to their new project).

Maybe this is obvious, but never forget to wrap your networking cover ops in real social ops. In a conference, go talk to people, than go drink a coffee, beer, discuss with them about anything but work at the lunch, etc.
If you do that, it is really simpler to send a thank you mail and to reactivate a contact afterwards.
Regarding the action of sending your work around, why not, but you must target carefully who you are contacting. Be sure that your work is really of interest for the person, this is what will make the difference between a spammer and an interesting contact.
Another way to ping is to ask genuine questions on new papers authored by your contacts.

Can you expand on reactivating contacts?

@Inquest, "Hi, I'm in town, want to grab coffee?"


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As a student, how much time should I spend helping other students with research?
Assume that one has a skill (assume, programming) and is one of the very few in the department to have it. The other students need a bit of this skill for their projects and you help out when you can.
Where do you set the limit for how much effort/time you spend on helping others this way?
Pros:

You get insight into other students' work.
It's a welcome break from our own research
Maybe earns you a second/third author for setting up the experiment
Networking !

Cons:

Effort/Time spent
You don't necessarily improve your skills (For instance, coding in Python for someone else doesn't augment my Python skills by much. What I do might be really routine)
You tend to have a soft commitment towards that project. For instance, if I start working on it as a favour, it doesn't really come off well if I leave 'em midway.

EDIT: I do enjoy the work so long as it is at least a little challenging. I often get really n00bish questions and that is when I start reconsidering my stance on helping people.

another pro I would add (similar to your first): you can learn a lot of new and fun things about sub-fields you are not familiar with.

You forget one consideration: do you enjoy using that skill? To take your own example, I enjoy programming, and even though I might not learn or get anything technical from it, I enjoy solving someone's else problem with some lines of code :) Of course, one should be careful of the time spent and the reward you can get from it, but doing something you enjoy is very different than doing something you can, but don't particularly enjoy doing.
For instance, I can probably install a server and manage a website (with a CMS) for someone, but I don't enjoy it, so I would be very explicit about the benefit (I do that for you if you do that for me). But I could do some cool coding on my free time.
On a completely different aspect, it also depends if you intend to stay in academia: it's a very small world, and the person you're doing a favor to today might be the one sitting on your tenure committee in 20 years (if you work in the same department and all stay in academia, it's quite likely that your paths will cross again at some point in the future).

+1 for "small world".  I'm routinely surprised by how often I keep running into the same people (both at conferences, and as I branch out into new areas of research).

It depends :)
I would invest effort proportionally to the potential benefit for me (in terms of number/quality of publications) as well as the fun I would have doing it. Where the limit lies depends only on you and I can't really give you any advice on it. You should also consider other factors, such as how your own research is progressing. If you have an important deadline coming up, you shouldn't be doing something unrelated, even if it might lead to a good publication or be a lot of fun.
In any case, you should be very upfront about the level of commitment. Leaving them hanging half way is IMHO a bad idea. Not only in terms of you not being able to participate in any successes after that, but also in terms of letting someone down who asked you to help them. If you don't particularly like the project or don't want to invest any effort in it, tell them.
That said, I've been involved with a few such projects and always found it very rewarding.

Usually when I help other students I put a large time investment making sure that they are capable of caring on the rest of the project without my help. I see my job is to help them get the ball rolling while maintaining the illusion that I'm the local "expert". An analogy would be dressing yourself up as a consultant rather than a mercenary.
Coding is a particularly good example of this. While you can just code away and have it do the job, claim victory, and get endless praise and thanks, I think that its worth writing quality well commented code so that your peer can take it and do their own thing mostly without your help. At the sacrifice of a lot of initial effort, you get some valuable practice time with teaching and writing interesting projects and they won't feel totally abandoned when you have to do something else.
An additional plus is that you can establish yourself at a person only to inquire when there is a worthwhile need for you skill and that should filter out the "hey dude, can you code this up real quick".

One idea: If you could ask the student if it's okay to record your sessions helping them (sort of like what Salman Khan did with Khan Academy), and then maybe save your sessions, then maybe you can then re-run those sessions for later students (so that you won't have to repeat yourself as much, and so that you can always show yourself when you're explaining the concept at your best). That way, you can do that and spend your time giving more personalized instruction to who-ever you're helping.

