Applied Mathematics

Departmental Office: 410 Mathematics; 212-854-2432
http://www.math.columbia.edu/

Director of Undergraduate Studies: Prof. Ovidiu Savin, 409 Mathematics; 212-854-8233;

Departmental Adviser: Prof. Michael Woodbury

Computer Science-Mathematics Adviser: Prof. Patrick X. Gallagher, 411 Mathematics; 212-854-4346; pxg@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

Calculus Director: Prof. Michael Woodbury

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:

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
  • 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
  • Chiu-Chu Liu
  • Dusa McDuff (Barnard)
  • Walter Neumann (Barnard
  • Andrei Okounkov
  • D. H. Phong
  • Henry Pinkham (Department Chair)
  • Ovidiu Savin
  • Michael Thaddeus
  • Eric Urban
  • Mu-Tao 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
  • Teng Fei
  • Bin Guo
  • David Hansen
  • Chao Li
  • Shotaro Makisumi
  • Joanna Nelson
  • Gus Schrader
  • Shrenik Shah
  • Hao Shen
  • Evan Warner
  • Hui Yu
  • Yihang Zhu

Senior Lecturers in Discipline

  • Lars Nielsen
  • Mikhail Smirnov
  • Peter Woit

Lecturers in Discipline

  • Michael Woodbury

On Leave

  • Profs. Daskalopoulos, Liu, Okounkov, Pinkham, Wang, Zhang (Fall 2017)
  • Profs. Daskalopoulos, Liu, Makisumi, Okounkov, Pinkham, Wang, Zhang (Spring 2018)

Major in Mathematics

The major requires 40-42 points as follows:

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 OptimizationMATH 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 2000-level (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 38-40 points as follows:

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 E4901Seminar: Problem in Applied Mathematics (junior year)
APMA E4903Seminar: 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
Honors Complex Variables
Functions of a Complex Variable
Ordinary Differential Equations
Partial Differential Equations
Partial Differential Equations
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.

Computer Science
COMS W1007Honors Introduction to Computer Science
COMS W3137Honors Data Structures and Algorithms
COMS W3157Advanced Programming
COMS W3203Discrete Mathematics: Introduction to Combinatorics and Graph Theory
COMS W3261Computer Science Theory
CSEE W3827Fundamentals of Computer Systems
Mathematics
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
MATH UN1101
 - MATH UN1102
 - MATH UN1207
 - MATH UN1208
Calculus I
and Calculus II
and Honors Mathematics A
and Honors Mathematics B
MATH UN3951Undergraduate Seminars in Mathematics I
or MATH UN3952 Undergraduate Seminars in Mathematics II
MATH GU4041Introduction 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.

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 Courses
Select one of the following courses:
STAT UN1001Introduction to Statistical Reasoning
STAT UN1101Introduction to Statistics
STAT UN1201Calculus-Based Introduction to Statistics
Required Courses
STAT GU4203PROBABILITY THEORY
STAT GU4204Statistical Inference
STAT GU4205Linear Regression Models
Select one of the following courses:
Elementary Stochastic Processes
Stochastic Processes for Finance
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 GU4281 Theory of Interest .


Concentration in Mathematics

The concentration requires the following:

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.

MATH UN1003 College Algebra and Analytic Geometry. 3 points.

Prerequisites: score of 550 on the mathematics portion of the SAT completed within the last year or the appropriate grade on the General Studies Mathematics Placement Examination.

Columbia College students do not receive any credit for this course and must see their CSA advising dean. For students who wish to study calculus but do not know analytic geometry. Algebra review, graphs and functions, polynomial functions, rational functions, conic sections, systems of equations in two variables, exponential and logarithmic functions, trigonometric functions and trigonometric identities, applications of trigonometry, sequences, series, and limits.

