Summer Internship to gain Research Exposure for Undergraduates

Question

I am a 1st year undergrad student in Maths and Computer Science.I wanted to do a summer internship in mathematics to gain some research exposure. However I am not sure what I can do after my 1st year.

     I wanted to ask what exactly is done in a summer internship by undergraduates? 

When applying for internships,at many institutes, you have to choose a topic/subject and write in about 250 word what you would be interested in doing.So if you choose some topic are you expected to know it beforehand and in the internship go further into that topic or can you say that you would like to learn a particular topic , which is new for you but you have the prerequisites for learning it. Or does it work both ways depending on the professor?

I would be interested in doing something in algebra, but I dont know what topic to choose? It could be something in Abstract Algebra or Linear Algebra .By the end of my 1st year I would have done the the following courses in maths-

1. Algebra 1 (Basically first 4-5 chapters from Artin)

2. Algebra 2

   The group of motions of the plane---finite and discrete groups of motions---abstract symmetry---group operations on cosets---permutation representations---the operations of a group on itself---class equation of icosahedral group---the Sylow theorems---free groups---generators and relations. Bilinear forms---symmetric and hermitian forms---spectral theorem ---conics and quadrics---skew-symmetric forms.

3. Calculus 1 (Single Variable)

4. Calculus 2 (Multi-Variable)

5. Discrete Mathematics

6. Probability Theory

So can you suggest some topic(s) that i can do ?

Answer 1

Here

http://www.nsf.gov/crssprgm/reu/list_result.cfm?unitid=5044

is a list of NSF-funded sites for research experiences for undergraduates (REU). You have to be a US citizen or permanent resident to be eligible. Applications are usually due in January or February.