I'm not sure I see the benefit here.  I think it's relatively rare for different people to ask for precisely the same help.  For example, most people asking for programming help aren't as interested in learning to program themselves (where Khan-Academy-style videos might help) as in having someone else write their specific program.

Hmm - those are good points. The OP did mention that he gets a lot of n00bish questions, so I think those are the questions that are most likely to cause fatigue, and also possibly the questions with answers that are most likely to be reusable.


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Preparing oneself for jobs after PhD
I'm a student doing PhD in North America. I wish to keep open the possibility of settling down in Europe, Australia or Asia. By settling down I mean getting a tenured professorship in good universities at any of the above mentioned places.
I'm just starting out with my research and I'd like to know if I can do anything right now, i.e. during my PhD to boost my chances.

You should familiarize yourself with recruiting procedures in Europe/Asia/Australia and how they might be different from the US. while its early now, you should (when your research direction starts crystallizing) familiarize yourself with research in your area being done in institutions of interest, and identify people at conferences to chat with.
The main disadvantage you could face is a language barrier, local competition, and general lack of awareness because of distance. But except for the first, the others are less and less of a problem.
But at least for the first few years of your Phd just focus on finding interesting research problems to work on and do work that people will pay attention to. Thats much more important  :)


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Learning/Reviewing New Material Quickly
My question is : What is a time effective way to learn new material/ reviewing old material you don't completely remember?
Motivation:
For my research in Computer Science, I need lots of elements of hypothesis testing that I learnt in Statistics 101 (Roughly 10 years ago). I haven't taken a single stats class ever since. I vaguely remember the terms involved in the subject and am somewhat conversant with the basics of statistics (so I don't need to retake the course). But all in all, it's as good as a blank slate right now.
I have read How to efficiently read mathematically and theoretically dense books in STEM fields? but that targets a specific topic that the OP wants to learn, not an entire book. The top 3 answers talk about reading only those sections which you are interesting in. I, on the other hand, need to understand the whole of hypothesis testing which by itself is a book.
My options are (As far as I can see):

Take out the book I used / "Best" book of the subject (i.e. the one best suited for my background) and read it from page 1 till I am comfortable. Too time consuming
Read the chapter I need and then go back to the terms I don't understand. Time spent in confusion and direction-less reading exceeds the time actually spent adding value.
Take course again/Read cover to cover. God No.
Any other?

EDIT : I wish to add that I can sacrifice depth for familiarity and savings in time. (Is this a good idea but? Assuming that it's not a central part of my research)

Note that (1) and (3) follow a likely incorrect assumption that what the book/course deems important is what you need to know. Once a course is over, most books/courses are best used as reference material for you to look up material on a need-to-know basis.

I disagree with the characterization "Time spent in confusion and direction-less reading exceeds the time actually spent adding value".  First, the process you're describing isn't directionless at all; you're explicitly moving in the direction of greatest ignorance!  Second, confusion is valuable.  It's uncomfortable as hell, sure, but that's precisely what makes it valuable.  When you finally come out of the fog, you own the ideas you were confused about.  Failure is the best teacher.

Also, why do you prefer learning fast over learning well?

@JeffE. I would love to study it well but I'm not sure I'd want to spend a month reading something that is only tangentially useful.

@Inquest: And how do you know that something you don't understand is only tangentially useful?

@JeffE I can't be too sure but I feel that it won't be too close to my actual research but then as you are pointing out, I can never be 100% sure.

First of all, you will spend virtually your entire career doing this (trying to remember stuff you forgot), so this is a question that's worth really working on. You'll hopefully receive a good few answers; try them all over the course of a year or two in different situations and choose the one that works.

My personal method is very similar to your (2); review the specific parts of the material that you need to know for that specific situation. In most cases, you'll find this to be sufficient for solving a particular problem. If you don't understand your material, then either Google the topic and try to read someone else's overview, or simply take a step back and go over the more fundamental material, working your way back up to the topic at hand.
Note that this will not work for situations where you need everything, such as interviews. For those, you'll want to read outlines and try to summarize contents for yourself. I've also gone back and looked at homework problems, which (assuming they were well-written assignments) can provide you with a good overview of the material that you need to know about the topic.

after textbooks and google, one or more SE sites can help fill in the gaps for a specific question. I find this often very helpful for more specific academic applications.