Spring 2017: MATH UN1003
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1003 001/62274 M W 6:10pm - 8:00pm
417 Mathematics Building
Qirui Li 3 15/30
MATH 1003 002/26568 T Th 12:10pm - 2:00pm
603 Hamilton Hall
Feiqi Jiang 3 30/30
Fall 2017: MATH UN1003
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1003 001/26477 M W 6:10pm - 8:00pm
407 Mathematics Building
Dmitry Korb 3 16/30
MATH 1003 002/26739 T Th 12:10pm - 2:00pm
103 Knox Hall
Darren Gooden 3 25/30

MATH UN1101 Calculus I. 3 points.

Prerequisites: (see Courses for First-Year Students). Functions, limits, derivatives, introduction to integrals, or an understanding of pre-calculus will be assumed.

,

The Help Room in 333 Milbank Hall (Barnard College) is open during the day, Monday through Friday, to students seeking individual help from the teaching assistants. (SC)

Spring 2017: MATH UN1101
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1101 001/69002 M W 8:40am - 9:55am
407 Mathematics Building
Ashwin Deopurkar 3 22/30
MATH 1101 002/18872 M W 10:10am - 11:25am
407 Mathematics Building
Mitchell Faulk 3 26/30
MATH 1101 003/11439 M W 6:10pm - 7:25pm
407 Mathematics Building
Minghan Yan 3 24/30
MATH 1101 004/14059 T Th 11:40am - 12:55pm
207 Mathematics Building
Yu-Shen Lin 3 83/100
MATH 1101 005/24071 T Th 1:10pm - 2:25pm
407 Mathematics Building
Changjian Su 3 18/30
MATH 1101 006/12207 M W 11:40am - 12:55pm
407 Mathematics Building
Xiaowei Tan 3 16/30
MATH 1101 007/11790 T Th 10:10am - 11:25am
507 Mathematics Building
Beomjun Choi 3 10/30
MATH 1101 008/29542 T Th 6:10pm - 7:25pm
307 Mathematics Building
Zhechi Cheng 3 8/30
Fall 2017: MATH UN1101
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1101 001/07384 M W 8:40am - 9:55am
Room TBA
Dusa McDuff 3 29/100
MATH 1101 002/17570 M W 10:10am - 11:25am
312 Mathematics Building
Chao Li 3 44/100
MATH 1101 003/29604 M W 11:40am - 12:55pm
312 Mathematics Building
Chao Li 3 67/100
MATH 1101 004/73071 M W 2:40pm - 3:55pm
417 Mathematics Building
Michael Woodbury 3 21/64
MATH 1101 005/18565 M W 4:10pm - 5:25pm
417 Mathematics Building
Michael Woodbury 3 8/64
MATH 1101 006/73884 T Th 10:10am - 11:25am
407 Mathematics Building
Oleksandr Kravets 3 15/30
MATH 1101 007/26909 T Th 11:40am - 12:55pm
407 Mathematics Building
Shuai Wang 3 16/30
MATH 1101 008/64016 T Th 1:10pm - 2:25pm
203 Mathematics Building
Alexander Perry 3 9/100
MATH 1101 009/21826 T Th 4:10pm - 5:25pm
203 Mathematics Building
Ila Varma 3 25/100
MATH 1101 010/67061 T Th 6:10pm - 7:25pm
207 Mathematics Building
Linh Truong 3 9/100

MATH UN1102 Calculus II. 3 points.

Prerequisites: MATH UN1101 or the equivalent.

Methods of integration, applications of the integral, Taylor's theorem, infinite series. (SC) 

