<p>What do freshmen usually do during the summer after freshman year? I feel like we don't have enough actual knowledge to begin research but are there any Summer programs I can apply to? By the time of Summer I would know Calc I through Calc IV and have taken a theoretical linear algebra course. So I was just wondering what my options are. I'm mostly interested in geometry.</p>
<p>The best thing to do is to talk to your professors and see if they hve a summer position for an undergraduate. You won’t be competitive for an REU until after your Sophomore or Junior year.</p>
<p>Look for any volunteer experience pertaining to your subject matter. For example, as a plant scientist, I would volunteer with local nurseries and parks. Then, use that experience to piggyback into a starting REU, offering to do grunt work. You can express your experience in whatever area your volunteer work was in. Then piggyback off of that into assisting with a grad student or post doc’s research, contributing as much as you can. By the end of the summer, you should be ready for a decent REU. It’s about getting experience where you can, and building off of that.</p>
<p>You could also look for campus programs that are specifically for integrating undergraduates into research positions.</p>
<p>Math research as an early undergraduate might be tough because so many areas have a high barrier to entry. Most people I know who did math research in high school (they were very smart people; this is not typical) did their research in combinatorics, which generally has the advantage that the problem statements can be understood by people with less higher math courses (not to say the problems are necessarily easy to solve). I’ve also heard of a few people doing algebra. The problem with areas such as analysis and topology is that you probably haven’t learned enough to do anything until you’re a graduate student (maybe a junior or senior if you’re a bit ahead). </p>
<p>If you’re talking about classical geometry, that’s not really an active field of research any more. Differential geometry and topology are probably the closest things, but you probably don’t have the necessary background at the moment (well, calc III would give you the very basics of differential geometry, but probably not enough).</p>
<p>You can try contacting professors at your school’s math department (as I’ve said I suspect you’ll have more luck with professors doing combo research), or you can look into REUs, though I don’t know what they expect math students to know.</p>