# Undergraduate Math Program Description

Mathematics combines abstract logical thought with concrete investigations into symmetry and order. It looks for patterns, makes analogies and formalizes the conclusions succinctly, in a way that they are useful in a variety of places where there is structure. Those who major in math go well beyond the basics to study calculus, differential equations and statistics, the basic structures in algebra, analysis and geometry, as well as rudiments of combinatorics and set theory. Juniors and seniors are given the opportunity to focus their studies in an area of interest by taking advanced topics courses and through reserach with faculty members. Students learn to write coherent and complete proofs of various assertions, to work out non-trivial examples and, when possible, use computational tools. Our undergraduate program has a special emphasis on equipping the students with needed tools for a successful research career. The knowledge and skills acquired here are consistent with admissions to graduate programs in peer institutions. Seniors are encouraged, but not required, to do a bachelor's thesis in an advanced topic.

#### Mission of the Undergraduate Mathematics Program

Mathematics combines abstract logical thought with concrete investigations into symmetry and order. It looks for patterns, makes analogies and formalizes the conclusions succinctly, in a way that they are useful in a variety of places where there is structure. Eugene Wigner spoke of the "unreasonable effectiveness of mathematics in the sciences" and engineering, and an integral part of our undergraduate program at Caltech is a thorough grounding in basic mathematics. Those who major in math go much further and study, besides calculus, differential Equations and statistics, the basic structures in algebra, analysis and geometry, as well as rudiments of combinatorics and set theory. In addition, these students take advanced courses in some of these subjects, depending on their interest. They absorb the theoretical underpinnings from a critical thinking viewpoint and learn to write coherent and complete proofs of various assertions. They also learn to work out non-trivial examples and when possible use computational tools. The mission is fulfilled through course work and research opportunities, the latter primarily through our SURF (Summer Undergraduate Research Fellowships) program.

Our undergraduate program has a special emphasis on equipping the students with needed tools for a successful research career. The program requires the math majors to take, in addition to the required core and advanced classes, courses in oral presentation (Ma 10) and a writing class (Ma 11); the latter is actually an Institute-wide requirement. Moreover, the seniors in math are encouraged, but not required, to do a bachelor's thesis on an advanced topic. The students are also exposed to some innovative ideas through topics classes, a freshman seminar, and a pizza course.

#### Course Work

The four-year undergraduate program in mathematics leads to the degree of Bachelor of Science. The purpose of the undergraduate option is to give students an understanding of the broad outlines of modern mathematics, to stimulate their interest in research, and to prepare them for later work, either in pure mathematics or allied sciences. Unless students have done very well in mathematics courses in their freshman and sophomore years, they should not contemplate specializing in mathematics.

Since the more interesting academic and industrial positions open to mathematicians require training beyond a bachelor's degree, students who intend to make mathematics their profession must normally plan to continue with graduate study. Some students use their background in mathematics as an entry to other fields such as physics, computer science, statistics, economics, business, finance, medicine, or law.

The schedule of courses in the undergraduate mathematics option is flexible. It enables students to adapt their programs to their needs and mathematical interests and gives them the opportunity of becoming familiar with creative mathematics early in their careers. In particular, students are encouraged to consider courses in areas such as applied and computational mathematics, physics, finance, economics, control and dynamical systems, computer science, electrical engineering, and computation and neural systems.

During each term of their junior and senior years, students normally take 18 units of courses in mathematics or applied and computational mathematics, including the required courses Ma 108abc and 109abc. Any course listed under applied and computational mathematics is regarded as an elective in mathematics and not as an elective in science, engineering, or humanities. Those who have not taken Ma 5 as sopohomores must do so as juniors. Overloads in course work are strongly discouraged; students are advised instead to deepen and supplement their course work by independent reading.

A student whose grade-point averages are less than 1.9 at the end of the academic year in the subjects under mathematics and applied and computational mathematics may, at the discretion of the department, be refused permission to continue the work in the mathematics option.

#### Learning Outcomes

By graduation time, our students are expected to have the following:

1. A substantial knowledge of the basic areas of mathematics, namely algebra, analysis, geometry/ topology, and discrete math.

2. Basics of probability and statistics (Ma 3), as well as basic physics (Ph 2 or Ph 12).

3. The equivalent of several quarters of advanced math and research work.

4. Some exposure to computations.

5. A broad range of problem solving experience.

The knowledge and skills acquired here are consistent with admissions to graduate programs in peer institutions.

#### Means of Evaluation

The outcomes of our program are regularly evaluated through several channels. Students provide course feedback through the online TQFR system; graduating students fill out an exit survey; each of the residence houses has an ombudsperson for each of the introductory math classes; the option representative meets with students periodically; every other year the students majoring in math collectively discuss the program with representatives of the faculty as part of the Student Faculty Conference. In addition, alumni outcomes are regularly monitored. The information gathered is used to improve class teaching and professor assignments, and to motivate curriculum changes.