# Mathematics (2016 - 2018)

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Joint Major in Computer Science and Mathematics

Joint Major in Economics and Mathematics

## CAS Mathematics Requirement (Quantitative Reasoning)

To satisfy the College Core Curriculum requirement in Quantitative Reasoning (QR), all College of Arts and Science students must either take one semester of an approved course with mathematical content, or present qualifying advanced standing credit or SAT Subject Examination scores to exempt from the requirement. (There is no CAS examination to exempt students from QR.)

Advanced Placement (AP), International Baccalaureate (IB; HL only), and Advanced Level (A Level; not AS) credit in calculus, statistics, and mathematics satisfies the QR requirement, as does similar credit in selected international examinations. Consult the admission section of this Bulletin or a CAS adviser for details on which approved examinations and minimum scores confer credit. In addition, a score of 700 or higher on the SAT Subject Examination in Mathematics (either level 1 or 2) satisfies the Core QR requirement (no credit is awarded).

Students in the following majors or tracks of study are required to take calculus and/or statistics courses which also satisfy the Core QR requirement: computer science, economics, engineering, global public health, international relations, mathematics, the natural sciences, the prehealth track, psychology, and sociology. (These areas of study differ in whether and how they accept advanced standing credit toward their quantitative requirements; consult the appropriate sections of this Bulletin.)

CAS students who are not pursuing one of these courses of study, and who cannot present advanced standing credit or SAT Subject Examination scores for exemption from the QR requirement, must take one of the Quantitative Reasoning (CORE-UA 1XX) courses offered in the College Core Curriculum.

Alternatively, students who meet the prerequisites or take a placement exam may register for an appropriate calculus course at the level of Calculus I (MATH-UA 121) or above. Other courses that satisfy the QR requirement (in statistics, e.g.) are posted on the Core Curriculum website, www.core.cas.nyu.edu.

## Placement into Calculus Courses

Students meeting any of the following criteria may enter Calculus I (MATH-UA 121) or Mathematics for Economics I (MATH-UA 211). The latter course is intended for majors in economics.

- SAT general test mathematics score of 670 or higher (for SAT taken in and after March 2016) or 650 or higher (for SAT taken before March 2016).

- SAT Subject Examination in Mathematics (level 1 or level 2) score of 650 or higher
- ACT mathematics score of 30 or higher
- Advanced Placement (AP) Calculus AB exam score of 3 or higher (must be 4 or 5 to earn credit)
- AB subscore on the AP Calculus BC exam of 3 or higher (must be 4 or 5 to earn credit)
- AP Calculus BC exam score of 3 or higher (must be 4 or 5 to earn credit)
- A Level Mathematics score of C or higher (must be B or higher to earn credit; anyone who took Further Mathematics should consult the mathematics department for placement)
- AS Level Mathematics score of B or higher (no credit is awarded for AS exams)
- International Baccalaureate (IB) HL Mathematics score of 5 or higher (must be 6 or higher to earn credit)
- IB SL Mathematics score of 6 or higher (no credit is awarded for SL exams)
- IB SL Mathematics Studies score of 7 (no credit is awarded for SL exams)
- Algebra and Calculus (MATH-UA 9) with a grade of C or higher, or equivalent
- Passing score on the departmental calculus placement exam

Students who do not meet any of these prerequisites must take Algebra and Calculus (MATH-UA 9) before proceeding to Calculus I or Mathematics for Economics I.

## Advanced Placement with Credit

Freshmen seeking advanced placement in the mathematics major or minor may present results of the Advanced Placement (AP) Calculus AB or BC Examination.

- A student who earns a 4 or 5 on the Calculus AB exam (or AB subscore) or a 4 on the Calculus BC exam will receive 4 points equivalent to Calculus I (MATH-UA 121) and will be placed into Calculus II (MATH-UA 122).
- A student who earns a score of 5 on the Calculus BC exam will receive 8 points, equivalent to both Calculus I (MATH-UA 121) and Calculus II (MATH-UA 122), and will be placed into Calculus III (MATH-UA 123).

For calculus equivalencies and placement for advanced standing credit in mathematics from International Baccalaureate (HL only), A Level, and other approved international examinations, please consult the admission section of this Bulletin or a CAS adviser.

## Advanced Placement without Credit

The department periodically gives its own advanced placement exams for students who know the material covered in Calculus I (MATH-UA 121) and/or Calculus II (MATH-UA 122) and who wish to enter Calculus II (MATH-UA 122) or Calculus III (MATH-UA 123). There is also an examination to pass out of Calculus III (MATH-UA 123). If a student passes any of these exams, he or she is placed into the next course of the sequence; however, no college credit is given for the courses that are skipped.