Let me answer a different (but related) question.  How can you learn material now so that you'll be better able to recall it years from now?  Here I recommend that you explore to discover your own learning style, which has a couple key advantages.

Using the right style can help you to absorb information more quickly and retain it longer.
Perhaps more importantly, by recreating the techniques you used to learn the material, you can more easily recall it.

A simple example of (2) is that if you learn certain material while listening to classical music, then later you'll be better able to recall it while listening to similar music.  This may sound obvious, but far too few people make use of it.  Let me give a personal example.  Like many people, I retain information far better if I interact with it: take notes, draw diagrams etc.  This process can be time intensive, so you have to find a balance.  What was terrible though, was that I would often lose my notes, so that when trying to recall the information later I'd be stuck.
Only after I was well into grad school did I settle on marking extensively in my books.  I underline, draw arrows, mark key ideas with boxes and stars, etc.  The magic is that when I reread later, my markup helps me to quickly remember what I was thinking when I first processed the concepts.  So whatever epiphanies I had the first time around are more readily accessible.  My point isn't to tell you to mark in books (though I find it invaluable).  My point is to tell you to recall information in a way similar to how you learn information.

This is a very useful point - a preventative approach

You didn't really mention WHY you need the material for your research.
One case is when you have this vague feeling that there's a concept from statistics (in your case, hypothesis testing) that you need for your work. In that case, eykanal's approach and strategy (2) is best. Focus on what you need, and do deep search, taking detours when needed to understand ancillary concepts.
Another case is when you're moving into a new area and expect that you'll need a firmer base in the material before you can continue (to develop the right intuitions and so on). In that case, strategy (2) is still ok, but it helps to do some background work on the side. I find that working through problems is a good way of keeping you focused on learning by doing, and also tests your fundamentals.

I need it because I feel my thesis is drifting in that direction but as of now, it's fairly basic stuff.

Tablets make it a lot easier to go through old materials. If you can find ways to download your textbooks and lecture notes as PDFs and then use a tablet to heavily annotate your textbook (you can even insert extra pages too), you can come up with an easily-searchable annotated version of your textbook that you can re-use again for years to come.
I use a Lenovo ThinkPad because of its large screen size and stylus pen, which makes it very easy to annotate and highlight textbooks and lecture notes. I also put my textbooks in my Dropbox folder so that my annotations are auto-saved in my Dropbox (and in case I ever lose my tablet, I won't lose my textbooks along with them).


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Popular proprietary program or obscure open source substitute for reproducible research?
For a long time, I had been using open source software for my work to make it reproducible. I believed that if I made my code open source and the software to run the code was also open source (or at least free), my research would be reproducible. However, recently, in a discussion, it came through that research is more reproducible if one uses "popular" software instead of "unpopular" free ones.
For instance:
I had been using Scilab (free) for a lot of my work and distributed my files to others. But I was surprised that more people had MATLAB ($$) and preferred if I sent  them MATLAB files instead (little modifications).
My question is :
Assuming I'm starting a new project and I wish to make it as reproducible as possible. Should I be using relatively unpopular free software or extremely popular proprietary ones?

I don't usually deal with data, so that might be a naive question, but isn't it possible to keep everything in an open format (such as a big table of value in a text file), and then import in one's preferred software?

@CharlesMorisset. In my field, the exchange has more to do with codes rather than data. For instance, codes for solving a mathematical problem. There are very few inputs.

In the particular case you describe, I strongly suggest using Octave instead of Scilab. It's another free Matlab alternative, but unlike Scilab it has a strong and explicit focus on Matlab compatibility. The last time I received a Matlab script from a collaborator, it ran perfectly under Octave with no modifications.

I think there are two kinds of reproducibility:

The ability of someone else to run your code and obtain the same output.
The ability of someone else to write their own code that does the same thing as yours based on your description and on examination of your code (reproduction from scratch).

The second kind of reproducibility is much more convincing, since the main point of scientific reproducibility is to verify correctness of the result.  For science that relies on code, it is usually impossible to include every detail of the code in the paper, so verification requires examination of the code.
If you use proprietary software, your code probably makes use of closed source code, and therefore it cannot be verified or reproduced from scratch.  If you use open source software, then all of the code that your code calls is probably open source, so it can all be verified or reproduced by someone else from scratch.
At present, it is probably true that the first kind of reproducibility is more achievable with proprietary, widely-used software.  I am optimistic that the current trend will lead to open-source software catching up in terms of wide use (consider SAGE, for example).