Spring 2017: MATH UN1102
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1102 001/22373 M W 11:40am - 12:55pm
307 Mathematics Building
Zijun Zhou 3 15/35
MATH 1102 002/71505 M W 2:40pm - 3:55pm
520 Mathematics Building
Noah Arbesfeld 3 28/30
MATH 1102 003/76382 M W 10:10am - 11:25am
417 Mathematics Building
Zhijie Huang 3 5/30
MATH 1102 004/75150 T Th 1:10pm - 2:25pm
203 Mathematics Building
Wei Zhang 3 78/100
MATH 1102 005/16760 T Th 6:10pm - 7:25pm
407 Mathematics Building
Elliott Stein 3 33/35
Fall 2017: MATH UN1102
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1102 001/67192 M W 10:10am - 11:25am
417 Mathematics Building
Bin Guo 3 19/64
MATH 1102 002/70122 M W 2:40pm - 3:55pm
312 Mathematics Building
Li-Cheng Tsai 3 19/100
MATH 1102 003/26180 M W 4:10pm - 5:25pm
407 Mathematics Building
Yi Sun 3 8/35
MATH 1102 004/22286 T Th 10:10am - 11:25am
203 Mathematics Building
Vivek Pal 3 20/100
MATH 1102 005/29410 T Th 6:10pm - 7:25pm
407 Mathematics Building
Xuan Wu 3 5/30

MATH UN1201 Calculus III. 3 points.

Prerequisites: MATH UN1101 or the equivalent

Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers. (SC) 

Spring 2017: MATH UN1201
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1201 001/21153 M W 8:40am - 9:55am
207 Mathematics Building
Galyna Dobrovolska 3 60/100
MATH 1201 002/25373 M W 11:40am - 12:55pm
312 Mathematics Building
Gabriele Di Cerbo 3 89/100
MATH 1201 003/71946 M W 1:10pm - 2:25pm
312 Mathematics Building
Gabriele Di Cerbo 3 82/100
MATH 1201 004/70892 M W 6:10pm - 7:25pm
312 Mathematics Building
Teng Fei 3 33/100
MATH 1201 005/05518 T Th 8:40am - 9:55am
304 Barnard Hall
Daniela De Silva 3 94/100
MATH 1201 006/07691 T Th 10:10am - 11:25am
405 Milbank Hall
Daniela De Silva 3 87/100
MATH 1201 007/67220 T Th 11:40am - 12:55pm
717 Hamilton Hall
Yoel Groman 3 51/100
Fall 2017: MATH UN1201
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1201 001/27988 M W 10:10am - 11:25am
203 Mathematics Building
Joanna Nelson 3 77/100
MATH 1201 002/15820 M W 11:40am - 12:55pm
203 Mathematics Building
Teng Fei 3 32/100
MATH 1201 003/62151 M W 1:10pm - 2:25pm
203 Mathematics Building
Joanna Nelson 3 54/100
MATH 1201 004/68024 M W 4:10pm - 5:25pm
312 Mathematics Building
Jeffrey Kuan 3 61/100
MATH 1201 005/06129 T Th 8:40am - 9:55am
Room TBA
Daniela De Silva 3 15/100
MATH 1201 006/63259 T Th 4:10pm - 5:25pm
312 Mathematics Building
Akram Alishahi 3 23/100
MATH 1201 007/27974 T Th 5:40pm - 6:55pm
312 Mathematics Building
Akram Alishahi 3 19/100

MATH UN1202 Calculus IV. 3 points.

Prerequisites: MATH UN1102 or MATH UN1201 or the equivalent

Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series. (SC)

Spring 2017: MATH UN1202
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1202 001/27342 M W 8:40am - 9:55am
312 Mathematics Building
Ovidiu Savin 3 62/100
MATH 1202 003/66692 M W 10:10am - 11:25am
312 Mathematics Building
Ovidiu Savin 3 59/100
MATH 1202 004/71308 T Th 11:40am - 12:55pm
312 Mathematics Building
Robert Friedman 3 81/100
Fall 2017: MATH UN1202
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1202 001/24333 M W 6:10pm - 7:25pm
312 Mathematics Building
Mikhail Smirnov 3 100/100
MATH 1202 002/87449 T Th 2:40pm - 3:55pm
203 Mathematics Building
Hao Shen 3 30/100

MATH UN1205 Accelerated Multivariable Calculus. 4 points.

