# Mathematics

333 Milbank Hall

212-854-3577

Department Assistant: Marsha Peruo

## General Information

Students who have special placement problems, or are unclear about their level, should make an appointment with a faculty member or the chair.

Two help rooms, one in 404 Mathematics and one in 333 Milbank, will be open all term (hours will be posted on the door and the online) for students seeking individual help and counseling from the instructors and teaching assistants. No appointments are necessary. However, resources are limited and students who seek individual attention should make every effort to come during the less popular hours and to avoid the periods just before midterm and final exams.

## Courses for First-Year Students

The systematic study of Mathematics begins with one of the following alternative sequences:

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

Calculus I, II, III, IV | ||

MATH UN1101 | Calculus I | |

MATH UN1102 | Calculus II | |

MATH UN1201 | Calculus III | |

MATH UN1202 | Calculus IV | |

Honors Math A-B | ||

MATH UN1207 | Honors Mathematics A | |

MATH UN1208 | Honors Mathematics B |

Credit is allowed for only one of the calculus sequences. The calculus sequence is a standard course in differential and integral calculus. *Honors Mathematics A-B * is for exceptionally well-qualified students who have strong advanced placement scores. It covers second-year Calculus (MATH UN1201 Calculus III–MATH UN1202 Calculus IV) and MATH UN2010 Linear Algebra, with an emphasis on theory.

*Calculus II *is NOT a prerequisite for *Calculus III*, so students who plan to take only one year of calculus may choose between I and II or I and III. The latter requires a B or better in Calculus I and is a recommended option for some majors.

MATH UN2000 An Introduction to Higher Mathematics is a course that can be taken in their first or second year by students with an aptitude for mathematics who would like to practice writing and understanding mathematical proofs.

## Placement in the Calculus Sequence

**College Algebra and Analytical Geometry **is a refresher course for students who intend to take Calculus but do not have adequate background for it.

**Advanced Placement: **Students who have passed the advanced placement test for Calculus AB with a grade of 4 or 5 or BC with a grade of 4 receive 3 points of credit. Those who passed Calculus BC with a grade of 5 will receive 4 points of credit or 6 points on placing into Calculus III or Honors Math A and completing with a grade of C or better.

**Calculus I, II, III:** Students who have not previously studied calculus should begin with Calculus I. Students with 4 or higher on the Calculus AB or BC advanced placement test may start with Calculus II. Students with 5 on the Calculus BC test should start with Calculus III.

**Honors Mathematics A:** Students who have passed the Calculus BC advanced placement test with a grade of 5, and who have strong mathematical talent and motivation, should start with Honors Mathematics A. This is the most attractive course available to well-prepared, mathematically talented first-year students, whether or not they intend to be mathematics majors. Students who contemplate taking this course should consult with the instructor. If this is not possible ahead of time, they should register and attend the first class.

**Chair:** David A. Bayer (Professor)

**Professors:** Dusa McDuff (Helen Lyttle Kimmel Chair), Walter D. Neumann

**Associate Professor: **Daniela De Silva

**Research Professor and Professor Emerita:** Joan Birman

Other officers of the University offering courses in Mathematics:

**Professors:** Panagiota Daskalopoulos, Aise Johan de Jong, Robert Friedman, Patrick X. Gallagher, Dorian Goldfeld, Brian Greene, Richard Hamilton, Michael Harris, Troels Jørgensen, Ioannis Karatzas, Mikhail Khovanov, Igor Krichever, Chiu-Chu Liu, Davesh Maulik, Andrei Okounkov, D. H. Phong, Henry Pinkham, Ovidiu Savin, Eric Urban, Mu-Tao Wang

**Associate Professors:** Mohammed Abouzaid, Ivan Corwin, Julien Dubedat, Robert Lipshtiz, Michael Thaddeus, Wei Zhang

**Assistant Professors:** Marcel Nutz, Rachel Ollivier

**Visiting Assistant Professors:** Christopher Jankowski, Paul Siegel

**J.F. Ritt Assistant Professors:** Salim Altug, Hector Chang, Po-Nig Chen, Qile Chen, Anand Deopurkar, Gabriele Di Cerbo, Luis Diogo, Alexander Drewitz, Sachin Gautam, Evgeny Gorskiy, David Hansen, Jennifer Hom, BoGwang Jeon, Paul Siegel, Xin Wan, Michael Woodbury, Anton Zeitlin, Xiangwen Zhang

