Why should I major in this subject?
Many students are attracted to the discipline of mathematics for the same reason as the faculty, because of its great intrinsic beauty, its abstract and logical nature, and the insights that it gives into the physical world, as well as because of its many applications, such as to cryptography or finance. The mathematics major will introduce you to some of the highlights of the development of theoretical mathematics over the last four hundred years, from a modern perspective, as well as ways in which this study can be applied to understand many problems, both internal to mathematics and arising in other disciplines. If you choose to major in mathematics, you will be prepared to go on to many challenging and rewarding careers in scientific, technical or other quantitative fields, and the analytical skills this major will help you develop can be useful in many other contexts, such as law, business, or finance.
What are some useful first courses that I should take in order to get to know this field of study? When should I take them?
Well-prepared, mathematically talented first-year students should seriously consider Honors Mathematics A, the most challenging of the courses open to them. A prerequisite is a score of 5 on the Calculus BC test and students receive 6 points of advanced credit upon successful completion of Honors Math A. The course is a rigorous, proof-oriented introduction to linear algebra and calculus in several variables, and it satisfies the linear algebra requirement. If you do not take this course but are interested in the mathematics major, you should complete the standard calculus sequences. For a detailed discussion of the calculus sequences, consult the Calculus Classes page.
Before attempting the upper division core courses (Introduction to Modern Algebra and Introduction to Modern Analysis), you should take a course that will provide you with practice in reading and writing proofs, especially if you have not taken Honors Math; Introduction to Higher Mathematics is a course especially designed to do so. Other examples of such courses are: Linear Algebra, Combinatorics, Number Theory and Cryptography, Making Breaking Codes, Differential Geometry, and Complex Variables.
What are the major requirements?
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Major |
To major in Mathematics you are required to complete at least 41 or 42 points. The major includes the following courses:
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Concentration |
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Interdisciplinary Majors |
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Why do the requirements take this form?
Calculus represents the beginning of modern mathematics and is one of the most profound applications of mathematics to the physical and social sciences. Thus, the major begins with the study of calculus in one and several variables, as well as with linear algebra, which will teach you how to manipulate linear equations in many variables and how to connect them to geometry. Starting in the second year, you will begin to gain experience in the three main branches of mathematics, algebra, analysis, and geometry, as well as in some of their subdivisions and hybrids, e.g., number theory, complex analysis, and differential geometry. As you continue in your studies, the courses will become increasingly less computational and more proof-oriented. Aside from the courses offered by the Mathematics Department, cognate courses in physics, probability, logic, and computer science are often part of the major, and are examples of the wealth of mathematical thinking present in other disciplines.
Whom do I speak to about this major? How does the department structure its faculty for advising purposes?
Any questions you have about majoring in mathematics should be addressed to the Director of Undergraduate Studies for the department or to one of the designated Departmental Advisors.
You are welcome to approach any of them with your questions about the major. While the department does not have a system of assigning individual advisors to students, you may consult the bulletin page for a list of our undergraduate advisors.
The department also organizes a series of information sessions and events in the fall and spring terms. The presentation slides from the virtual Math Open House Spring 2022 can be found here.
When should I declare my major?
Although most students generally declare their major during the major declaration period in the second semester of their sophomore year, you should definitely begin a calculus sequence in your first year if you are seriously considering majoring in mathematics.
What research opportunities exist in or through the department?
The department runs several undergraduate research programs aimed at math majors. Opportunities are available during the academic year and summer terms.
Will study abroad enhance this major?
Although study abroad is not an integral part of your studies in mathematics, it can provide you with exposure to a different culture and a different educational system, and, as such, can be very fulfilling. You may also want to participate in the Budapest Mathematical Seminar or similar programs in your junior year. Keep in mind, however, that study abroad requires careful planning. If you are seriously considering studying abroad, you should consult with the Director of Undergraduate Studies as early in your program as possible in order to plan your major accordingly and to incorporate study abroad courses that are compatible with your major in mathematics.
How might a sample track or course of study look?
In the sample tracks listed below courses required for the major are listed in bold face. You should understand that these sample tracks are just models and that your program might look quite different.
