Undergraduate research in mathematics

Question

I am a European student applying for a PhD in mathematics in the US. Browsing the internet, I have come across quite a lot of American PhD applicants claiming that they have several years of research experience, or even publications. I am rather astonished because in Europe first research attempts normally begin when a student is writing his or her Master's thesis. On the other hand the level of mathematics taught in European undergrad programmes is typically higher than in the US an normally involves courses which are classed as graduate courses in the US. Therefore I would like to ask the following questions:

1. What does "undergraduate research" really mean in most cases? Original research, contribution to a senior mathematician's original research through some calculations/programming work, independent study of a difficult topic + a paper/report but without any original findings?

2. Am I right being skeptical that someone who has only taken some courses in linear algebra, real analysis, etc. is ready to do research?

3. US PhD programmes typically involve 2 years of taught courses and a qualifying exam prior to beginning the work on one's thesis, i.e. research proper. Is undergraduate research really an important prerequisite in the eyes of the admissions committee?

4. How common is it to have publications as an undergraduate in the US?

5. As a student coming from Europe, where there is little opportunity for undergrad research and it is generally not encouraged, is my application at an disadvantage? While I have not done any "undergrad research", I have done a lot of independent work. I have written a Bachelor's thesis, given talks at many seminars, and am now working on a Master's thesis which will hopefully contain some original results. Can this experience be regarded on a par with a US applicant's research experience?

Answer 1

To address a few of your points.

1. There is no standard definition of undergraduate research. In many US universities one can graduate with a degree without doing an independent thesis, so in effect it is possible to obtain a bachelor's degree without having anything resembling "research experience". Therefore some people would be generous and count the research for bachelor's theses as "undergraduate research". For some people the phrase excludes theses but includes REU experiences, and for some only original research leading to publications count.

   It is not too unreasonable to expect that the work put into independent projects, even an undergraduate thesis that basically is just a review of existing results, can be beneficial to graduate research.

2. Yes; but where did you get the idea that undergraduates only learn some linear algebra and real analysis? Combinatorics is a field in which the barrier to entry is somewhat lower, and is especially approachable to the Math Olympiad types; and discrete mathematics is certainly on the rise in the past decade or so in terms of undergraduate education. And many undergraduates do have quite a bit more under their belt than you seem to believe.

   Having worked in Europe for the past 5 years, I feel comfortable asserting that the claim

   the level of mathematics taught in European undergrad programmes is typically higher than in the US an normally involves courses which are classed as graduate courses in the US.

   is not quite true. What is true is that the mathematics in European undergraduate programs often start out at a higher level and with more rigour, but at the advanced undergraduate level the US schools usually have caught up. (Note also that the US degrees are usually 4 years compared to 3 years in Europe, so they have a bit more time to build.) Couple that with the liberal arts tradition you'll often find elective classes at the top level of undergraduate education in the US which have no analogue at all in Europe.

   While it is certainly true that good European universities offer undergraduate courses that are at the level of graduate courses in mediocre US universities, the reverse swapping European/US is also true.

3. It depends on the answer to (1). The rare individual who actually did original research and obtained publishable results will likely get some bonus points when the admission committee deliberates, but it is certainly not a norm and one is not expected to have done such necessarily.

4. I don't have any statistics. Hearsay suggests that original research leading to something published in a research journal is somewhat rare. (Publications resulting from REU or similar programs, or publications in undergraduate journals are less rare.)

5. If your master's thesis contains original results, then you are likely more than on par with the typical applicant to US PhD programs, in terms of research experience.

Answer 2

Is undergraduate research really an important prerequisite in the eyes of the admissions committee?

It's certainly not mandatory: every US graduate program in mathematics is willing to admit students who have not done original research or published a paper. On the other hand, having impressive accomplishments helps, and it's important to have letters of recommendation from faculty who have seen you engage deeply with something difficult. It's easier to get such a letter from a supervisor on a substantial independent project, but it doesn't have to be a research project. Your background as described in the question sounds strong, and I don't think you need to worry about a lack of research experience.

The worrisome scenario is someone who has never focused on any question for more than a few hours. This is entirely possible for an undergraduate who hasn't written a thesis or worked on any comparable project. The problem is that doing mathematics on a long time scale is qualitatively different from short-term problem solving. It's psychologically different: you need to be patient and flexible, while still maintaining focus over months or years. Some people are effective and enthusiastic over short time scales but aren't temperamentally suited to research, and it can be a heavy blow to discover in graduate school that you don't enjoy this sort of work or have trouble focusing. Other people are intimidated by the idea of research and assume they would be worse at it than at short-term problem solving, while in fact they might be much better.

So the key advantage of undergraduate research is that it gives students a chance to discover their preferences and talents, with faculty mentors who can judge their performance and vouch for it. On the other hand, there are other ways to achieve the same goals, and it's the goals that matter.

Historically, the initial push for undergraduate research in the US was actually as a recruiting tool. The idea was that encouraging students to do research would help uncover people who should go to graduate school but might not have realized it (either from being underconfident or from not having had the opportunity to shine). REUs were not intended as a prerequisite for graduate school, but this view has become more common among applicants as the programs have become more popular.

What does "undergraduate research" really mean in most cases? Original research, contribution to a senior mathematician's original research through some calculations/programming work, independent study of a difficult topic + a paper/report but without any original findings?

Preferably one of the first two, and ideally the first, but it is sometimes used to refer to the third as well. Both applicants and supervisors have incentives to describe work as "undergraduate research" regardless of whether they intended or tried to do original research, in the hope that it will sound more impressive than just saying it was an independent project. This broader usage gets on my nerves, so I'd recommend against it, but it's widespread enough that there's not much risk of standing out as being misleading or manipulative.

How common is it to have publications as an undergraduate in the US?

It's relatively common among those who attend an REU or similar research program, although the publications are not necessarily impressive in absolute terms (i.e., not taking into account that the authors are undergraduates). Publishing is rare among undergraduate students who have not participated in such a program.