## Major in Mathematics

The major consists of thirteen 4-point courses (52 points) numbered MATH-UA 120 or higher.

**Calculus and Linear Algebra Requirement**

- Calculus I (MATH-UA 121) or advanced standing credit
- Calculus II (MATH-UA 122) or advanced standing credit
- Calculus III (MATH-UA 123)
- Linear Algebra (MATH-UA 140)

**Analysis and Algebra Requirement**

- Analysis (MATH-UA 325)
- Algebra (MATH-UA 343)

Either of these courses can be substituted by its more intensive counterpart:

- Honors Analysis I (MATH-UA 328)
- Honors Algebra I (MATH-UA 348)

Students are strongly advised to take analysis before algebra.

**Advanced Electives Requirement**

The rest of the thirteen required courses must include at least three of the following:

- Theory of Probability (MATH-UA 233)
- Mathematical Statistics (MATH-UA 234)
- Numerical Analysis (MATH-UA 252)
- Partial Differential Equations (MATH-UA 263)
- Functions of a Complex Variable (MATH-UA 282)
- Honors Analysis II (MATH-UA 329)
- Honors Algebra II (MATH-UA 349)
- Differential Geometry (MATH-UA 377)
- Honors I (MATH-UA 393)
- Honors II (MATH-UA 394)
- Honors III (MATH-UA 397)
- Honors IV (MATH-UA 398)

**Policies Applying to the Major in Mathematics**

- Mathematics for Economics I, II, III (MATH-UA 211, 212, 213) do not count toward the major in mathematics as substitutions for the Calculus I, II, III sequence (MATH-UA 121, 122, 123). Exceptions may be made for students who have already taken Mathematics for Economics for their declared or intended economics major; they must seek the approval of the Department of Mathematics for this substitution. All other students must follow the regular calculus sequence. Students may not register simultaneously for separate courses within the two sequences.
- Transformations and Geometries (MATH-UA 270) does not count toward the major.
- Up to two computer science courses at the level of Introduction to Computer Science (CSCI-UA 101) or higher, or up to two graduate data science (DS-GA) courses, may be counted toward the thirteen courses required for the major. The prerequisites for DS-GA courses are Calculus III (MATH-UA 123), Linear Algebra (MATH-UA 140), and programming experience in Python (preferred) or MATLAB. Probability is a recommended prerequisite.
- Students who complete the prehealth program may substitute at most two MATH-UA courses by any two of the following: General Physics I, II (PHYS-UA 11, 12) or Physics I, II (PHYS-UA 91, 93).
- However, if these physics courses are used towards the mathematics major, the computer science or data science courses will not apply towards the major.
- Students may double-count no more than two courses toward both the mathematics major and the requirements of another major or minor.
- Courses taken under the pass/fail option cannot count toward the major. A grade of C or better is required in all courses used to fulfill major requirements. In addition, majors must maintain a 2.0 mathematics GPA.
- Students may petition to enroll in graduate mathematics courses and apply them to the undergraduate major. Permission is not granted until the student has completed Analysis (MATH-UA 325) and the available undergraduate course(s) on the same topic.
- All mathematics majors and minors are required to see an undergraduate faculty adviser to review their course of study and be advised on appropriate courses for each term. Inquire at the department office, Warren Weaver Hall, 251 Mercer Street, Room 625 or 627, or call 212-998-3005 for more information.

## Joint Major in Computer Science and Mathematics

This is an interdisciplinary major (eighteen courses/72 points) offered by the Department of Computer Science with the Department of Mathematics. Students must complete one CSCI-UA course with a recorded grade of C or better before they can declare this joint major (this policy applies to all NYU students, not just to those matriculated in CAS).

Students may double-count no more than two courses towards both this joint major and the requirements of another major or minor. A grade of C or better is necessary in all courses used to fulfill joint major requirements (courses taken under the pass/fail option cannot be counted toward the major). Interested students should consult with the director of undergraduate studies in both departments for additional information.

The mathematics requirements (ten courses/40 points) are as follows. Note that students must choose one calculus sequence or the other and cannot mix courses from both; they also cannot register simultaneously for separate courses within the two sequences.

