# Mathematics

**Departmental Undergraduate Office:** 410 Mathematics; 212-854-2432

http://www.math.columbia.edu/

**Director of Undergraduate Studies:** Prof. Ovidiu Savin, 409 Mathematics; 212-854-8233; savin@math.columbia.edu

**Calculus Director:** Prof. Michael Woodbury; 525 Mathematics; 212-854-2849; woodbury@math.columbia.edu

**Computer Science-Mathematics Adviser:** Prof. Eric Urban, 608 Mathematics; 212-854-6362; urban@math.columbia.edu

**Economics-Mathematics Advisers:**

*Mathematics*: Prof. Julien Dubedat, 601 Mathematics; 212-854-8806; jd2653@columbia.edu

*Economics*: Dr. Susan Elmes, 1006 International Affairs Building; 212-854-9124; se5@columbia.edu

**Mathematics-Statistics Advisers:**

*Mathematics*: Prof. Julien Dubedat, 601 Mathematics; 212-854-8806; dubedat@math.columbia.edu

*Statistics*: Prof. Banu Baydil, 611 Watson; 212-851-2132; bb2717@columbia.edu

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The major in mathematics is an introduction to some of the highlights of the development of theoretical mathematics over the past four hundred years from a modern perspective. This study is also applied to many problems, both internal to mathematics and arising in other disciplines such as physics, cryptography, and finance.

Majors begin by taking either Honors mathematics or the calculus sequence. Students who do not take MATH UN1207 Honors Mathematics A and MATH UN1208 Honors Mathematics B normally take MATH UN2010 Linear Algebra in the second year. Following this, majors begin to learn some aspects of the main branches of modern mathematics: algebra, analysis, and geometry; as well as some of their subdivisions and hybrids (e.g., number theory, differential geometry, and complex analysis). As the courses become more advanced, they also become more theoretical and proof-oriented and less computational.

Aside from the courses offered by the Mathematics Department, cognate courses in areas such as astronomy, chemistry, physics, probability, logic, economics, and computer science can be used toward the major. A cognate course must be a *2000*-level (or higher) course and must be approved by the director of undergraduate studies. In general, a course not taught by the Mathematics Department is a cognate course for the mathematics major if either (a) it has at least two semesters of calculus as a stated prerequisite, or (b) the subject matter in the course is mathematics beyond an elementary level, such as PHIL UN3411 Symbolic Logic, in the Philosophy Department, or COMS W3203 Discrete Mathematics: Introduction to Combinatorics and Graph Theory, in the Computer Science Department.

Another requirement for majors is participation in an undergraduate seminar, usually in the junior or senior year. In these seminars, students gain experience in learning an advanced topic and lecturing on it. In order to be eligible for departmental honors, majors must write a senior thesis.

## Courses for First-Year Students

The systematic study of mathematics begins with one of the following three alternative calculus and linear algebra sequences:

Code | Title | Points |
---|---|---|

MATH UN1101 - MATH UN1102 - MATH UN1201 - MATH UN1202 - MATH UN2010 | Calculus I and Calculus II and Calculus III and Calculus IV and Linear Algebra | |

MATH UN1101 - MATH UN1102 - MATH UN1205 - MATH UN2010 | Calculus I and Calculus II and Accelerated Multivariable Calculus and Linear Algebra | |

MATH UN1101 - MATH UN1102 - MATH UN1207 - MATH UN1208 | Calculus I and Calculus II and Honors Mathematics A and Honors Mathematics B |

Credit is allowed for only one calculus and linear algebra sequence.

*Calculus I, II * is a standard course in single-variable differential and integral calculus; *Calculus III, IV *is a standard course in multivariable differential and integral calculus; *Accelerated Multivariable Calculus *is an accelerated course in multivariable differential and integral calculus.

While *Calculus II* is no longer a prerequisite for *Calculus III*, students are strongly urged to take it before taking *Calculus III*. In particular, students thinking of majoring or concentrating in mathematics or one of the joint majors involving mathematics should take *Calculus II* before taking *Calculus III*. Note that* Calculus II* is a prerequisite for Accelerated Multivariable Calculus, and both *Calculus II* and *Calculus III* are prerequisites for *Calculus IV*.

The third sequence, *Honors Mathematics A- B*, is for exceptionally well-qualified students who have strong Advanced Placement scores. It covers multivariable calculus (MATH UN1201 Calculus III- MATH UN1202 Calculus IV) and linear algebra (MATH UN2010 Linear Algebra), with an emphasis on theory.

MATH UN1003 College Algebra and Analytic Geometry does not count toward the degree. Students who take this course do not receive college credit.