Prerequisites: (MATH UN1101 and MATH UN1102)

Vectors in dimensions 2 and 3, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, optimization, Lagrange multipliers, double and triple integrals, line and surface integrals, vector calculus.  This course is an accelerated version of MATH UN1201-1202.  Students taking this course may not receive credit for MATH UN1201 and MATH UN1202.

Fall 2017: MATH UN1205
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1205 001/86696 M W 2:40pm - 3:55pm
207 Mathematics Building
Robert Friedman 4 12/120

MATH UN1207 Honors Mathematics A. 4 points.

Prerequisites:  (see Courses for First-Year Students).  The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC)

Fall 2017: MATH UN1207
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1207 001/11993 M W 1:10pm - 2:25pm
417 Mathematics Building
David Hansen 4 9/64
MATH 1207 002/26797 T Th 1:10pm - 2:25pm
520 Mathematics Building
Evan Warner 4 4/49

MATH UN1208 Honors Mathematics B. 4 points.

Prerequisites: (see Courses for First-Year Students).

The second term of this course may not be taken without the first. Multivariable calculus and linear algebra from a rigorous point of view. Recommended for mathematics majors. Fulfills the linear algebra requirement for the major. (SC)

Spring 2017: MATH UN1208
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 1208 001/27564 M W 2:40pm - 3:55pm
203 Mathematics Building
David Hansen 4 57/100

MATH UN2000 An Introduction to Higher Mathematics. 3 points.

Introduction to understanding and writing mathematical proofs.  Emphasis on precise thinking and the presentation of mathematical results, both in oral and in written form.  Intended for students who are considering majoring in mathematics but wish additional training. CC/GS: Partial Fulfillment of Science Requirement.  BC: Fulfillment of General Education Requirement: Quantitative and Deductive Reasoning (QUA).

Spring 2017: MATH UN2000
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 2000 001/07922 M W 1:10pm - 2:25pm
504 Diana Center
Dusa McDuff 3 34/64
Fall 2017: MATH UN2000
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 2000 001/04201 M W 8:40am - 9:55am
Room TBA
Walter Neumann 3 24/64

MATH UN2002 The Magic of Numbers. 3 points.

BC: Fulfillment of General Education Requirement: Quantitative and Deductive Reasoning (QUA).

In this class, we will cover many interesting aspects of math that can be used in everyday life.  The goal will be to cover fun, exciting topics that don't require any prerequisites, but still capture some of the mystery of mathematics.  We will emphasize discovering concepts in combinatorics (the mathematics of counting), geometry (the mathematics of shapes), number theory (the mathematics of whole numbers) and more.  This class will be interactive and include demonstrations when possible.

Fall 2017: MATH UN2002
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 2002 001/71448 T Th 11:40am - 12:55pm
520 Mathematics Building
Vivek Pal 3 10/40

MATH UN2010 Linear Algebra. 3 points.

Prerequisites: MATH UN1201 or the equivalent.

Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. (SC)

Spring 2017: MATH UN2010
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 2010 001/22319 T Th 8:40am - 9:55am
417 Mathematics Building
Henry Pinkham 3 38/64
MATH 2010 002/74185 T Th 10:10am - 11:25am
312 Mathematics Building
Henry Pinkham 3 47/64
MATH 2010 003/67337 T Th 1:10pm - 2:25pm
614 Schermerhorn Hall
Eric Urban 3 77/110
MATH 2010 004/19348 M W 10:10am - 11:25am
207 Mathematics Building
Guillaume Barraquand 3 86/110
Fall 2017: MATH UN2010
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 2010 001/27023 M W 4:10pm - 5:25pm
207 Mathematics Building
Nathan Dowlin 3 99/100
MATH 2010 002/25355 M W 1:10pm - 2:25pm
312 Mathematics Building
Gus Schrader 3 45/100
MATH 2010 003/03818 T Th 8:40am - 9:55am
Room TBA
David Bayer 3 73/100
MATH 2010 004/02940 T Th 10:10am - 11:25am
Room TBA
David Bayer 3 100/100
MATH 2010 005/18445 T Th 6:10pm - 7:25pm
203 Mathematics Building
Elliott Stein 3 100/100

MATH V2020 Honors Linear Algebra. 3 points.

