The only geometry and topology REUs I know are Williams and U of Tennessee. There is a reason why there are so many algebra/discrete math and math modeling REUs but hardly any geometry ones: most students do not learn enough topology and geometry before their junior year to have a productive REU experience. Williams is among the most selective REUs in the country and their geometry group is not going to accept you with just a basic course in point-set topology. To even stand a chance you would need one of your references to say that you know a *lot* more about those subjects.

Talking about references, your letters of recommendation are the single most important element of the REU application. If your Calc III professor is the one professor who has been more impressed by your performance than anyone else, go ahead and ask him for a letter. If professors in your higher-level courses will be as excited about you, they might be a better option because higher-level classes tend to reveal more about the math skills that are relevant to REUs. I don't think you have to decide right now who you are going to ask for a letter. You can ask the professor now if he would be willing to write you a letter for your REU applications next spring, and then e-mail him a list of the programs you want to apply to later. If you change your mind about the references, you could give him only a partial list of the programs you are applying to.

I just finished my sophomore year and I was advised to apply to ~10 programs because many REUs prefer juniors. I was lucky and was admitted to 7 programs. Another sophomore at my college got into 3/8 programs and she had taken a decent amount of math as well. (Something like 2 courses each in linear algebra, abstract algebra and real analysis; number theory, cryptology, graph theory, diff eq and multivariable calc. She is interested in algebra and discrete math.)

I think it is a good idea to apply to programs of varying selectivity. A few REUs regularly publish in standard professional journals, but most don't publish at all or "only" in undergraduate research journals or some obscure journal that no one ever reads. If you are concerned about a publication, apply to programs that work in areas you have never heard of before rather than doing standard math. The reason is that all of the easy problems in standard math have been solved, and the remaining ones are too hard for undergraduates to tackle. Many REUs try to admit applicants with a similar level of mathematical maturity because otherwise the students will have a hard time working together. Surprisingly I ended up not getting into the less selective programs I applied to, but typically it's the other way round. It's hard to estimate beforehand how competitive of an applicant you are, so it is safest to apply to a variety of programs.

The following report contains a lot of info about individual REU programs that may not be published on each program's website. Some of them specify exactly how selective they are, how they select applicants, how they approach undergraduate research, etc.

http://www.ams.org/employment/PURMproceedings.pdf There is also a version from 1999 that has some reports from REUs that are not in the 2007 proceedings. You can probably find it on google. When reading the latter, keep in mind that most programs have become a lot more selective since 1999 because REUs only became popular very recently.

I am just curious, why would you need more math courses than currently possible at your institution to take algebraic topology and differential geometry in your senior year? At my college the first graduate courses in those fields realistically only require linear algebra, multivariable calculus, abstract algebra and point-set topology (and officially also real analysis but one would be fine w/o). That doesn't sound all that impossible. I took both classes as a sophomore.