**Senior Lecturers:** Lars Nielsen, Mikhail Smirnov, Peter Woit

## Requirements for the Major

The major programs in both Mathematics and Applied Mathematics are appropriate for students who plan to continue their training in graduate school. The major in Mathematical Sciences combines the elements of Mathematics, Computer Science and Statistics. It is designed to prepare students for employment in business, administration, and finance, and also give excellent background for someone planning graduate study in a social science field. Students who plan to obtain a teaching qualification in mathematics should plan their course of study carefully with an advisor, since courses that are too far from mathematics do not count towards certification.

**For a major in Mathematics: **14 courses as follows:

Four courses in calculus or Honors Mathematics A-B, including Advanced Placement Credit. Six courses in mathematics numbered at or above 2000, and four courses in any combination of mathematics and cognate courses. The courses in mathematics must include:

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

MATH UN2010 | Linear Algebra (also satisfied by Honors Math A-B) | |

MATH GU4041 | Introduction to Modern Algebra I (I) | |

MATH GU4042 | Introduction to Modern Algebra II (II) | |

MATH GU4061 | Introduction To Modern Analysis I (I) | |

MATH GU4062 | Introduction To Modern Analysis II (II) | |

MATH UN3951 | Undergraduate Seminars in Mathematics I (at least one term) | |

or MATH UN3952 | Undergraduate Seminars in Mathematics II |

* | Note: It is strongly recommended that the sequences MATH GU4041 Introduction to Modern Algebra I - MATH GU4062 Introduction To Modern Analysis II and MATH GU4061 Introduction To Modern Analysis I - MATH GU4062 Introduction To Modern Analysis II be taken in separate years. |

However, students who are not contemplating graduate study in mathematics may replace one or both of the two terms of MATH GU4061 Introduction To Modern Analysis I - MATH GU4062 Introduction To Modern Analysis II by one or two of the following courses: MATH UN2500 Analysis and Optimization, MATH UN3007 Complex Variables, or MATH GU4032 Fourier Analysis and may replace MATH GU4042 Introduction to Modern Algebra II by one of MATH UN3020 Number Theory and Cryptography or MATH UN3025 Making, Breaking Codes. In exceptional cases, the chair will approve the substitution of certain more advanced courses for those mentioned above.

**For a major in Applied Mathematics:** 14 courses

Four courses in calculus or Honors Mathematics A-B, including Advanced Placement Credit.

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

MATH UN2010 | Linear Algebra (also satisfied by Honors Math A-B) | |

MATH GU4061 | Introduction To Modern Analysis I | |

APMA E4901 | Seminar: Problem in Applied Mathematics | |

APMA E4903 | Seminar: Problems in Applied Mathematics | |

APMA E3900 | Undergraduate Research in Applied Mathematics (APMA E3900 may be replaced, with approval, by another technical elective for seniors that involves an undergraduate thesis or creative research report) |

Additional electives, to be approved by the Applied Math Committee, e.g.:

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

MATH UN2500 | Analysis and Optimization | |

MATH UN3007 | Complex Variables | |

or MATH GU4065 | Honors Complex Variables | |

or APMA E4204 | Functions of a Complex Variable | |

MATH UN3027 | Ordinary Differential Equations | |

or MATH UN2030 | Ordinary Differential Equations | |

MATH UN3028 | Partial Differential Equations | |

or APMA E4200 | Partial Differential Equations | |

MATH GU4032 | Fourier Analysis | |

APMA E4300 | Computational Math: Introduction to Numerical Methods | |

APMA E4101 | Introduction to Dynamical Systems | |

APMA E4150 | Applied Functional Analysis |

**For a major in Mathematical Sciences: **14 courses:

6 from Mathematics, 5 from a combination of Statistics and Computer Science and 3 electives from a combination of Mathematics, Statistics, Computer Science.