CC/GS: Partial Fulfillment of Science Requirement
Not offered during 2017-18 academic year.

Prerequisites: MATH V1201.

A more extensive treatment of the material in Math V2010, with increased emphasis on proof. Not to be taken in addition to Math V2010 or Math V1207-Math V1208.

MATH UN2030 Ordinary Differential Equations. 3 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent.

Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications.

Spring 2017: MATH UN2030
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 2030 001/21760 T Th 11:40am - 12:55pm
203 Mathematics Building
Mu-Tao Wang 3 85/100
MATH 2030 002/66814 T Th 2:40pm - 3:55pm
203 Mathematics Building
Mu-Tao Wang 3 81/100
Fall 2017: MATH UN2030
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 2030 001/23233 M W 5:40pm - 6:55pm
203 Mathematics Building
Hector Chang-Lara 3 29/100
MATH 2030 002/12064 T Th 11:40am - 12:55pm
312 Mathematics Building
Guillaume Barraquand 3 46/100

MATH UN2500 Analysis and Optimization. 3 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent and MATH UN2010.

Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control. (SC)

Spring 2017: MATH UN2500
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 2500 001/60299 T Th 8:40am - 9:55am
203 Mathematics Building
Daniel Halpern-Leistne 3 26/100
MATH 2500 002/14794 T Th 10:10am - 11:25am
203 Mathematics Building
Daniel Halpern-Leistne 3 44/100
Fall 2017: MATH UN2500
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 2500 001/73444 M W 11:40am - 12:55pm
207 Mathematics Building
Shotaro Makisumi 3 52/100
MATH 2500 002/26047 M W 2:40pm - 3:55pm
203 Mathematics Building
Shotaro Makisumi 3 65/100

MATH UN3007 Complex Variables. 3 points.

Prerequisites: MATH UN1202 An elementary course in functions of a complex variable.

Fundamental properties of the complex numbers, differentiability, Cauchy-Riemann equations. Cauchy integral theorem. Taylor and Laurent series, poles, and essential singularities. Residue theorem and conformal mapping.(SC)

Spring 2017: MATH UN3007
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 3007 001/73623 T Th 2:40pm - 3:55pm
312 Mathematics Building
Patrick Gallagher 3 80/116

MATH UN3020 Number Theory and Cryptography. 3 points.

Prerequisites: one year of calculus.

Prerequisite: One year of Calculus. Congruences. Primitive roots. Quadratic residues. Contemporary applications.

Spring 2017: MATH UN3020
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 3020 001/76902 M W 10:10am - 11:25am
203 Mathematics Building
Bogwang Jeon 3 42/100

MATH UN3025 Making, Breaking Codes. 3 points.

Prerequisites: (MATH UN1101 and MATH UN1102 and MATH UN1201) and and MATH UN2010.

A concrete introduction to abstract algebra. Topics in abstract algebra used in cryptography and coding theory.

Fall 2017: MATH UN3025
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 3025 001/15106 T Th 2:40pm - 3:55pm
312 Mathematics Building
Dorian Goldfeld 3 97/100

MATH UN3027 Ordinary Differential Equations. 3 points.

Prerequisites: MATH UN1102 and MATH UN1201 or the equivalent.
Corequisites: MATH UN2010

Equations of order one; systems of linear equations. Second-order equations. Series solutions at regular and singular points. Boundary value problems. Selected applications. 

Fall 2017: MATH UN3027
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 3027 001/74190 M W 4:10pm - 5:25pm
203 Mathematics Building
Hector Chang-Lara 3 52/100

MATH UN3028 Partial Differential Equations. 3 points.

Prerequisites: MATH UN3027 and MATH UN2010 or the equivalent

Introduction to partial differential equations. First-order equations. Linear second-order equations; separation of variables, solution by series expansions. Boundary value problems. 