- Discrete Mathematics (MATH-UA 120)
- Calculus I (MATH-UA 121) or Mathematics for Economics I (MATH-UA 211)
- Calculus II (MATH-UA 122) or Mathematics for Economics II (MATH-UA 212)
- Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213)
- Linear Algebra (MATH-UA 140)
- Analysis (MATH-UA 325) or Honors Analysis I (MATH-UA 328)
- Algebra (MATH-UA 343) or Honors Algebra I (MATH-UA 348)
- The remaining three mathematics courses must include two of the following: Theory of Probability (MATH-UA 233), Mathematical Statistics (MATH-UA 234), Mathematical Modeling (MATH-UA 251), Numerical Analysis (MATH-UA 252), Partial Differential Equations (MATH-UA 263), Functions of a Complex Variable (MATH-UA 282), Honors Analysis II (MATH-UA 329), Honors Algebra II (MATH-UA 349), or Differential Geometry (MATH-UA 377).
- All mathematics electives for the joint major must be numbered MATH-UA 120 or higher, and may not include Transformations and Geometries (MATH-UA 270).

The computer science requirements (eight courses/32 points) are as follows:

- Introduction to Computer Science (CSCI-UA 101)
- Data Structures (CSCI-UA 102)
- Computer Systems Organization (CSCI-UA 201)
- Operating Systems (CSCI-UA 202)
- Basic Algorithms (CSCI-UA 310)
- Numerical Computing (CSCI-UA 421)
- Two computer science electives at the 400 level

Students who take Numerical Analysis (MATH-UA 252) as one of their math electives for this major must contact the director of undergraduate studies in computer science before registering for Numerical Computing (CSCI-UA 421).

## Joint Major in Economics and Mathematics

An interdisciplinary major (eighteen courses/72 points) offered jointly by the Departments of Economics and Mathematics. In the economics department, joint majors with mathematics must follow the theory concentration. Nine courses must be taken in each department. Students may double-count no more than two courses towards both this joint major and the requirements of another major or minor. Courses taken under the pass/ fail option cannot be counted toward the major. A grade of C or better is necessary in all courses used to fulfill joint major requirements. Interested students should consult with the director of undergraduate studies in both departments for additional information.

The nine mathematics courses (36 points) must all be numbered MATH-UA 120 or higher, and must include these five courses. Note that students must choose one calculus sequence or the other and cannot take courses from both; the Mathematics for Economics sequence is the preferred sequence for this joint major. Students also cannot register simultaneously for separate courses within the two sequences.

- Mathematics for Economics I (MATH-UA 211) or Calculus I (MATH-UA 121)
- Mathematics for Economics II (MATH-UA 212) or Calculus II (MATH-UA 122)
- Mathematics for Economics III (MATH-UA 213) or Calculus III (MATH-UA 123)
- Linear Algebra (MATH-UA 140)
- Analysis (MATH-UA 325) or Honors Analysis I (MATH-UA 328)

Four additional courses must be completed from the following choices: Theory of Probability (MATH-UA 233), Mathematical Statistics (MATH-UA 234), Probability and Statistics (MATH-UA 235), Combinatorics (MATH-UA 240), Theory of Numbers (MATH-UA 248), Mathematics of Finance (MATH-UA 250), Mathematical Modeling (MATH-UA 251), Numerical Analysis (MATH-UA 252), Ordinary Differential Equations (MATH-UA 262), Partial Differential Equations (MATH-UA 263), Chaos and Dynamical Systems (MATH-UA 264), Functions of a Complex Variable (MATH-UA 282), Honors Analysis II (MATH-UA 329), or Algebra (MATH-UA 343) or Honors Algebra I (MATH-UA 348).

The economics requirements (nine courses/36 points) are as follows:

- Introduction to Macroeconomics (ECON-UA 1)
- Introduction to Microeconomics (ECON-UA 2)
- Microeconomics (ECON-UA 11)
- Macroeconomics (ECON-UA 13)
- Analytical Statistics (ECON-UA 20), if not taking Mathematical Statistics (MATH-UA 234)
- Introduction to Econometrics (ECON-UA 266)
- Plus any three economics elective courses, at least two of which must be theory electives numbered ECON-UA 300 to 399. Note that students who take MATH-UA 234 instead of ECON-UA 20 for the statistics requirement must take a total of four ECON-UA electives.

## Minor in Mathematics

The requirements are four 4-point courses (16 points) in the department, numbered MATH-UA 120 or higher. Courses from the sequences Calculus I, II, III (MATH-UA 121, 122, 123) and Math for Economics I, II, III (MATH-UA 211, 212, 213) cannot both be applied to the mathematics minor; students must choose one calculus sequence or the other and cannot take courses from both. They also cannot register simultaneously for separate courses within the two sequences. Although courses transferred from other colleges may count towards the minor with departmental approval, at least two courses for the minor must be taken at New York University.