1. Starting with Calculus I
If you start with Calculus I-II your first year, you may either continue with Calculus III – IV, or, ideally, switch to Honors Math A and B in your sophomore year. Junior year you should either take the Introduction to Modern Algebra or the Introduction to Modern Analysis. Some students find it preferable to substitute other analysis courses for Introduction to Modern Analysis. The following is a possible program if you were to choose to complete the entire Calculus I sequence and substitute Partial Differential Equations (PDE) and Complex Variables for the Modern Analysis sequence:
| First Year | Second Year | Third Year | Fourth Year | |
| Fall | Calculus I | Calculus III | Differential Geometry | Modern Algebra |
| Introduction to Higher Mathematics | Ordinary Differential Equations | Seminar | ||
| Spring | Calculus II | Calculus IV | Complex Variables | Modern Algebra |
| Linear Algebra | PDE (Analysis Sequence) | Making, Breaking Codes |
Your program might look something like this if you were to choose to switch to Honors Math in your second year (note that Linear Algebra is not required):
| First Year | Second Year | Third Year | Fourth Year | |
| Fall | Calculus I | Honors Math A | Modern Algebra I | Modern Analysis |
| Number Theory and Cryptography | Ordinary Differential Equations | Honors Complex Variables | ||
| Spring | Calculus II | Honors Math B | Modern Algebra II | Modern Analysis II |
| Making, Breaking Codes | PDE | Seminar |
2. Starting with Honors Math A
If you begin with Honors Math A, your program will be a very accelerated one, and you will not need to take Linear Algebra, as the material is covered in Honors Math. Because you will receive six points of advanced credit and because Honors Math A and B are worth 4 points each, you will need only 10 additional math and cognate courses to complete the major.
| First Year | Second Year | Third Year | Fourth Year | |
| Fall | Honors Math A | Modern Algebra I | Modern Analysis I | Graduate Class 1 |
| Honors Complex Variables | Topology | Seminar | ||
| Spring | Honors Math B | Modern Algebra II | Modern Analysis II | Graduate Class 2 |
| Probability Theory | Differentiable Manifolds |
3. Starting with Calculus III
If you start with Calculus III and take Linear Algebra in your sophomore year, your program might look something like this:
| First Year | Second Year | Third Year | Fourth Year | |
| Fall | Calculus III | Linear Algebra | Modern Analysis I | Modern Algebra I |
| Number Theory and Cryptography | Differential Geometry | |||
| Spring | Calculus IV | Combinatorics | Modern Analysis II | Modern Algebra II |
| Complex Variables | Seminar |
You can accelerate this program by taking Linear Algebra in the first year, an option that will allow you to sample graduate courses in your senior year and/or to take more mathematics courses than required by the major.
| First Year | Second Year | Third Year | Fourth Year | |
| Fall | Calculus III | Modern Algebra I | Modern Analysis I | Graduate Class |
| Differential Geometry | Seminar | Topology | ||
| Spring | Calculus IV | Modern Algebra II | Modern Analysis II | Differentiable Manifolds |
| Linear Algebra | Complex Variables |
How does one receive departmental honors?
To be recommended to the College Committee on Honors, Awards, and Prizes, which makes the final decisions on all honors’ recipients, you must have a GPA of 3.63 in the major and have completed a senior thesis of merit. For more information on researching and writing the senior thesis and on departmental honors, you should consult with the Director of Undergraduate Studies.
What awards and prizes are sponsored by the department?
The following cash prizes are awarded annually to students with the highest score on a special departmental exam:
- Professor Van Amringe Mathematical Prize
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John Dash Van Buren Jr. Prize in Mathematics
Are there any student clubs, committees, and/or activities offered within or through the department?
Yes. The Undergraduate Mathematics Society is the department’s undergraduate club. Detailed information on membership, Society-sponsored seminars and activities, and archival resources are available on the Society’s Web site. The department also sponsors workshops and weekly seminars in mathematics, and posts information about special lectures, conferences, and seminars at nearby schools.
In addition, the Association for Women in Mathematics Columbia Chapter connects students and professors interested in mathematics at Columbia University and Barnard College as part of a broader effort to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment of women and girls in the STEM fields.
What career opportunities follow upon study in this field?
If you graduate with a bachelor’s degree in mathematics you will have the foundation for a broad range of positions in business, industry, government, and education. Companies in the computer and communications industries employ many mathematicians, as do oil companies, banks, insurance companies, and consulting firms. Almost every bureau and branch of the federal government, including the Department of Health and Human Services, the General Accounting Office, the Department of Energy, the Department of Defense, the National Aeronautics and Space Administration, and the National Security Agency, employ mathematicians in various capacities. Your Columbia College foundation in general education combined with your major in mathematics will qualify you for a vast array of employment opportunities in nearly any field. You may also decide to continue your study of mathematics at the graduate level, especially if you are interested in research and/or teaching at the college level.
Whom should I contact about graduate study in this field?
Departmental advisors can offer advice about and help with graduate school applications. The Mathematics department also runs a Master’s degree program in mathematical finance and a Ph.D. program in mathematics.
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