Spring 2017: MATH UN3028
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 3028 001/60863 M W 11:40am - 12:55pm
614 Schermerhorn Hall
Simon Brendle 3 59/100

MATH UN3050 Discrete Time Models in Finance. 3 points.

Prerequisites: (MATH UN1102 and MATH UN1201) or (MATH UN1101 and MATH UN1102 and MATH UN1201) and MATH UN2010 Recommended: MATH UN3027 (or MATH UN2030 and SIEO W3600).

Elementary discrete time methods for pricing financial instruments, such as options. Notions of arbitrage, risk-neutral valuation, hedging, term-structure of interest rates.

Spring 2017: MATH UN3050
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 3050 001/17362 M W 6:10pm - 7:25pm
203 Mathematics Building
Mikhail Smirnov 3 46/64

MATH UN3386 Differential Geometry. 3 points.

Prerequisites: MATH UN1202 or the equivalent.

Local and global differential geometry of submanifolds of Euclidiean 3-space. Frenet formulas for curves. Various types of curvatures for curves and surfaces and their relations. The Gauss-Bonnet theorem.

Fall 2017: MATH UN3386
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 3386 001/77535 T Th 11:40am - 12:55pm
417 Mathematics Building
Richard Hamilton 3 30/49

MATH UN3951 Undergraduate Seminars in Mathematics I. 3 points.

Prerequisites: Two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission.

The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow.

Fall 2017: MATH UN3951
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 3951 001/02944  
Daniela De Silva 3 38/49

MATH UN3952 Undergraduate Seminars in Mathematics II. 3 points.

Prerequisites: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission.

The subject matter is announced at the start of registration and is different in each section. Each student prepares talks to be given to the seminar, under the supervision of a faculty member or senior teaching fellow. Prerequisite: two years of calculus, at least one year of additional mathematics courses, and the director of undergraduate studies' permission.

Spring 2017: MATH UN3952
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 3952 001/00853  
David Bayer 3 41

MATH GU4007 Analytic Number Theory. 3 points.

Prerequisites: MATH UN3007

A one semeser course covering the theory of modular forms, zeta functions, L -functions, and the Riemann hypothesis. Particular topics covered include the Riemann zeta function, the prime number theorem, Dirichlet characters, Dirichlet L-functions, Siegel zeros, prime number theorem for arithmetic progressions, SL (2, Z) and subgroups, quotients of the upper half-plane and cusps, modular forms, Fourier expansions of modular forms, Hecke operators, L-functions of modular forms.

Spring 2017: MATH GU4007
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4007 001/76901 T Th 11:40am - 12:55pm
520 Mathematics Building
Dorian Goldfeld 3 8/49

MATH GU4032 Fourier Analysis. 3 points.

Prerequisites: three terms of calculus and linear algebra or four terms of calculus.

Prerequisite: three terms of calculus and linear algebra or four terms of calculus. Fourier series and integrals, discrete analogues, inversion and Poisson summation formulae, convolution. Heisenberg uncertainty principle. Stress on the application of Fourier analysis to a wide range of disciplines. 

Spring 2017: MATH GU4032
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4032 001/15388 T Th 1:10pm - 2:25pm
417 Mathematics Building
Peter Woit 3 7/49

MATH GU4041 Introduction to Modern Algebra I. 3 points.

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent

The second term of this course may not be taken without the first. Groups, homomorphisms, rings, ideals, fields, polynomials, field extensions, Galois theory.

Spring 2017: MATH GU4041
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4041 001/17587 M W 1:10pm - 2:25pm
203 Mathematics Building
Mikhail Khovanov 3 29/100
Fall 2017: MATH GU4041
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4041 001/08005 M W 10:10am - 11:25am
Room TBA
Walter Neumann 3 70/100

MATH GU4042 Introduction to Modern Algebra II. 3 points.

Prerequisites: MATH UN1102 and MATH UN1202 and MATH UN2010 or the equivalent.