Students pursuing majors that require mathematics courses may follow this minor, as long as two of the four courses do not apply simultaneously to the requirements for their major.

Courses taken under the pass/fail option are not counted toward the minor. A grade of C or better is required in all courses counting for the minor. Advisers are available for consultation on minor requirements before and during registration; students should consult an adviser if they have any doubt about which courses fulfill their requirements.

## Joint Minor in Computer Science and Mathematics

The requirements are these four courses (16 points):

- Calculus I and II (MATH-UA 121, 122), or Mathematics for Economics I and II (MATH-UA 211, 212). Students must choose one calculus sequence or the other and cannot take courses from both. They also cannot register simultaneously for separate courses within the two sequences.
- Introduction to Computer Science (CSCI-UA 101)
- Data Structures (CSCI-UA 102)

Students must complete one CSCI-UA course with a recorded grade of C or better before they can declare this joint minor (this policy applies to all NYU students, not just to those matriculated in CAS). A grade of C or better is required for courses to count toward the minor. Courses taken under the pass/fail option are not counted towards the minor. Students who wish to double-count courses for the math portion of the minor and another requirement may count at most two such courses toward the minor. At least two of the courses in total and at least one of the math courses must be taken in residence at New York University.

## Minor in Advanced Mathematical Methods (for Students in Stern)

The advanced mathematical methods minor (open only to Stern undergraduates) consists of four courses (typically 15 points) completed with a grade of C or better. It provides students with mathematical tools to handle complex business problems. The requirements are as follows:

- Linear Algebra (MATH-UA 140)
- Numerical Analysis (MATH-UA 252) or Numerical Methods I (MATH-GA 2010)
- Introduction to the Theory of Probability (STAT-UB 14)
- One course chosen from:
- Ordinary Differential Equations (MATH-UA 262)
- Partial Differential Equations (MATH-UA 263)
- Analysis (MATH-UA 325)
- Statistical Inference and Regression Analysis (STAT-UB 15)
- Introduction to Stochastic Processes (STAT-UB 21)

Students must take at least one Stern course (typically 3 points) to complete the minor. If a student has completed a CAS course in probability, STAT-UB 14 should not be taken. Either STAT-UB 15 or STAT-UB 21 should be substituted. Students who have the equivalent of MATH-UA 140 should substitute a more advanced course from the list above.

## Honors Program in Mathematics

The honors program is designed for students with a strong commitment to mathematics and is recommended for those who intend to pursue graduate study in this field. The requirements for admission into the honors program are (1) a GPA of 3.65 or higher in the major (including joint honors requirements), (2) an overall GPA of 3.65 or better, and (3) approval of the director of the honors program. Interested students should consult with the faculty honors adviser.

Like the regular major, the honors major consists of thirteen 4-point courses (52 points). However, students in the honors program must fulfill the requirements of the regular program together with the following additional requirements.

**Honors electives:** Honors majors must take at least four of the following 4-point courses:

- Honors Analysis I (MATH-UA 328)
- Honors Analysis II (MATH-UA 329)
- Honors Algebra I (MATH-UA 348)
- Honors Algebra II (MATH-UA 349)
- Honors I (MATH-UA 393)
- Honors II (MATH-UA 394)
- Honors III (MATH-UA 397)
- Honors IV (MATH-UA 398)

Where applicable, the same course counts toward both the advanced electives requirement of the regular major and the honors electives. Students who have taken Analysis (MATH-UA 325) or Algebra (MATH-UA 343) may not take the corresponding Honors Analysis I (MATH-UA 328) or Honors Algebra I (MATH-UA 348) to fulfill this requirement. Under special circumstances and with the permission of the department, certain graduate courses may be substituted for Honors I-IV.

**Honors research project:** Honors students must also complete a senior research project and present it at the College’s Undergraduate Research Conference in the spring. Students must register for two semesters of independent study (MATH-UA 997, 998) under faculty supervision, and obtain approval of their research project from the faculty honors advisor, director of undergraduate studies, or vice chair of undergraduate affairs. The research project can also be completed through the mathematics summer research program (S.U.R.E.). Students who participate in this program are required to present their research at the undergraduate research forum at Courant in the fall semester of their senior year.

