Computer Science  Mathematics
Departmental Office: 410 Mathematics; 2128542432
http://www.math.columbia.edu/
Director of Undergraduate Studies: Prof. ChiuChu (Melissa) Liu, 623 Mathematics; 2128542499; ccliu@math.columbia.edu
Departmental Adviser: Prof. MuTao Wang, 514 Mathematics; 2128543052; mtwang@math.columbia.edu
Computer ScienceMathematics Adviser: Prof. Patrick X. Gallagher, 411 Mathematics; 2128544346; pxg@math.columbia.edu
EconomicsMathematics Advisers:
Mathematics: Prof. Julien Dubedat, 601 Mathematics; 2128548806; jd2653@columbia.edu
Economics: Dr. Susan Elmes, 1006 International Affairs Building; 2128549124; se5@columbia.edu
MathematicsStatistics Advisers:
Mathematics: Prof. Julien Dubedat, 601 Mathematics; 2128548806; dubedat@math.columbia.edu
Statistics: Prof. Banu Baydil, 611 Watson; 2128512132; bb2717@columbia.edu
Calculus Director: Prof. Michael Thaddeus, 414 Mathematics; 2128544308; thaddeus@math.columbia.edu
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 prooforiented 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 2000level (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 FirstYear Students
The systematic study of mathematics begins with one of the following two alternative sequences:
Code  Title  Points 

MATH UN1101  MATH UN1102  MATH UN1201  MATH UN1202 
Calculus I and Calculus II and Calculus III and Calculus IV 

MATH UN1207  MATH UN1208 
Honors Mathematics A and Honors Mathematics B 
Credit is allowed for only one calculus sequence.
The first sequence, Calculus I, II, III, IV, is a standard course in 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 both Calculus II and Calculus III are prerequisites for Calculus IV.
The second sequence, Honors Mathematics A B, is for exceptionally wellqualified 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 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.
Honors Mathematics A
Students who want a prooforiented 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. 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

Panagiota Daskalopoulos

Aise Johan de Jong

Robert Friedman

Patrick X. Gallagher

Dorian Goldfeld

Brian Greene

Richard Hamilton

Michael Harris

Ioannis Karatzas

Mikhail Khovanov

Igor Krichever

ChiuChu Liu

Dusa McDuff (Barnard)

Davesh Maulik

Walter Neumann (Barnard)

Andrei Okounkov

D. H. Phong

Henry Pinkham (Department Chair)

Ovidiu Savin

Michael Thaddeus

Eric Urban

MuTao Wang

Wei Zhang
Associate Professors
 Ivan Corwin
 Daniela De Silva (Barnard)
 Julien Dubedat
Assistant Professors
 n/a
J.F. Ritt Assistant Professors
 Akram Alishahi
 Guillaume Barraquand
 Hector Chang
 Gabriele Di Cerbo
 Luis Diogo
 Galyna Dobrovolska
 Teng Fei
 Bin Guo
 Daniel HalpernLeistner
 David Hansen
 BoGwang Jeon
 Chao Li
 Hao Shen
 Anton Zeitlin
Senior Lecturers in Discipline

Lars Nielsen

Mikhail Smirnov

Peter Woit
On Leave
 Profs. Abouzaid, Dubedat, Goldfeld, Maulik, Krichever, Zhang (Fall 2016)
 Profs. Corwin, Maulik, Krichever (Spring 2017)
Major in Mathematics
The major requires 42 points as follows (or 41 points if Honors Mathematics AB is taken):
Code  Title  Points 

12 points in Calculus or Honors Mathematics AB, including Advanced Placement credit.  
18 points in mathematics courses numbered 2000 and above, including the following courses:  
Linear Algebra (if Honors Mathematics AB is not taken)  
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 Introduction To Modern Analysis II 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 2000level (or higher) course, 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. In exceptional cases, the director of undergraduate studies may approve the substitution of certain more advanced courses for those mentioned above. 
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 40 points as follows:
Code  Title  Points 

12 points in Calculus or Honors Mathematics AB  
MATH UN2010  Linear Algebra  
Select one of the following 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 a total of at least 47 points: 20 points in computer science, 21 points in mathematics, and two 3point electives in either computer science or mathematics.
Code  Title  Points 

Computer Science  
COMS W1007  Honors Introduction to Computer Science  
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 sequences:  
Calculus I and Calculus II and Calculus III and Calculus IV 

Honors Mathematics A and Honors Mathematics B 

MATH UN2010  Linear Algebra  
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 EconomicsMathematics
For a description of the joint major in economicsmathematics, see the Economics section of this bulletin.
For a description of the joint major in economicsmathematics, see the Economics section of this bulletin.
Major in MathematicsStatistics
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 

MATH UN1207  MATH UN1208  MATH UN2500 
Honors Mathematics A and Honors Mathematics B and Analysis and Optimization (with approval from the adviser) 

Statistics  
Introductory Courses


Select one of the following courses:


STAT UN1001  Introduction to Statistical Reasoning  
STAT UN1101  Introduction to Statistics  
STAT UN1201  CalculusBased 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  
STAT W4262


Stochastic Processes and Applications  
Stochastic Methods in Finance  
Stochastic Control and Applications 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 W4281 .
Concentration in Mathematics
The concentration requires the following:
Code  Title  Points 

Mathematics  
Select one of the two following multivariable calculus and linear algebra sequences:  
Calculus III and Calculus IV 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.