The second term of this course may not be taken without the first. Groups, homomorphisms, rings, ideals, fields, polynomials, field extensions, Galois theory.

Spring 2017: MATH GU4042
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4042 001/16294 T Th 4:10pm - 5:25pm
602 Hamilton Hall
Michael Thaddeus 3 32/100
Fall 2017: MATH GU4042
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4042 001/23554 T Th 10:10am - 11:25am
312 Mathematics Building
Yihang Zhu 3 14/64

MATH GU4043 Algebraic Number Theory. 3 points.

Prerequisites: MATH GU4041 and MATH GU4042 or the equivalent

Algebraic number fields, unique factorization of ideals in the ring of algebraic integers in the field into prime ideals. Dirichlet unit theorem, finiteness of the class number, ramification. If time permits, p-adic numbers and Dedekind zeta function.

Fall 2017: MATH GU4043
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4043 001/18323 M W 10:10am - 11:25am
407 Mathematics Building
Michael Harris 3 13/30

MATH GU4044 Representations of Finite Groups. 3 points.

Prerequisites: MATH UN2010MATH GU4041 or the equivalent.

Finite groups acting on finite sets and finite dimensional vector spaces. Group characters. Relations with subgroups and factor groups. Arithmetic properties of character values. Applications to the theory of finite groups: Frobenius groups, Hall subgroups and solvable groups. Characters of the symmetric groups. Spherical functions on finite groups.

Spring 2017: MATH GU4044
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4044 001/29931 M W 2:40pm - 3:55pm
417 Mathematics Building
Patrick Gallagher 3 15/64

MATH GU4045 Algebraic Curves. 3 points.

Prerequisites: (MATH GU4041 and MATH GU4042) and MATH UN3007

Plane curves, affine and projective varieties, singularities, normalization, Riemann surfaces, divisors, linear systems, Riemann-Roch theorem.

MATH GU4051 Topology. 3 points.

Prerequisites: (MATH UN1202 and MATH UN2010) and rudiments of group theory (e.g., MATH GU4041). MATH UN1208 or MATH GU4061 is recommended, but not required.

Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces.

Fall 2017: MATH GU4051
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4051 001/74453 T Th 4:10pm - 5:25pm
417 Mathematics Building
Michael Thaddeus 3 47/64

MATH W4052 Introduction to Knot Theory. 3 points.

CC/GS: Partial Fulfillment of Science Requirement

Prerequisites: MATH W4051 Topology and / or MATH W4061 Introduction To Modern Analysis I (or equivalents) \nRecommended (can be taken concurrently): MATH V2010 linear algebra, or equivalent

The study of algebraic and geometric properties of knots in R^3, including but not limited to knot projections and Reidemeister's theorm, Seifert surfaces, braids, tangles, knot polynomials, fundamental group of knot complements. Depending on time and student interest, we will discuss more advanced topics like knot concordance, relationship to 3-manifold topology, other algebraic knot invariants.

MATH GU4053 Introduction to Algebraic Topology. 3 points.

Prerequisites: MATH UN2010MATH GU4041MATH GU4051

The study of topological spaces from algebraic properties, including the essentials of homology and the fundamental group. The Brouwer fixed point theorem. The homology of surfaces. Covering spaces.

Spring 2017: MATH GU4053
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4053 001/64661 M W 4:10pm - 5:25pm
417 Mathematics Building
Akram Alishahi 3 8/49

MATH GU4061 Introduction To Modern Analysis I. 3 points.

Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010. The second term of this course may not be taken without the first.

Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.

Spring 2017: MATH GU4061
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4061 001/10113 M W 4:10pm - 5:25pm
717 Hamilton Hall
Bin Guo 3 53/100
Fall 2017: MATH GU4061
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4061 001/62851 M W 8:40am - 9:55am
417 Mathematics Building
Bin Guo 3 34/64
MATH 4061 002/73447 T Th 2:40pm - 3:55pm
520 Mathematics Building
Patrick Gallagher 3 49/49

MATH GU4062 Introduction To Modern Analysis II. 3 points.

