# Mathematics

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

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

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

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

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

**Economics-Mathematics Advisers:**

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

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

**Mathematics-Statistics Advisers:**

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

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

----

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

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

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

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

## Courses for First-Year Students

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

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

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

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

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

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

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

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

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

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

## Advanced Placement

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

## Placement in the Calculus Sequences

### Calculus I

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

### Calculus II and III

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

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

### Accelerated Multivariable Calculus

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

### Honors Mathematics A

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

## Transfers Inside the Calculus Sequences

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

## Grading

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

## Departmental Honors

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

## Professors

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

## Associate Professors

- Daniela De Silva (Barnard)
- Julien Dubedat

## Assistant Professors

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

## J.F. Ritt Assistant Professors

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

## Senior Lecturers in Discipline

- Lars Nielsen
- Mikhail Smirnov
- Peter Woit

## Lecturers in Discipline

- Michael Woodbury

## On Leave

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

## Major in Mathematics

The major requires 40-42 points as follows:

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

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

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

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

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

15 points in the following required courses: | ||

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

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

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

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

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

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

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

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

## Major in Applied Mathematics

The major requires 38-40 points as follows:

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

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

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

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

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

Select one of the following three courses: | ||

Analysis and Optimization | ||

Fourier Analysis | ||

Introduction To Modern Analysis I | ||

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

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

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

Analysis and Optimization | ||

Complex Variables | ||

or MATH GU4065 | Honors Complex Variables | |

or APMA E4204 | Functions of a Complex Variable | |

Ordinary Differential Equations | ||

Partial Differential Equations | ||

or APMA E4200 | Partial Differential Equations | |

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

Fourier Analysis | ||

Computational Math: Introduction to Numerical Methods | ||

Introduction to Dynamical Systems | ||

Applied Functional Analysis | ||

Introduction to Biophysical Modeling |

## Major in Computer Science–Mathematics

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

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

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

Computer Science | ||

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

or COMS W1007 | Honors Introduction to Computer Science | |

COMS W3134 | Data Structures in Java | |

or COMS W3137 | Honors Data Structures and Algorithms | |

COMS W3157 | Advanced Programming | |

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

COMS W3261 | Computer Science Theory | |

CSEE W3827 | Fundamentals of Computer Systems | |

Mathematics | ||

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

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

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

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

MATH UN3951 | Undergraduate Seminars in Mathematics I | |

or MATH UN3952 | Undergraduate Seminars in Mathematics II | |

MATH GU4041 | Introduction to Modern Algebra I | |

Electives | ||

Select two of the following courses: | ||

Analysis of Algorithms I | ||

Numerical Algorithms and Complexity | ||

Combinatorics | ||

Analysis and Optimization | ||

Complex Variables | ||

Number Theory and Cryptography | ||

Differential Geometry | ||

Topology | ||

Introduction To Modern Analysis I |

## Major in Economics-Mathematics

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

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

## Major in Mathematics-Statistics

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

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

Mathematics | ||

Select one of the following sequences: | ||

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

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

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

Statistics | ||

Introductory Course | ||

STAT UN1201 | Calculus-Based Introduction to Statistics | |

Required Courses | ||

STAT GU4203 | PROBABILITY THEORY | |

STAT GU4204 | Statistical Inference | |

STAT GU4205 | Linear Regression Models | |

Select one of the following courses: | ||

Elementary Stochastic Processes | ||

Stochastic Processes for Finance | ||

STOCHASTC PROCSSES-APPLIC | ||

Stochastic Methods in Finance | ||

Computer Science | ||

Select one of the following courses: | ||

Introduction to Computer Science and Programming in Java | ||

Introduction to Computer Science and Programming in MATLAB | ||

Introduction to Computing for Engineers and Applied Scientists | ||

Honors Introduction to Computer Science | ||

or an advanced computer science offering in programming | ||

Electives | ||

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

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

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

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

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

## Concentration in Mathematics

The concentration requires the following:

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

Mathematics | ||

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

Calculus III and Calculus IV and Linear Algebra | ||

Accelerated Multivariable Calculus and Linear Algebra | ||

Honors Mathematics A and Honors Mathematics B | ||

Additional Courses | ||

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

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

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

**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 2020: MATH UN1003 |
|||||

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

MATH 1003 | 002/12023 | T Th 11:40am - 12:55pm 407 Mathematics Building |
Yier Lin | 3 | 10/30 |

MATH 1003 | 003/00593 | M W 6:10pm - 7:25pm 302 Barnard Hall |
Lindsay Piechnik | 3 | 26/36 |

Fall 2020: MATH UN1003 |
|||||

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

MATH 1003 | 001/11290 | M W 6:10pm - 7:25pm Room TBA |
3 | 0/30 | |

MATH 1003 | 002/11291 | T Th 2:40pm - 3:55pm Room TBA |
3 | 0/30 |

**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 2020: MATH UN1101 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 1101 | 001/13846 | M W 11:40am - 12:55pm 407 Mathematics Building |
Cailan Li | 3 | 21/30 |

MATH 1101 | 002/12024 | M W 2:40pm - 3:55pm 203 Mathematics Building |
Akash Sengupta | 3 | 72/110 |

MATH 1101 | 003/12025 | M W 6:10pm - 7:25pm 407 Mathematics Building |
Gerhardt Hinkle | 3 | 21/30 |

MATH 1101 | 004/12026 | T Th 10:10am - 11:25am 203 Mathematics Building |
Alexandra Florea | 3 | 85/110 |

MATH 1101 | 005/12027 | T Th 11:40am - 12:55pm 203 Mathematics Building |
William Chen | 3 | 44/110 |

Fall 2020: MATH UN1101 |
|||||

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

MATH 1101 | 002/11292 | M W 10:10am - 11:25am Room TBA |
Daniele Alessandrini | 3 | 0/116 |

MATH 1101 | 003/11293 | M W 11:40am - 12:55pm Room TBA |
Daniele Alessandrini | 3 | 0/116 |

MATH 1101 | 004/11294 | M W 1:10pm - 2:25pm Room TBA |
Akash Sengupta | 3 | 0/110 |

MATH 1101 | 005/11295 | M W 2:40pm - 3:55pm Room TBA |
Akash Sengupta | 3 | 0/110 |

MATH 1101 | 006/11296 | M W 4:10pm - 5:25pm Room TBA |
3 | 0/30 | |

MATH 1101 | 007/11297 | T Th 10:10am - 11:25am Room TBA |
George Dragomir | 3 | 0/100 |

MATH 1101 | 008/11298 | T Th 11:40am - 12:55pm Room TBA |
3 | 0/30 | |

MATH 1101 | 009/11299 | T Th 1:10pm - 2:25pm Room TBA |
George Dragomir | 3 | 0/100 |

MATH 1101 | 010/11300 | T Th 4:10pm - 5:25pm Room TBA |
3 | 0/100 |

**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 2020: MATH UN1102 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 1102 | 001/12029 | M W 1:10pm - 2:25pm 207 Mathematics Building |
Yi Sun | 3 | 43/125 |

MATH 1102 | 002/12030 | M W 2:40pm - 3:55pm 407 Mathematics Building |
Semen Rezchikov | 3 | 32/35 |

MATH 1102 | 003/12031 | T Th 11:40am - 12:55pm 207 Mathematics Building |
Michael Woodbury | 3 | 51/125 |

MATH 1102 | 004/12032 | T Th 6:10pm - 7:25pm 407 Mathematics Building |
Iakov Kononov | 3 | 20/30 |