## Advanced Placement

The department grants 3 credits for a score of 4 or 5 on the AP Calculus AB exam provided students complete MATH UN1102 Calculus II or MATH UN1201 Calculus III with a grade of C or better. The department grants 3 credits for a score of 4 on the AP Calculus BC exam provided students complete MATH UN1102 Calculus II or MATH UN1201 Calculus III with a grade of C or better. The department grants 6 credits for a score of 5 on the AP Calculus BC exam provided students complete MATH UN1201 Calculus III or MATH UN1205 Accelerated Multivariable Calculus MATH UN1207 Honors Mathematics A with a grade of C or better. Students can receive credit for only one calculus sequence.

## Placement in the Calculus Sequences

### Calculus I

Students who have essentially mastered a precalculus course and those who have a score of 3 or less on an Advanced Placement (AP) exam (either AB or BC) should begin their study of calculus with MATH UN1101 Calculus I.

### Calculus II and III

Students with a score of 4 or 5 on the AB exam, 4 on the BC exam, or those with no AP score but with a grade of A in a full year of high school calculus may begin with either MATH UN1102 Calculus II or* MATH UN1201 Calculus III. *
Note that such students who decide to start with *Calculus III *may still need to take* Calculus II *since it is a requirement or prerequisite for other courses. In particular, they MUST take *Calculus II *before going on to MATH UN1202 Calculus IV. Students with a score of 5 on the BC exam may begin with *Calculus III *and do not need to take *Calculus II*.

Those with a score of 4 or 5 on the AB exam or 4 on the BC exam may receive 3 points of AP credit upon completion of *Calculus II *with a grade of C or higher. Those students with a score of 5 on the BC exam may receive 6 points of AP credit upon completion of *Calculus III *with a grade of C or higher.

### Accelerated Multivariable Calculus

Students with a score of 5 on the AP BC exam or 7 on the IB HL exam may begin with MATH UN1205 Accelerated Multivariable Calculus. Upon completion of this course with a grade of C or higher, they may receive 6 points of AP credit.

### Honors Mathematics A

Students who want a proof-oriented theoretical sequence and have a score of 5 on the BC exam may begin with MATH UN1207 Honors Mathematics A, which is especially designed for mathematics majors. Upon completion of this course with a grade of C or higher, they may receive 6 points of AP credit.

## Transfers Inside the Calculus Sequences

Students who wish to transfer from one calculus course to another are allowed to do so beyond the date specified on the Academic Calendar. They are considered to be adjusting their level, not changing their program. However, students must obtain the approval of the new instructor and their advising dean prior to reporting to the Office of the Registrar.

## Grading

No course with a grade of D or lower can count toward the major, interdepartmental major, or concentration. Students who are doing a double major cannot double count courses for their majors.

## Departmental Honors

In order to be eligible for departmental honors, majors must write a senior thesis. To write a senior thesis, students must register for MATH UN3999 Senior Thesis in Mathematics in the fall semester of their senior year. Normally no more than 10% of graduating majors receive departmental honors in a given academic year.

## Professors

- Mohammed Abouzaid
- David A. Bayer (Barnard)
- Simon Brendle
- Ivan Corwin
- Panagiota Daskalopoulos
- Aise Johan de Jong
- Robert Friedman
- Dorian Goldfeld
- Brian Greene
- Richard Hamilton
- Michael Harris
- Ioannis Karatzas
- Mikhail Khovanov
- Igor Krichever
- Chiu-Chu Liu
- Dusa McDuff (Barnard)
- Walter Neumann (Barnard)
- Andrei Okounkov
- D. H. Phong
- Henry Pinkham
- Ovidiu Savin
- Michael Thaddeus (Department Chair)
- Eric Urban
- Mu-Tao Wang

## Associate Professors

- Daniela De Silva (Barnard)
- Julien Dubedat

## Assistant Professors

- Chao Li
- Francesco Lin
- Giulia Sacca
- Will Sawin

## J.F. Ritt Assistant Professors

- Konstantin Aleshkin
- Evgeni Dimitrov
- Nathan Dowlin
- Alexandra Florea
- Florian Johne
- Inbar Klang
- Shotaro Makisumi
- Konstantin Matetski
- S. Michael Miller
- Henri Roesch
- Nicholas Salter
- Gus Schrader
- Akash Sengupta
- Kyler Siegel
- Yi Sun
- Evan Warner
- Hui Yu
- Yihang Zhu