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

Mathematics | ||

Six required courses: | ||

MATH UN1101 | Calculus I | |

MATH UN1102 | Calculus II | |

MATH UN1201 | Calculus III | |

MATH UN2010 | Linear Algebra (also satisfied by Honors Math A-B) | |

MATH UN2000 | An Introduction to Higher Mathematics | |

MATH UN2030 | Ordinary Differential Equations | |

or MATH UN3027 | Ordinary Differential Equations | |

Possible further courses selected from the following: | ||

MATH UN1202 | Calculus IV | |

MATH UN2500 | Analysis and Optimization | |

MATH UN3020 | Number Theory and Cryptography | |

MATH UN3025 | Making, Breaking Codes | |

Any 3 credit MATH course numbered 2000 or above | ||

Statistics | ||

Select at least one of the following: | ||

STAT UN3105 | Applied Statistical Methods | |

STAT UN3107 | Undergraduate Research | |

Possible further courses selected from the following: | ||

Other courses from the Statistics major list | ||

Computer Science | ||

Select at least one of the following programming courses: | ||

Introduction to Computer Science and Programming in Java (preferred) | ||

Introduction to Computer Science and Programming in MATLAB | ||

Honors Introduction to Computer Science | ||

Possible further courses selected from the following: | ||

Other classes from the Computer Science Core | ||

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

COMS W3210 | Scientific Computation | |

COMS W3251 | Computational Linear Algebra |

More generally, electives may be any course with a prerequisite of at least one semester of Calculus, Statistics or Computer Science with the prior approval of the Mathematics Chair.

The Capstone Experience can be fulfilled by a significant thesis written under the supervision of faculty of any one of the three departments or by the Undergraduate Seminar in Mathematics.

**For a major in Mathematics-Statistics: **14 courses:

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

Mathematics | ||

MATH UN1101 | Calculus I | |

MATH UN1102 | Calculus II | |

MATH UN1201 | Calculus III | |

MATH UN2010 | Linear Algebra | |

MATH UN2500 | Analysis and Optimization | |

Statistics | ||

STAT UN1001 | Introduction to Statistical Reasoning | |

or STAT UN1101 | Introduction to Statistics | |

or STAT UN1211 | ||

STAT UN3105 | Applied Statistical Methods | |

or STAT GU4105 | ||

STAT UN3107 | Undergraduate Research | |

or STAT GU4107 | ||

STAT UN3315 | ||

or STAT W4315 | Linear Regression Models | |

STAT GU4606 | ||

or STAT GR5262 | Stochastic Processes for Finance | |

Computer Science | ||

Select one course that requires substantial work in programming | ||

Electives | ||

Select three electives from an approved selection of advanced courses in mathematics, statistics, applied mathematics, computer sciences or mathematical methods courses in physical or social sciences, including biology, economics, and physics |

Students should plan to include a senior thesis or the Undergraduate Seminar in Mathematics in their program, in consultation with their advisors.

*Note:* Students must obtain approval from an adviser in each of the two departments before selecting electives. Students should take MATH UN2010 Linear Algebra Linear Algebra in the second semester of the second year.

**For a major in Mathematics-Computer Science **15 courses:

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

Mathematics | ||

Four courses in calculus or Honors Mathematics A-B, including Advanced Placement Credit; and the 3 following courses: | ||

MATH UN2010 | Linear Algebra (also satisfied by Honors Math A-B) | |

MATH GU4041 | Introduction to Modern Algebra I | |

MATH UN3951 | Undergraduate Seminars in Mathematics I (at least one term) | |

or MATH UN3952 | Undergraduate Seminars in Mathematics II | |

Computer Science | ||

COMS W1007 | Honors Introduction to Computer Science ^{A} | |

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 |

*Note A: *AP Computer Science with a grade of 4 or 5 or similar experience (e.g., COMS W1004) is a prerequisite for COMS W1007

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

Electives: 2 of the following: | ||

CSOR W4231 | Analysis of Algorithms I | |

COMS W4241 | Numerical Algorithms and Complexity | |

MATH UN3020 | Number Theory and Cryptography | |

MATH BC2006 | Combinatorics | |

MATH GU4061 | Introduction To Modern Analysis I | |

MATH UN2500 | Analysis and Optimization | |

MATH UN3007 | Complex Variables | |

MATH UN3386 | Differential Geometry | |

MATH GU4051 | Topology |

Students seeking to pursue a Ph.D. program in either discipline are urged to take additional courses, in consultation with their advisers.