Fall 2020: MATH UN1102 |
|||||

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

MATH 1102 | 001/11302 | M W 11:40am - 12:55pm Room TBA |
3 | 0/30 | |

MATH 1102 | 002/11303 | M W 2:40pm - 3:55pm Room TBA |
3 | 0/30 | |

MATH 1102 | 003/11304 | M W 4:10pm - 5:25pm Room TBA |
3 | 0/110 | |

MATH 1102 | 004/11305 | T Th 10:10am - 11:25am Room TBA |
3 | 0/110 | |

MATH 1102 | 005/00434 | T Th 2:40pm - 3:55pm Room TBA |
Lindsay Piechnik | 3 | 0/100 |

MATH 1102 | 006/11306 | T Th 6:10pm - 7:25pm Room TBA |
Elliott Stein | 3 | 0/64 |

**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 2020: MATH UN1201 |
|||||

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

MATH 1201 | 001/12037 | M W 10:10am - 11:25am 207 Mathematics Building |
Carolyn Abbott | 3 | 45/125 |

MATH 1201 | 002/12039 | M W 11:40am - 12:55pm 602 Hamilton Hall |
Konstantin Aleshkin | 3 | 21/125 |

MATH 1201 | 003/12040 | M W 2:40pm - 3:55pm 312 Mathematics Building |
Igor Krichever | 3 | 99/120 |

MATH 1201 | 004/12041 | T Th 1:10pm - 2:25pm 312 Mathematics Building |
Stephen Miller | 3 | 87/116 |

MATH 1201 | 005/12042 | T Th 6:10pm - 7:25pm 207 Mathematics Building |
Inbar Klang | 3 | 130/130 |

Fall 2020: MATH UN1201 |
|||||

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

MATH 1201 | 001/11389 | M W 10:10am - 11:25am Room TBA |
Konstantin Aleshkin | 3 | 0/110 |

MATH 1201 | 002/11390 | M W 11:40am - 12:55pm Room TBA |
Konstantin Aleshkin | 3 | 0/110 |

MATH 1201 | 003/11394 | M W 1:10pm - 2:25pm Room TBA |
Ovidiu Savin | 3 | 0/110 |

MATH 1201 | 004/11398 | T Th 10:10am - 11:25am Room TBA |
Carolyn Abbott | 3 | 0/116 |

MATH 1201 | 005/11402 | T Th 11:40am - 12:55pm Room TBA |
Evan Warner | 3 | 0/116 |

MATH 1201 | 006/11407 | T Th 2:40pm - 3:55pm Room TBA |
Inbar Klang | 3 | 0/116 |

MATH 1201 | 007/11412 | T Th 4:10pm - 5:25pm Room TBA |
Inbar Klang | 3 | 0/116 |

MATH 1201 | 008/11417 | T Th 6:10pm - 7:25pm Room TBA |
Guillaume Remy | 3 | 0/116 |

**MATH UN1202 Calculus IV.** *3 points*.

Prerequisites: MATH UN1102 and 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 2020: MATH UN1202 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 1202 | 001/00067 | T Th 10:10am - 11:25am 202 Milbank Hall |
Daniela De Silva | 3 | 35/100 |

MATH 1202 | 002/00275 | T Th 2:40pm - 3:55pm 202 Milbank Hall |
Lindsay Piechnik | 3 | 37/100 |

Fall 2020: MATH UN1202 |
|||||

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

MATH 1202 | 001/11421 | T Th 10:10am - 11:25am Room TBA |
Stephen Miller | 3 | 0/64 |

MATH 1202 | 002/11424 | M W 6:10pm - 7:25pm Room TBA |
Mikhail Smirnov | 3 | 0/116 |

**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 - MATH UN1202. Students taking this course may not receive credit for MATH UN1201 and MATH UN1202.

