The National Science Foundation has a page which describes all the mathematical Research Experiences for Undergraduates which are summer programs designed for students who want to try doing research. Some are designed for those who just want a taste of the feeling of research to see if it is what they might like to do. Others are designed for those who are already quite advanced and desire a head start into research. Many of these programs have themes or topics, like applied mathematics, group theory, or combinatorics and graph theory; others have a wide variety of subjects. Most require little background as the necessary material will be covered when you get there. The important prerequisite is a desire to learn and to think about mathematics.
The Mathematics Advanced Study Semester (MASS) at Penn State is a full semester-long program which includes three core courses, a weekly seminar, and computer related mathematical projects.
The Carleton College Summer Mathematics Program for Women, the Summer Program for Women in Mathematics (SPWM) at George Washington University, the EDGE Program at Bryn Mawr and Spelman Colleges, and the Junior Year for Women at Smith College are programs for talented undergraduate women in mathematics. There are seminars at various levels to get students more interested in different types of mathematics and mathematical research. Women who are contemplating or unsure about their future in the mathematical sciences should consider these programs. As with many others, these programs carry a stipend and provide housing.
The Barry M. Goldwater Scholarship and Excellence in Education Program was created to encourage outstanding students to pursue careers in mathematics, the natural sciences, or engineering and to foster excellence in those fields. The application materials include the statement: "Nominees for Goldwater Scholarships must include in their nomination materials an essay discussing a significant issue or problem in their field of study that is of particular interest to them. The essay must include a description of the issue or problem, a discussion of an idea for research that could have significant impact on the issue or problem, a description of an aspect of the research in which the nominee would be involved, and an explanation of the relevance of the issue or problem to the nominee as a mathematician or scientist." For more information, see our Goldwater Scholarship Program page.
The K-State Career and Employment Services Internships and Co-ops = Experiential Learning page lists opportunities to learn and work in business, government, and industry. An internship is a paid or unpaid work experience completed over a summer or semester. A Co-op is an extended (semester and summer) full-time paid internship. Cooperative education alternates summers and/or semesters of work and school in paid positions and provides the opportunity to test-drive a career.
The National Security Agency has a Career Opportunities page which describes their Opportunities and Programs for Undergraduates. For example, the Director's Summer Program is the National Security Agency's premier outreach effort to the very best undergraduate mathematics majors in the country. The students participating in the cryptologic mathematics program will work on a broad range of problems involving applications of abstract algebra, analysis, combinatorics, geometry, number theory, graph theory, probability, and statistics.
The US State Department has a number of Student Programs including Student Internships in which mathematics students have the opportunity to gain firsthand knowledge of American foreign affairs. The duties may include areas such as research, report writing, correspondence, computer science, and analysis.
The United States Department Of Energy (DOE) sponsors a number of Student Programs and Internships at Ames Laboratory, Argonne National Laboratory, Lawrence Livermore National Laboratory, Oak Ridge National Laboratory, and Sandia National Laboratories. These offer unique scientific experiences for qualified undergraduates in science, mathematics, computer science, and engineering. For example, Argonne National Laboratory has a number of Undergraduate Programs, and the Oak Ridge Institute for Science and Education (ORISE) has a number of Undergraduate Research Opportunities. Many research projects are available, and great care is taken to ensure that students are placed in programs which best meet their interests and needs. Appointments are usually for the fall or spring semester.
The K-State Office of International Programs home page describes opportunities to study abroad for a semester or year at a university in the Czech Republic, France, Germany, Iceland, Italy, Mexico, Sweden, and other countries. You will also find listings for thousands of study abroad programs in more than 100 countries throughout the world on the Study Abroad home page.
Students interested in traveling abroad and studying mathematics in an exciting and challenging environment, should consider the Budapest Semesters in Mathematics program. This is a semester or year long program and can be done either in the fall or the spring. The courses are taught in English and the program is an exceptional place to meet students and professors who are intelligent and excited about mathematics. It is also a unique cultural experience, as Hungary has a rich historical tradition and wonderful artistic and musical displays and events. The prerequisites are not much more than the calculus sequence. It would be very helpful to have done at least a semester of junior-level mathematics like advanced calculus, linear algebra, or modern algebra. The most important prerequisite is an enthusiasm about learning mathematics. There are courses suitable for all levels, from beginning modern algebra and complex analysis, to more advanced combinatorics and number theory (the Hungarians' specialties!). This one does cost money, though not considerably more than a semester of study at Kansas State. There is also the plane ticket to buy, but the experience is definitely worth it!