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:
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?
Am I right being skeptical that someone who has only taken some courses in linear algebra, real analysis, etc. is ready to do research?
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?
How common is it to have publications as an undergraduate in the US?
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?