Spring 2020: MATH UN1205 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 1205 | 001/12046 | M W 2:40pm - 3:55pm 520 Mathematics Building |
Florian Johne | 4 | 24/35 |

Fall 2020: MATH UN1205 |
|||||

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

MATH 1205 | 001/11425 | M W 2:40pm - 3:55pm Room TBA |
4 | 0/49 |

**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 2020: MATH UN1207 |
|||||

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

MATH 1207 | 001/11430 | T Th 1:10pm - 2:25pm Room TBA |
Evan Warner | 4 | 0/110 |

**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 2020: MATH UN1208 |
|||||

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

MATH 1208 | 001/12047 | M W 4:10pm - 5:25pm 312 Mathematics Building |
Evan Warner | 4 | 39/100 |

**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 2020: MATH UN2000 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 2000 | 001/00068 | M W 2:40pm - 3:55pm 805 Altschul Hall |
Dusa McDuff | 3 | 23/55 |

Fall 2020: MATH UN2000 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 2000 | 001/11446 | M W 11:40am - 12:55pm Room TBA |
Gus Schrader | 3 | 0/49 |

**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 2020: MATH UN2010 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 2010 | 001/12050 | M W 10:10am - 11:25am 203 Mathematics Building |
Alexis Drouot | 3 | 37/110 |

MATH 2010 | 002/12051 | M W 11:40am - 12:55pm 203 Mathematics Building |
Gus Schrader | 3 | 105/110 |

MATH 2010 | 003/12052 | T Th 10:10am - 11:25am 312 Mathematics Building |
Henry Pinkham | 3 | 17/116 |

MATH 2010 | 004/12053 | T Th 2:40pm - 3:55pm 207 Mathematics Building |
Nathan Dowlin | 3 | 102/125 |

MATH 2010 | 005/12054 | T Th 6:10pm - 7:25pm 520 Mathematics Building |
Elliott Stein | 3 | 18/45 |

Fall 2020: MATH UN2010 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 2010 | 001/00117 | T Th 8:40am - 9:55am Room TBA |
David Bayer | 3 | 0/100 |

MATH 2010 | 002/00118 | T Th 10:10am - 11:25am Room TBA |
David Bayer | 3 | 0/100 |

MATH 2010 | 003/11450 | M W 4:10pm - 5:25pm Room TBA |
Francesco Lin | 3 | 0/116 |

MATH 2010 | 004/11453 | T Th 11:40am - 12:55pm Room TBA |
Kyle Hayden | 3 | 0/100 |

MATH 2010 | 005/11455 | T Th 6:10pm - 7:25pm Room TBA |
Giulia Sacca | 3 | 0/116 |

**MATH UN2020 Honors Linear Algebra.** *3 points*.

**Not offered during 2019-20 academic year.**

*Prerequisites: MATH UN1201. * A more extensive treatment of the material in MATH UN2010, with increased emphasis on proof. Not to be taken in addition to MATH UN2010 or MATH UN1207-MATH UN1208.

**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 2020: MATH UN2030 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 2030 | 001/12103 | T Th 4:10pm - 5:25pm 312 Mathematics Building |
Kyler Siegel | 3 | 96/116 |

MATH 2030 | 002/12104 | T Th 6:10pm - 7:25pm 312 Mathematics Building |
Kyler Siegel | 3 | 37/116 |

Fall 2020: MATH UN2030 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 2030 | 001/11457 | M W 1:10pm - 2:25pm Room TBA |
Florian Johne | 3 | 0/116 |

MATH 2030 | 002/11461 | M W 2:40pm - 3:55pm Room TBA |
Florian Johne | 3 | 0/116 |

**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 2020: MATH UN2500 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 2500 | 001/12105 | M W 1:10pm - 2:25pm 312 Mathematics Building |
Kanstantsin Matetski | 3 | 29/110 |

MATH 2500 | 002/12107 | M W 4:10pm - 5:25pm 207 Mathematics Building |
Kanstantsin Matetski | 3 | 43/125 |

Fall 2020: MATH UN2500 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 2500 | 001/11464 | T Th 1:10pm - 2:25pm Room TBA |
Kanstantsin Matetski | 3 | 0/64 |