Addendum, in light of Epigrad's answer below, which I mainly agree with:
The problem with relying on closed-source code isn't that someone else won't know what that closed-source code is expected to do.
The problem is that if you have two closed-source implementations of the same algorithm and they give different results (trust me, they usually will), then you have no way of determining which (if either) is correct.
In other words, closed-source code would be fine for reproducibility if it were bug-free.  But it's not.

I think this answer should indicate it is limited to cases where the "output" of the software is or indicates "the result", rather than constituting a part of the experimental setup for the actual evaluation.

To supplement @David Ketcheson's answer with a "Yes, but..."
I agree that there are two types of reproducibility - CrossValidated discusses them with some degree of frequency. There is, as has been mentioned, "Can I click 'Run' and get the same answer you did" reproducibility, which I generally don't find very compelling.
There's also "Could I repeat your analysis from what you have provided from Step 1 to Step End, and get the same or a similar answer?" I think this is the one we should be aiming for.
That is often helped by using accessible, non-proprietary code, but not always. Consider the following example of an infectious disease dynamics model, expressed as a system of ODEs:
Here, in order to replicate (or fail to replicate) my findings, the software I used doesn't matter. What matters is the equations and parameter values I chose. If I provide those, then the only reason for code being needed is because someone doesn't want to implement the study from scratch, and does want to just run the code and see if the results match, tinker with the assumptions a bit, etc. In that case, everyone benefits from the code being in a form people use.
I think the same is often true for statistical analysis that doesn't use novel methods. At this point, what matters is that the data is available, and that the code is implemented in a language people understand and use. If 95% of people use SAS, even if it is proprietary, then the way to make your results most accessible, and most easy to replicate, is to have an implementation in SAS. Because if you pick an obscure but free language, what you've done is replaced the "Money" barrier with a "Time to understand" barrier - which for most people equates to the same thing.
The summary is this: I don't think "Free/Open" vs. "Proprietary/Closed" is necessarily the deciding distinction. I think that distinction is accessibility, and trying to maximize that. If there is both an open, free and popular software package that's used (R for example) then great! - use that. But if the field uses primarily one commercial package, picking an obscure alternative just because its free doesn't fix accessibility, it just shifts the burden.

Whether code is open or closed, an algorithm that is key to a paper needs to be specified clearly enough to replicate it. That effort can be greatly assisted by having additional "test case" data for at least some of the steps. That can allow an independent implementer to increase their confidence that they understood what was going on, and it can also aid in the sometimes difficult task of explaining what some step actually does. I'd agree this isn't likely to belong in the body of the paper, but that is clearly what an SI is for.

As someone who works on making ODE solvers efficient and reliable, I love that you use them as an example of something that will never fail.  Of course, there are still lots of situations in which they do fail.

Let me start with a disclaimer. I generally subscribe to the free software community perspective that proprietary software is questionable ethically, and best avoided if possible. I realise this perspective is not commonly held in scientific circles. Having said that, sometimes proprietary software is a necessary, or at least not easily avoided evil, and I'm generally pragmatic about using proprietary software when no good alternatives exist. I've used proprietary software in the past, though currently the only proprietary software I'm currently (sporadically) using is Skype, for which no good free alternatives exist.
However, special considerations apply in a scientific context. One of thse has already been covered by @David, namely that in general you can't "see inside" proprietary software to see how something is implemented. Having said that, sometimes proprietary software is written in an interpreted language, as in Splus, and one may be able to see part or all of an algorithm implementation. Regardless, the point holds generally.
A separate and obvious issue, which I don't think anyone has raised, is that using proprietary software forces others who want to use your software to buy the proprietary product you use. These products can be quite expensive, especially for people from poor countries. For example, Matlab, which has been mentioned in this thread, runs to thousands of dollars if one has to pay for a license oneself. Western academic institutions often have site licenses for such popular software, so researchers don't have to pay for it themselves. I personally am quite unhappy when I am expected to use a piece of software written using some proprietary language or package that has to be purchased.
A related issue is that much, if not most research, is done using public funding, i.e. taxpayer money. It seems undesirable to me to use such funds to buy proprietary software, thus adding to the profit of some corporation. In general, there is some movement to make academic work that is done using public funding free. And one can easily make the argument that the usage of proprietary software makes ones scientific product less free. For example, I believe the NIH now has some such policies in place. Similar arguments could be applied to the usage of software tools.
A tangential technical issue is that it is often difficult to get proprietary software to play nice on free software platforms such as the free Unix-like systems currently popular in scientific circles, e.g. the Linux based systems, and the BSD systems. These difficulties include, but are not restricted to
a) ABI problems. If one wants to compile a C/C++ extension for Matlab, for example, one has to use exactly the version of the compiler that the Matlab program has been compiled with
b) The program requires obsolete libraries or requires libraries to be in non-standard places.
I mention this issue in part because my understanding of the question is that it is asking about proprietary vs free in the context of pragmatic usage.
So, to respond to the question directly:

Assuming I'm starting a new project and I wish to make it as
  reproducible as possible. Should I be using relatively unpopular free
  software or extremely popular proprietary ones?

I don't think there is a clear answer. If there is no viable alternative, then one would have to use the proprietary software, as I do with Skype. If there a viable free version, I would use it. Bear in mind that if more people start using the "relatively unpopular free software" it will become more popular. :-)

Agreeing with most that has been said by EnergyNumbers and David Ketcheson, I'd like to add some slightly different points:

the fact that code is written in a particular language (Matlab) does not make it closed source per se.
Just as using Scilab on closed source Windows (or using a closed source BLAS) doesn't make Scilab closed source.
There are journals that require reproducible research and open code and accept Matlab code.
likewise, popular and free do not exclude each other, nor does proprietary imply that it is popular
neither of the two imply that the respective software is suited for the reproducible data analysis.
The choice on the language should be based on several factors

how suitable is it for the problem at hand
here also: how suitable is it for reproducible data analysis (I'm experimental scientist, so reproducing a data analysis is just a part of reproducible research for me)
particularly if talking about sharing code: infrastructure considerations (can code be packaged into libraries? How can data and code and the text be bundled into a reproducible paper?)
popularity = size of peer group using this software (which somehow includes the cost of a license)
you may also want to consider what the peer group that is interested in reproducible research is using (In my field(s), the reproducible research crowd coincides much more with the open source (R) crowd, while of the peer group working on the same kind of problem probably the majority uses Matlab)

My old supervisor used to say that the cost of a license is no scientific argument.
But of course you may need to consider it.
Likewise, popularity is not a scientific argument, but you should critically examine whether the propularity does actually indicate that lots of good reasons exist to use that software.

Examples:

In my field, R is increasingly popular (by now popular enough to publish in R) and to a certain extent replacing Matlab.

So R is both free and popular (yet most people from my field don't make use of the fact that you can look into R's source if they discuss reproducible data analysis)

Reasons IMHO include

most importantly: R is well suited for our kind of problems (I think it is better suited than Matlab, others differ slightly in their opinion and use Matlab. Even others do think it may actually be better suited but not as much as to outweigh learning R right now)
Being well suited includes the availability of methods we use at CRAN vs. commercial Matlab toolboxes and Matlab file repository
Being better suited includes the fact that I cannot adapt the code of proprietary Matlab toolboxes (p files) to particular needs.
Being well suited includes the ease of report generation
infrastructure: e.g. R's package concept enforcing a minimal standard and allowing to rely on examples and tests actually being runable vs. a folder full of .m files (I heard that recently a more package like concept was introduced). This helps a lot with sharing the code (whether to reproduce your findings or to use it on their own data)
license costs come in indirectly: it is more that you don't need to worry if you install it on your private computer as well and you can tell students to install it on their laptops than worrying about the cost of a few licenses for computers at work.
probably far more costly than license fees themselves are the time to get the license manager working or to transfer licenses between computers, and if you just want to give a toolbox a try before deciding whether to buy it or not, the hassle of a) asking the vendor for a demo version and later on b) of filling out order forms and writing justifications why you need to spend money on that license.