## Senior Lecturers in Discipline

- Lars Nielsen
- Mikhail Smirnov
- Peter Woit

## Lecturers in Discipline

- Michael Woodbury

## On Leave

- Profs. Florea, Krichever, Neumann, Roesch, Sacca, Salter, Sawin
*(Fall 2019)* - Profs. Abouzaid, Hamilton, Neumann, Roesch, Sacca, Salter, Sawin
*(Spring 2020)*

## Major in Mathematics

The major requires 40-42 points as follows:

Code | Title | Points |
---|---|---|

Select one of the following three calculus and linear algebra sequences (13-15 points including Advanced Placement Credit): | ||

Calculus I and Calculus II and Calculus III and Calculus IV and Linear Algebra | ||

Calculus I and Calculus II and Accelerated Multivariable Calculus and Linear Algebra | ||

Calculus I and Calculus II and Honors Mathematics A and Honors Mathematics B | ||

15 points in the following required courses: | ||

Undergraduate Seminars in Mathematics I and Undergraduate Seminars in Mathematics II (at least one term) | ||

Introduction to Modern Algebra I and Introduction to Modern Algebra II | ||

Introduction To Modern Analysis I and Introduction To Modern Analysis II | ||

12 points in any combination of mathematics and cognate courses. ^{**} |

* | Students who are not contemplating graduate study in mathematics may replace one or both of the two terms of MATH GU4061- MATH GU4062 by one or two of the following courses: MATH UN2500 Analysis and Optimization, MATH UN3007 Complex Variables, MATH UN3028 Partial Differential Equations, or MATH GU4032 Fourier Analysis. |

** | A course not taught by the Mathematics Department is a cognate course for the mathematics major if either (a) it has at least two semesters of calculus as a stated prerequisite and is a |

The program of study should be planned with a departmental adviser before the end of the sophomore year. Majors who are planning on graduate studies in mathematics are urged to obtain a reading knowledge of one of the following languages: French, German, or Russian.

Majors are offered the opportunity to write an honors senior thesis under the guidance of a faculty member. Interested students should contact the director of undergraduate studies.

## Major in Applied Mathematics

The major requires 38-40 points as follows:

Code | Title | Points |
---|---|---|

Select one of the following three calculus and linear algebra sequences (13-15 points including Advanced Placement Credit): | ||

Calculus I and Calculus II and Calculus III and Calculus IV and Linear Algebra | ||

Calculus I and Calculus II and Accelerated Multivariable Calculus and Linear Algebra | ||

Calculus I and Calculus II and Honors Mathematics A and Honors Mathematics B | ||

Select one of the following three courses: | ||

Analysis and Optimization | ||

Fourier Analysis | ||

Introduction To Modern Analysis I | ||

APMA E4901 | Seminar: Problem in Applied Mathematics (junior year) | |

APMA E4903 | Seminar: Problems in Applied Mathematics (senior year) | |

18 points in electives, selected from the following (other courses may be used with the approval of the Applied Mathematics Committee): | ||

Analysis and Optimization | ||

Complex Variables | ||

or MATH GU4065 | Honors Complex Variables | |

or APMA E4204 | Functions of a Complex Variable | |

Ordinary Differential Equations | ||

Partial Differential Equations | ||

or APMA E4200 | Partial Differential Equations | |

or APMA E6301 | Analytic methods for partial differential equations | |

Fourier Analysis | ||

Computational Math: Introduction to Numerical Methods | ||

Introduction to Dynamical Systems | ||

Applied Functional Analysis | ||

Introduction to Biophysical Modeling |

## Major in Computer Science–Mathematics

The goal of this interdepartmental major is to provide substantial background in each of these two disciplines, focusing on some of the parts of each which are closest to the other. Students intending to pursue a Ph.D. program in either discipline are urged to take additional courses, in consultation with their advisers.

The major requires 20 points in computer science, 19-21 points in mathematics, and two 3-point electives in either computer science or mathematics.

Code | Title | Points |
---|---|---|

Computer Science | ||

COMS W1004 | Introduction to Computer Science and Programming in Java | |

or COMS W1007 | Honors Introduction to Computer Science | |

COMS W3134 | Data Structures in Java | |

or COMS W3137 | Honors Data Structures and Algorithms | |

COMS W3157 | Advanced Programming | |

COMS W3203 | Discrete Mathematics: Introduction to Combinatorics and Graph Theory | |

COMS W3261 | Computer Science Theory | |

CSEE W3827 | Fundamentals of Computer Systems | |

Mathematics | ||

Select one of the following three calculus and linear algebra sequences (13-15 points including Advanced Placement Credit): | ||