MATH 2500 | 002/11466 | T 2:40pm - 3:55pm Room TBA |
Kanstantsin Matetski | 3 | 0/64 |

**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)

Fall 2020: MATH UN3007 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3007 | 001/11470 | M W 2:40pm - 3:55pm Room TBA |
Nicholas Salter | 3 | 0/64 |

**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 2020: MATH UN3020 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3020 | 001/12108 | M W 10:10am - 11:25am 312 Mathematics Building |
Shotaro Makisumi | 3 | 94/116 |

**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 2020: MATH UN3025 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3025 | 001/11471 | T Th 1:10pm - 2:25pm Room TBA |
Dorian Goldfeld | 3 | 0/116 |

**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 2020: MATH UN3027 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3027 | 001/11478 | T Th 11:40am - 12:55pm Room TBA |
Simon Brendle | 3 | 0/116 |

**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 2020: MATH UN3028 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3028 | 001/12110 | T Th 11:40am - 12:55pm 312 Mathematics Building |
Panagiota Daskalopoulos | 3 | 43/100 |

**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 2020: MATH UN3050 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3050 | 001/12111 | M W 6:10pm - 7:25pm 203 Mathematics Building |
Mikhail Smirnov | 3 | 57/100 |

**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 2020: MATH UN3386 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3386 | 001/11484 | T Th 11:40am - 12:55pm Room TBA |
Richard Hamilton | 3 | 0/49 |

**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 2020: MATH UN3951 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3951 | 001/00120 | |
Daniela De Silva | 3 | 0/64 |

MATH 3951 | 002/00121 | M W 6:10pm - 7:25pm Room TBA |
3 | 0/15 |

**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 2020: MATH UN3952 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 3952 | 002/12112 | |
Daniele Alessandrini | 3 | 48/100 |

**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 2020: MATH GU4007 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4007 | 001/12113 | M W 2:40pm - 3:55pm 307 Mathematics Building |
Evan Warner | 3 | 5/20 |

**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 2020: MATH GU4032 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4032 | 001/12115 | M W 11:40am - 12:55pm 520 Mathematics Building |
Peter Woit | 3 | 22/50 |

**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 2020: MATH GU4041 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4041 | 001/12116 | T Th 10:10am - 11:25am 520 Mathematics Building |
Michael Harris | 3 | 44/55 |

Fall 2020: MATH GU4041 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 4041 | 001/11487 | M W 2:40pm - 3:55pm Room TBA |
Robert Friedman | 3 | 0/110 |

**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 2020: MATH GU4042 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4042 | 001/12121 | T Th 1:10pm - 2:25pm 417 Mathematics Building |
Yihang Zhu | 3 | 42/50 |

Fall 2020: MATH GU4042 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 4042 | 001/11488 | M W 1:10pm - 2:25pm Room TBA |
Mikhail Khovanov | 3 | 0/35 |

**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.

**MATH GU4044 Representations of Finite Groups.** *3 points*.

Prerequisites: MATH UN2010 and MATH 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.

Fall 2020: MATH GU4044 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4044 | 001/11490 | T Th 1:10pm - 2:25pm Room TBA |
Chao Li | 3 | 0/19 |

**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.

Spring 2020: MATH GU4045 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4045 | 001/12122 | M W 4:10pm - 5:25pm 528 Mathematics Building |
Akash Sengupta | 3 | 9/20 |

**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 2020: MATH GU4051 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4051 | 001/11491 | T Th 11:40am - 12:55pm Room TBA |
Stephen Miller | 3 | 0/64 |

**MATH GU4053 Introduction to Algebraic Topology.** *3 points*.

Prerequisites: MATH UN2010 and MATH GU4041 and MATH 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 2020: MATH GU4053 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4053 | 001/12123 | T Th 2:40pm - 3:55pm 307 Mathematics Building |
Oleg Lazarev | 3 | 8/50 |