Prerequisites: MATH UN1202 or the equivalent, and MATH UN2010. The second term of this course may not be taken without the first.

Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.

Spring 2017: MATH GU4062
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4062 001/60905 M W 8:40am - 9:55am
203 Mathematics Building
Hector Chang-Lara 3 13/100
Fall 2017: MATH GU4062
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4062 001/13821 M W 1:10pm - 2:25pm
520 Mathematics Building
Hui Yu 3 20/100

MATH GU4065 Honors Complex Variables. 3 points.

Prerequisites: (MATH UN1207 and MATH UN1208) or MATH GU4061

A theoretical introduction to analytic functions. Holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy's integral formula, poles, Laurent series, residue theorem. Other topics as time permits: elliptic functions, the gamma and zeta function, the Riemann mapping theorem, Riemann surfaces, Nevanlinna theory.

Fall 2017: MATH GU4065
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4065 001/75532 T Th 1:10pm - 2:25pm
417 Mathematics Building
Julien Dubedat 3 22/64

MATH W4071 Introduction to the Mathematics of Finance. 3 points.

CC/GS: Partial Fulfillment of Science Requirement, BC: Fulfillment of General Education Requirement: Quantitative and Deductive Reasoning (QUA).

Prerequisites: MATH V1202, MATH V3027, STAT W4150, SEIOW4150, or their equivalents.

The mathematics of finance, principally the problem of pricing of derivative securities, developed using only calculus and basic probability. Topics include mathematical models for financial instruments, Brownian motion, normal and lognormal distributions, the BlackûScholes formula, and binomial models.

MATH GU4081 Introduction to Differentiable Manifolds. 3 points.

Prerequisites: (MATH GU4051 or MATH GU4061) and MATH UN2010

Concept of a differentiable manifold. Tangent spaces and vector fields. The inverse function theorem. Transversality and Sard's theorem.  Intersection theory. Orientations. Poincare-Hopf theorem. Differential forms and Stoke's theorem.

Spring 2017: MATH GU4081
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4081 001/64475 M W 10:10am - 11:25am
520 Mathematics Building
Luis Diogo 3 7/49

MATH GU4155 Probability Theory. 3 points.

Prerequisites: MATH GU4061 or MATH UN3007

A rigorous introduction to the concepts and methods of mathematical probability starting with basic notions and making use of combinatorial and analytic techniques. Generating functions. Convergence in probability and in distribution. Discrete probability spaces, recurrence and transience of random walks. Infinite models, proof of the law of large numbers and the central limit theorem. Markov chains.

Spring 2017: MATH GU4155
Course Number Section/Call Number Times/Location Instructor Points Enrollment
MATH 4155 001/14556 T Th 4:10pm - 5:25pm
520 Mathematics Building
Ioannis Karatzas 3 14/35

MATH W4391 Intro to Quantum Mechanics: An Introduction for Mathematicians and Physicists I. 3 points.

CC/GS: Partial Fulfillment of Science Requirement
Not offered during 2017-18 academic year.

Prerequisites: MATH V1202 or the equivalent and MATH V2010.

This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant for undergraduates with no previous formal training in quantum theory. The measurement problem and issues of non-locality will be stressed.

MATH W4392 Quantum Mechanics: An Introduction for Mathematicians and Physicists II. 3 points.

Not offered during 2017-18 academic year.

Prerequisites: MATH V1202 or the equivalent, MATH V2010, and MATH W4391.

This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant for undergraduates with no previous formal training in quantum theory. The measurement problem and issues of non-locality will be stressed.

Of Related Interest

Computer Science
COMS W3203Discrete Mathematics: Introduction to Combinatorics and Graph Theory
COMS W3251Computational Linear Algebra
COMS W4203Graph Theory
Industrial Engineering and Operations Research
CSOR E4010Graph Theory: A Combinatorial View