MATH UN1101 - MATH UN1102 - MATH UN1201 - MATH UN1202 - MATH UN2010 | Calculus I and Calculus II and Calculus III and Calculus IV and Linear Algebra | |

MATH UN1101 - MATH UN1102 - MATH UN1205 - MATH UN2010 | Calculus I and Calculus II and Accelerated Multivariable Calculus and Linear Algebra | |

MATH UN1101 - MATH UN1102 - MATH UN1207 - MATH UN1208 | Calculus I and Calculus II and Honors Mathematics A and Honors Mathematics B | |

MATH UN3951 | Undergraduate Seminars in Mathematics I | |

or MATH UN3952 | Undergraduate Seminars in Mathematics II | |

MATH GU4041 | Introduction to Modern Algebra I | |

Electives | ||

Select two of the following courses: | ||

Analysis of Algorithms I | ||

Numerical Algorithms and Complexity | ||

Combinatorics | ||

Analysis and Optimization | ||

Complex Variables | ||

Number Theory and Cryptography | ||

Differential Geometry | ||

Topology | ||

Introduction To Modern Analysis I |

## Major in Economics-Mathematics

For a description of the joint major in economics-mathematics, see the *Economics* section of this bulletin.

For a description of the joint major in economics-mathematics, see the *Economics* section of this bulletin.

## Major in Mathematics-Statistics

The program is designed to prepare the student for: (1) a career in industries such as finance and insurance that require a high level of mathematical sophistication and a substantial knowledge of probability and statistics, and (2) graduate study in quantitative disciplines. Students choose electives in finance, actuarial science, operations research, or other quantitative fields to complement requirements in mathematics, statistics, and computer science.

Code | Title | Points |
---|---|---|

Mathematics | ||

Select one of the following sequences: | ||

Calculus I and Calculus II and Calculus III and Linear Algebra and Analysis and Optimization | ||

Calculus I and Calculus II and Accelerated Multivariable Calculus and Linear Algebra and Analysis and Optimization | ||

Honors Mathematics A and Honors Mathematics B and Analysis and Optimization (with approval from the adviser) | ||

Statistics | ||

Introductory Course | ||

STAT UN1201 | Calculus-Based Introduction to Statistics | |

Required Courses | ||

STAT GU4203 | PROBABILITY THEORY | |

STAT GU4204 | Statistical Inference | |

STAT GU4205 | Linear Regression Models | |

Select one of the following courses: | ||

Elementary Stochastic Processes | ||

Stochastic Processes for Finance | ||

STOCHASTC PROCSSES-APPLIC | ||

Stochastic Methods in Finance | ||

Computer Science | ||

Select one of the following courses: | ||

Introduction to Computer Science and Programming in Java | ||

Introduction to Computer Science and Programming in MATLAB | ||

Introduction to Computing for Engineers and Applied Scientists | ||

Honors Introduction to Computer Science | ||

or an advanced computer science offering in programming | ||

Electives | ||

An approved selection of three advanced courses in mathematics, statistics, applied mathematics, industrial engineering and operations research, computer science, or approved mathematical methods courses in a quantitative discipline. At least one elective must be a Mathematics Department course numbered 3000 or above. |

Students interested in modeling applications are recommended to take MATH UN3027 Ordinary Differential Equations and MATH UN3028 Partial Differential Equations.

Students interested in finance are recommended to take MATH GR5010 Introduction to the Mathematics of Finance, STAT GU4261 Statistical Methods in Finance, and STAT GU4221 Time Series Analysis.

Students interested in graduate study in mathematics or in statistics are recommended to take MATH GU4061 Introduction To Modern Analysis I and MATH GU4062 Introduction To Modern Analysis II.

Students preparing for a career in actuarial science are encouraged to replace STAT GU4205 Linear Regression Models with STAT GU4282 Linear Regression and Time Series Methods , and to take among their electives STAT GU4281 Theory of Interest .

## Concentration in Mathematics

The concentration requires the following:

Code | Title | Points |
---|---|---|

Mathematics | ||

Select one of the following three multivariable calculus and linear algebra sequences: | ||

Calculus III and Calculus IV and Linear Algebra | ||

Accelerated Multivariable Calculus and Linear Algebra | ||

Honors Mathematics A and Honors Mathematics B | ||

Additional Courses | ||

Select at least 12 additional points from any of the courses offered by the department numbered 2000 or higher. |

For mathematics courses taken in other departments, consult with the director of undergraduate studies.

Any course given by the Mathematics department fulfills the General Studies quantitative reasoning requirement when passed with a satisfactory letter grade.

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