**For a major in Economics and Mathematics, see the catalogue.**

### Requirement for the Minor in Mathematics

For a minor in Mathematics or Applied Mathematics: Six courses from any of the courses offered by the department except MATH UN1003 College Algebra and Analytic Geometry, MATH UN1101 Calculus I / MATH UN1102 Calculus II. Some cognate courses are also acceptable with prior approval from the department chair.

### Requirements for the Minor in Mathematical Sciences

The minor in Mathematical Sciences comprises 6 courses, at least two from Mathematics and one from each of Statistics and Computer Science. There should be a minimum of three courses in Statistics and Computer Science. Eligible courses are any listed in the Mathematical Sciences Major with the exception of Calculus I and II.

**MATH BC2001 Perspectives in Mathematics.** *1 point*.

Prerequisites: some calculus or the instructor's permission.

Intended as an enrichment to the mathemathics curriculum of the first years, this course introduces a variety of mathematical topics (such as three dimensional geometry, probability, number theory) that are often not discussed until later, and explains some current applications of mathematics in the sciences, technology and economics.

Fall 2017: MATH BC2001 |
|||||

Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 2001 | 001/07748 | W 6:10pm - 7:25pm 202 Milbank Hall |
Dusa McDuff | 1 | 15/64 |

**MATH BC2006 Combinatorics.** *3 points*.

Corequisites: *MATH V2010* is helpful as a corequisite, but not required.

Honors-level introductory course in enumerative combinatorics. Pigeonhole principle, binomial coefficients, permutations and combinations. Polya enumeration, inclusion-exclusion principle, generating functions and recurrence relations.

**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 405 Milbank Hall |
Dusa McDuff | 3 | 102/110 |

MATH 1101 | 002/17570 | M W 10:10am - 11:25am 312 Mathematics Building |
Chao Li | 3 | 111/116 |

MATH 1101 | 003/29604 | M W 11:40am - 12:55pm 312 Mathematics Building |
Chao Li | 3 | 116/116 |

MATH 1101 | 004/73071 | M W 2:40pm - 3:55pm 417 Mathematics Building |
Michael Woodbury | 3 | 52/64 |

MATH 1101 | 005/18565 | M W 4:10pm - 5:25pm 417 Mathematics Building |
Michael Woodbury | 3 | 56/64 |

MATH 1101 | 006/73884 | T Th 10:10am - 11:25am 407 Mathematics Building |
Oleksandr Kravets | 3 | 28/35 |

MATH 1101 | 007/26909 | T Th 11:40am - 12:55pm 407 Mathematics Building |
Shuai Wang | 3 | 26/30 |

MATH 1101 | 008/64016 | T Th 1:10pm - 2:25pm 203 Mathematics Building |
Alexander Perry | 3 | 77/100 |

MATH 1101 | 009/21826 | T Th 4:10pm - 5:25pm 203 Mathematics Building |
Ila Varma | 3 | 88/100 |

MATH 1101 | 010/67061 | T Th 6:10pm - 7:25pm 207 Mathematics Building |
Linh Truong | 3 | 32/100 |

**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 | 64/64 |

MATH 1102 | 002/70122 | M W 2:40pm - 3:55pm 312 Mathematics Building |
Li-Cheng Tsai | 3 | 52/100 |

MATH 1102 | 003/26180 | M W 4:10pm - 5:25pm 407 Mathematics Building |
Yi Sun | 3 | 34/35 |

MATH 1102 | 004/22286 | T Th 10:10am - 11:25am 203 Mathematics Building |
Vivek Pal | 3 | 85/100 |

MATH 1102 | 005/29410 | T Th 6:10pm - 7:25pm 407 Mathematics Building |
Renata Picciotto | 3 | 26/30 |

**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 | 101/105 |

MATH 1201 | 002/15820 | M W 11:40am - 12:55pm 203 Mathematics Building |
Teng Fei | 3 | 60/100 |

