There are a wealth of research opportunities for Yale students both at Yale and elsewhere. Here's some ideas for summer work of all kinds in mathematics:
Research Experiences for Undergraduates (REU)
Research Experiences for Undergraduates are summer programs funded by the NSF in a variety of sciences, including pure mathematics. They vary a great deal as to style, duration, topic, and location. There are REUs for people of all backgrounds and ability levels--browse the NSF website and you're sure to find one that's a good fit. Note that some of them (Williams in particular) will provide funding for non-citizens as well.
NSF REU list: Math
Duluth: Graph Theory, Combinatorics, Discrete Math
Williams (SMALL): Diverse topics
Wisconson-Madison: Modular Forms and Number Theory
ETSU: Probability and Discrete Math
Rutgers DIMACS: Discrete Math
A Yale prize providing money for many undergraduates to pursue some kind of summer math activity (usually under supervision of some faculty member here). Applications are easy (a one page proposal), but come very late in the year (ie May).Analysis
Cornell offers a program teaching analysis which includes a small research component as well. http://www.math.cornell.edu/~smi/
Hedge funds like to hire the brightest young minds, but can be hard to find. DE Shaw (http://www.deshaw.com/)
is known for liking math people (especially Putnam competitors), and Goldman Sachs hires quants (try the recruiting database thru UCS). Some statistics/programming skills are probably be useful.
Caltech's SURF program hires tons and tons of undergrads to do all kinds of research projects, some which are pretty math-y (http://www.surf.caltech.edu/)
. You can also look at REUs in physics (just google).
Math Summer Camp Counselor
Cryptography and Communications
Summer Research at Yale
Email the DUS, Andrew Casson, for more information on Yale's summer opportunities