Undergraduate Program in Mathematics

Mission of the Caltech mathematics undergraduate 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 (ma10) and a writing class (ma11); 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.


Option Requirements

  1. Ma 2.
  2. Ma 3 or Ma 144a.
  3. Either Ph 2bc or Ph 12abc (the department recommendation is the Ph 12abc sequence).
  4. Ma 5abc, Ma 10, Ma 108abc, Ma 109abc.
  5. Ma/CS 6a or Ma 121a.
  6. Ma/CS 6c or Ma 116a or Ma/CS 117a.
  7. 45 additional units in Ma or ACM numbered 95 or above (other than Ma 98). Courses in other options with high mathematical content may be used to fulfill this requirement with the approval of the executive officer for mathematics. Of these 45 units at most 18 can be in ACM or other courses outside Ma. Math courses taken elsewhere and allowed (such as in a study abroad program) are included in this 18 units outside of Caltech Ma courses.
  8. Math majors must take two quarters (18 units) of a single course, chosen from the mathematics course listings with numbers between 110 and 190, inclusive. (In years where one of these courses is given as a one-term course only, it cannot be used to satisfy this requirement.) These two quarters may be used to meet requirements 2, 5, 6, or 7.
  9. Unlike courses satisfying requirements 7 and 8, which may be taken pass/fail, none of the courses satisfying requirements 1-6 may be taken on a pass/fail basis.
  10. Passing grades must be earned in a total of 486 units, including the courses listed above.

Typical Course Schedule

 

 

 

Units per term

 

 

 

1st

 

2nd

 

3rd

Second Year

Ma 2, 3

Sophomore Mathematics

 

9

 

9

 

-

Ph 2bc

Sophomore Physics

 

-

 

9

 

9

Ma 5abc

Introduction to Abstract Algebra 

 

9

 

9

 

9

 

HSS Electives

 

9

 

9

 

9

 

Electives*

 

18

 

9

 

18

 

Total

 

45

 

45

 

45

Third Year

Ma 10

Oral Presentation

 

3

 

-

 

-

Ma 108abc

Classical Analysis

 

9

 

9

 

9

Ma/CS 6ac

Introduction to Discrete Mathematics

 

9

 

-

 

9

 

HSS Electives

 

9

 

9

 

9

 

Electives*

 

18

 

27

 

18

 

Total

 

48

 

45

 

45

Fourth Year

Ma 109abc

Introduction to Geometry and Topology 

 

9

 

9

 

9

 

HSS Electives

 

9

 

9

 

9

 

Electives*

 

27

 

27

 

27

 

Total

 

45

 

45

 

45

* Includes menu course (2nd year, if not taken in freshman year). Also must include courses to meet items 7, 8 under option requirements.


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 (ma3), as well as basic physics (ph2 or ph12).

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.


Summer Research Opportunities

Students interested in summer research should refer to the Summer Undergraduate Research Fellowship (SURF) website. Past SURF project titles are available here. If you would like to read the abstracts for these project, please refer to the SURF abstract books.