MATH 1201 | 003/62151 | M W 1:10pm - 2:25pm 203 Mathematics Building |
Joanna Nelson | 3 | 100/105 |

MATH 1201 | 004/68024 | M W 4:10pm - 5:25pm 312 Mathematics Building |
Jeffrey Kuan | 3 | 109/116 |

MATH 1201 | 005/06129 | T Th 8:40am - 9:55am 304 Barnard Hall |
Daniela De Silva | 3 | 107/110 |

MATH 1201 | 006/63259 | T Th 4:10pm - 5:25pm 312 Mathematics Building |
Akram Alishahi | 3 | 66/100 |

MATH 1201 | 007/27974 | T Th 5:40pm - 6:55pm 312 Mathematics Building |
Akram Alishahi | 3 | 23/100 |

**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 |
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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 |
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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 | 85/100 |

MATH 1202 | 002/87449 | T Th 2:40pm - 3:55pm 203 Mathematics Building |
Hao Shen | 3 | 40/100 |

**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 |
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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 | 45/64 |

MATH 1207 | 002/26797 | T Th 1:10pm - 2:25pm 520 Mathematics Building |
Evan Warner | 4 | 15/49 |

**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 |
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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 |
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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 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 2000 | 001/04201 | M W 8:40am - 9:55am Ll103 Diana Center |
Walter Neumann | 3 | 16/64 |

**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 |
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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 |
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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 | 107/120 |

MATH 2010 | 002/25355 | M W 1:10pm - 2:25pm 312 Mathematics Building |
Gus Schrader | 3 | 67/100 |

MATH 2010 | 003/03818 | T Th 8:40am - 9:55am 405 Milbank Hall |
David Bayer | 3 | 78/100 |

MATH 2010 | 004/02940 | T Th 10:10am - 11:25am 405 Milbank Hall |
David Bayer | 3 | 93/100 |

MATH 2010 | 005/18445 | T Th 6:10pm - 7:25pm 203 Mathematics Building |
Elliott Stein | 3 | 97/108 |

**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 |
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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 |
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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 | 35/100 |

MATH 2030 | 002/12064 | T Th 11:40am - 12:55pm 312 Mathematics Building |
Guillaume Barraquand | 3 | 62/100 |

**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 | 54/100 |

MATH 2500 | 002/26047 | M W 2:40pm - 3:55pm 203 Mathematics Building |
Shotaro Makisumi | 3 | 73/100 |

**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 |
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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 | 84/100 |

**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 | 47/100 |

**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 | 24/49 |

**MATH UN3901 Supervised Readings in Mathematics I.** *2-3 points*.

*Prerequisites:* The written permission of the staff member who agrees to act as sponsor (sponsorship limited to full-time instructors on the staff list), as well as the permission of the Director of Undergraduate Studies. *The written permission must be deposited with the Director of Undergraduate Studies before registration is completed.* Guided reading and study in mathematics. A student who wishes to undertake individual study under this program must present a specific project to a member of the staff and secure his or her willingness to act as sponsor. Written reports and periodic conferences with the instructor.

**MATH UN3902 Supervised Readings in Mathematics II.** *2-3 points*.

*Prerequisites:* The written permission of the staff member who agrees to act as sponsor (sponsorship limited to full-time instructors on the staff list), as well as the permission of the Director of Undergraduate Studies. *The written permission must be deposited with the Director of Undergraduate Studies before registration is completed. * Guided reading and study in mathematics. A student who wishes to undertake individual study under this program must present a specific project to a member of the staff and secure his or her willingness to act as sponsor. Written reports and periodic conferences with the instructor.

Spring 2017: MATH UN3902 |
|||||

Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3902 | 001/05662 | |
Dusa McDuff | 2-3 | 6 |

MATH 3902 | 002/22214 | |
Akram Alishahi | 2-3 | 1 |

MATH 3902 | 003/93672 | |
David Hansen | 2-3 | 1 |

MATH 3902 | 004/75032 | |
Mikhail Khovanov | 2-3 | 3 |

MATH 3902 | 005/84535 | |
Peter Woit | 2-3 | 1 |

MATH 3902 | 006/87701 | |
Patrick Gallagher | 2-3 | 1/1 |

**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 | 34/49 |

**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 V3997 Supervised Individual Research.** *3 points*.

Prerequisites: the written permission of the faculty member who agrees to act as a supervisor, and the director of undergraduate studies' permission.