**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 2020: MATH GU4061 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4061 | 001/12124 | M W 1:10pm - 2:25pm 417 Mathematics Building |
Hui Yu | 3 | 45/64 |

MATH 4061 | 002/12125 | M W 4:10pm - 5:25pm 417 Mathematics Building |
Hui Yu | 3 | 39/64 |

Fall 2020: MATH GU4061 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 4061 | 001/11494 | T Th 2:40pm - 3:55pm Room TBA |
Henri Roesch | 3 | 0/100 |

MATH 4061 | 002/11495 | T Th 4:10pm - 5:25pm Room TBA |
Henri Roesch | 3 | 0/100 |

**MATH GU4052 Introduction to Knot Theory.** *3 points*.

CC/GS: Partial Fulfillment of Science Requirement

Prerequisites: MATH GU4051 Topology and / or MATH GU4061 Introduction To Modern Analysis I (or equivalents). Recommended (can be taken concurrently): MATH UN2010 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 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 2020: MATH GU4062 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4062 | 001/12126 | M W 1:10pm - 2:25pm 407 Mathematics Building |
Evgeni Dimitrov | 3 | 22/65 |

Fall 2020: MATH GU4062 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |

MATH 4062 | 001/11498 | M W 4:10pm - 5:25pm Room TBA |
Hui Yu | 3 | 0/49 |

**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 2020: MATH GU4065 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4065 | 001/11503 | T Th 10:10am - 11:25am Room TBA |
Julien Dubédat | 3 | 0/20 |

**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 Stokes' theorem.

Fall 2020: MATH GU4081 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4081 | 001/00119 | M W 10:10am - 11:25am Room TBA |
Dusa McDuff | 3 | 0/40 |

**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 2020: MATH GU4155 |
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Course Number | Section/Call Number | Times/Location | Instructor | Points | Enrollment |
---|---|---|---|---|---|

MATH 4155 | 001/12127 | T Th 4:10pm - 5:25pm 520 Mathematics Building |
Ioannis Karatzas | 3 | 25/55 |

**MATH GU4391 Intro to Quantum Mechanics: An Introduction for Mathematicians and Physicists I.** *3 points*.

**Not offered during 2019-20 academic year.**

*Prerequisites: MATH UN1202 or the equivalent and MATH UN2010*. 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 GU4392 Quantum Mechanics: An Introduction for Mathematicians and Physicists II.** *3 points*.

**Not offered during 2019-20 academic year.**

*Prerequisites: MATH UN1202 or the equivalent, MATH UN2010 and MATH GU4391.* 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 GR5010 Introduction to the Mathematics of Finance.** *3 points*.

Prerequisites: MATH UN1102 and MATH UN1201 , or their equivalents.

Introduction to mathematical methods in pricing of options, futures and other derivative securities, risk management, portfolio management and investment strategies with an emphasis of both theoretical and practical aspects. Topics include: Arithmetic and Geometric Brownian ,motion processes, Black-Scholes partial differential equation, Black-Scholes option pricing formula, Ornstein-Uhlenbeck processes, volatility models, risk models, value-at-risk and conditional value-at-risk, portfolio construction and optimization methods.

,

Spring 2020: MATH GR5010 |
|||||

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

MATH 5010 | 001/12129 | M W 7:40pm - 8:55pm 207 Mathematics Building |
Mikhail Smirnov | 3 | 123/150 |

Fall 2020: MATH GR5010 |
|||||

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

MATH 5010 | 001/12050 | M W 7:40pm - 8:55pm Room TBA |
Mikhail Smirnov | 3 | 0/140 |

## Of Related Interest

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

Computer Science | ||

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

COMS W3251 | ||

COMS W4203 | Graph Theory | |

Industrial Engineering and Operations Research | ||

CSOR E4010 | Graph Theory: A Combinatorial View |