Note how many of these arguments are "soft" and in fact have more to do with being used to one system or the other or using a feature is easier, more obvious, or more common among the peer group of that software's users: noweb can work with Matlab, documentation of m-files is possible and encouraged (though not really enforced), unit tests are possible in Matlab, Matlab Central has lots of free code, etc. Just R users always know of CRAN, whereas not all Matlab users know of Matlab Central, there's a good culture of citing scientific papers about the implemented method in R help files, shipping code together with example data sets and/or giving actually running example code.
Examples of invalid arguments:

if my peers did not use version control for coding, I wouldn't consider that an argument against using version control (as there are lots of good reasons for using it)
Or, if the ones refusing to use version control were programming in Matlab, neither  would that be an argument against Matlab - but I'd check whether there is any reason that prohibits using version control for Matlab code.

Listen to your colleagues & peers.
They've already told you what is most suitable for them, in order for them to be able to reproduce your results.
That answer, in your particular case, is Matlab.
There will be some others who want to port it to Octave, SciLab, Excel, Fortran or whatever. That's fine too - but if you're flexible about which platform you code in, and coding in Matlab won't make you less productive (or the small reduction in productivity is a price worth paying for the increased reproducibility), then code in Matlab. Because your colleagues have already told you that that's what enables them to reproduce your work, easiest.
There are plenty of good reasons (and maybe some bad ones) why many of your colleagues prefer Matlab. Sometimes the cheapest things can cost you most.
For anyone else reading this, with a similar problem, the gist of the answer is the same: listen to your peers.

but if you use Octave, it should be reproducible in Matlab. (the reverse is not always true, Matlab has many features that are not supported by Octave)

@Abe compatibility in the other direction is also not ensured. Matlab and Octave have a common feature set, but are a lot different when you scratch the surface. You can write programs that run on both, but you would have to avoid most of the advanced features.

My two cents:
I often compare code with a scientific paper. The purpose of a paper is to describe your results and approach to the problem in such a way that your peers can validate/refute your findings, in whatever way they see fit, so that a collaborative effort can take place to make progress in the field.
Who cares whether the paper's author has used LaTeX to write his paper, or MS  Word? Who cares if the data was processed with MATLAB, or Excel, or Pascal? It is the truth(s) in the paper that count(s), not the tools used to get there (although many would gladly jump in here and start a fierce discussion on this point...but in my experience, the arguments used in such discussions tend to be more like religion than anything else).
What is very important however, is the means of getting through to your peers. For example, if you write and publish a paper in Esperanto (supposing for the moment that would get accepted), simply because you think it's beautiful and elegant and everyone should speak it. Plus there are many books about how to convey meaning in Esperanto right?
Not many of your peers will be able to understand the paper, let alone get the message and reproduce your findings. You'll have to wait until someone comes along in your field who shares your views on Esperanto, which might take half a lifetime. Altogether this is a very poor way of making progress in your field.
This I think is the crux of your predicament -- if your peers mostly use MATLAB, you'd best stick to MATLAB, because you'll reach the most peers the quickest. Leave it to (other) engineers to sort out whether MATLAB actually produces good (enough) results for your case, and leave it to one of your peers half a world away to translate your code to C++ and verify your findings.
It's not the code that counts -- it's the knowledge that is contained in it.

You can do whatever you damn well please, but there are a few considerations:
1) You might have an obligation to disseminate your work. If you are supported by an external funding agency such as the NSF or NIH then dissemination is an obligation. Many private foundations and other funding sources also provide support with the intent of dissemination, whether it's stated explicitly or not.
If you have such an obligation, then absent any other consideration this would suggest you use the most accessible software possible, regardless of whether that is proprietary or free.
2) Open source-ness is important for some kinds of research, but for most research it's irrelevant. Unless you're doing computer systems research, where it really matters how the computer arrives at an answer, the method does not matter so much as correctness. Sometimes it can matter (e.g. if your work is heavily dependent on numerical methods), but it probably doesn't.
3) Communities establish standards of validity and integrity. If everyone else uses MATLAB, then the community has deemed it accurate. Absent evidence to the contrary, using open-source software does not make your results seem more correct or more verifiable in anyone's eyes.
As as side note, Mathworks has a strong reputation for working with researchers. If you really felt that MATLAB was giving you incorrect results, and you had examples to show it, they would be knocking down your door to fix the issue. I've had Mathworks issue me a custom support patch the very same day I called about an obscure hardware incompatibility that was causing incorrect behavior.

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