For specially selected mathematics majors, the opportunity to write a senior thesis on a problem in contemporary mathematics under the supervision of a faculty member.

**MATH V3998 Supervised Individual Research.** *3 points*.

Prerequisites: the written permission of the faculty member who agrees to act as a supervisor, and the director of undergraduate studies' permission.

For specially selected mathematics majors, the opportunity to write a senior thesis on a problem in contemporary mathematics under the supervision of a faculty member.

**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 |
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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 | 26/35 |

MATH 1003 | 002/26739 | T Th 12:10pm - 2:00pm 103 Knox Hall |
Darren Gooden | 3 | 32/30 |

**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 |
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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 405 Milbank Hall |
Walter Neumann | 3 | 62/100 |

**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 |
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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 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 4042 | 001/23554 | T Th 10:10am - 11:25am 310 Fayerweather |
Yihang Zhu | 3 | 15/64 |

**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 |
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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 | 10/30 |

**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 |
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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 W4046 Introduction to Category Theory.** *3 points*.

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

Prerequisites: *MATH W4041*.

Categories, functors, natural transformations, adjoint functors, limits and colimits, introduction to higher categories and diagrammatic methods in algebra.

**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 |
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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 |
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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 | 46/64 |

MATH 4061 | 002/73447 | T Th 2:40pm - 3:55pm 417 Mathematics Building |
Patrick Gallagher | 3 | 45/64 |

**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 |
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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 | 13/100 |

**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 | 11/64 |

**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.

## Cross-Listed Courses

### Computer Science

**COMS W3203 Discrete Mathematics: Introduction to Combinatorics and Graph Theory.** *3 points*.

Lect: 3.

Prerequisites: Any introductory course in computer programming.

Logic and formal proofs, sequences and summation, mathematical induction, binomial coefficients, elements of finite probability, recurrence relations, equivalence relations and partial orderings, and topics in graph theory (including isomorphism, traversability, planarity, and colorings).

Spring 2017: COMS W3203 |
|||||

Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

COMS 3203 | 001/63319 | M W 10:10am - 11:25am 501 Schermerhorn Hall |
Ansaf Salleb-Aouissi | 3 | 141/150 |

COMS 3203 | 002/29040 | M W 11:40am - 12:55pm 486 Computer Science Bldg |
Jessica Ouyang | 3 | 51/60 |

Fall 2017: COMS W3203 |
|||||

Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

COMS 3203 | 001/62348 | T Th 10:10am - 11:25am 209 Havemeyer Hall |
Ansaf Salleb-Aouissi | 3 | 101/100 |

COMS 3203 | 002/20951 | T Th 11:40am - 12:55pm 209 Havemeyer Hall |
Ansaf Salleb-Aouissi | 3 | 99/100 |

COMS 3203 | 003/69273 | M W 8:40am - 9:55am 203 Mathematics Building |
Antonio Moretti | 3 | 71/100 |

**COMS W3251 Computational Linear Algebra.** *3 points*.

Lect: 3.

Prerequisites: Two terms of calculus.

Computational linear algebra, solution of linear systems, sparse linear systems, least squares, eigenvalue problems, and numerical solution of other multivariate problems as time permits.

**COMS W4203 Graph Theory.** *3 points*.

Lect: 3.

Prerequisites: (COMS W3203)

General introduction to graph theory. Isomorphism testing, algebraic specification, symmetries, spanning trees, traversability, planarity, drawings on higher-order surfaces, colorings, extremal graphs, random graphs, graphical measurement, directed graphs, Burnside-Polya counting, voltage graph theory.

### Industrial Engineering and Operations Research

**CSOR E4010 Graph Theory: A Combinatorial View.** *3 points*.

Lect: 3.

Prerequisites: Linear algebra, or instructor's permission.

An introductory course in graph theory with emphasis on combinatorial aspects. Basic definitions, and some fundamental topics in graph theory and its applications. Topics include trees and forests graph coloring, connectivity, matching theory and others.