## Joint Honors Program in Computer Science and Mathematics

This is a twenty-course (80-point) interdisciplinary major offered by the Departments of Computer Science and Mathematics. The mathematics requirements (ten courses/40 points) are as follows (students must choose one calculus sequence and cannot mix courses from the two tracks):

- Calculus I (MATH-UA 121) or Mathematics for Economics I (MATH-UA 211)
- Calculus II (MATH-UA 122) or Mathematics for Economics II (MATH-UA 212)
- Calculus III (MATH-UA 123) or Mathematics for Economics III (MATH-UA 213)
- Linear Algebra (MATH-UA 140)
- Analysis (MATH-UA 325) or Honors Analysis I (MATH-UA 328)
- Honors Analysis II (MATH-UA 329)
- Algebra (MATH-UA 343) or Honors Algebra I (MATH-UA 348)
- Honors Algebra II (MATH-UA 349)
- Two honors courses chosen from Honors I (MATH-UA 393), Honors II (MATH-UA 394), Honors III (MATH-UA 397), or Honors IV (MATH-UA 398). With permission of the mathematics department, these honors courses may be replaced with two graduate courses.

The computer science requirements (ten courses/40 points) are as follows:

- Introduction to Computer Science (CSCI-UA 101)
- Data Structures (CSCI-UA 102)
- Computer Systems Organization (CSCI-UA 201)
- Operating Systems (CSCI-UA 202)
- Basic Algorithms (CSCI-UA 310)
- Numerical Computing (CSCI-UA 421)
- Theory of Computation (CSCI-UA 453)
- Three computer science courses listed at the CSCI-UA 400 level

Guided research, sponsored by either department, should be presented at the Dean’s Undergraduate Research Conference which takes place in late April. Students are expected to dedicate 10 to 20 hours per week toward their research.

## Accelerated B.A./M.S. Program in Mathematics

The College and the Graduate School of Arts and Science offer students the opportunity to obtain both bachelor’s and master’s degrees in mathematics in five years. Qualifying students are accepted into the program toward the end of the sophomore year or during the junior year before they reach 96 credits. Students must have a minimum 3.50 GPA in order to qualify for acceptance. During their last few undergraduate semesters, students must accelerate by taking one quarter of their graduate courses during regular terms and/or during the summer. Students in the program must satisfy all requirements of both the bachelor’s and master’s degrees; there is no double-counting of courses. In the graduate portion of the program, they can qualify for a scholarship covering up to 50 percent of the tuition for the master’s degree. For more information, please contact the College Advising Center, Silver Center, 100 Washington Square East, Room 905; 212-998-8130.

## Joint B.S./B.S. Program in Mathematics and Engineering

The College of Arts and Science, in cooperation with the NYU Tandon School of Engineering, offers a joint B.S./B.S. program in engineering. Students in the program receive the B.S. degree in mathematics from CAS and the B.S. degree in either civil, computer, electrical, or mechanical engineering from the Tandon School of Engineering. Joint mathematics/engineering students must complete the 13-course mathematics major. Students are allowed to substitute Introduction to Computer Science and Data Structures (CSCI-UA 101 and 102), or Physics I and II (PHYS-UA 91 and 93), for a maximum of two mathematics major courses. Further information and advisement are available from the College Advising Center, Silver Center, 100 Washington Square East, Room 905; 212-998-8130.

## Study Away

Students majoring and minoring in the Department of Mathematics can spend a semester studying abroad at one of the many academic centers run by NYU Global Programs. Currently, mathematics can be studied at NYU Berlin, NYU London, NYU Abu Dhabi, and NYU Shanghai. Students planning study away should make their plans and speak to an adviser early in their NYU careers.

## Departmental Activities and Awards

**Mathematics Society**

Open to all students interested in the study of mathematics. An organizational meeting is held shortly after classes begin in the fall to plan for the coming academic year. Activities include talks by faculty and guest speakers on a variety of topics as well as attending conferences.

**Association for Women in Mathematics (AWM)**

The mission and purpose of AWM’s NYU chapter is to increase interest in the mathematical sciences and their applications in various industries. It focuses on mentoring, encouraging and bringing together women undergraduates in mathematics to increase the visibility of women and their contributions in the discipline.

**William Lowell Putnam Competition**

The department participates in this annual competition open to all undergraduate mathematics students in the United States and Canada. Interested students should contact the department as early as possible in the school year, as the contest takes place in early December. A series of preparation sessions is held under the supervision of mathematics faculty.

**Mathematical Contest in Modeling (MCM)**

In this contest, teams of undergraduates use mathematical modeling to present their solutions to real-world problems. Interested students should contact the department as early as possible in the school year.

**Interdisciplinary Contest in Modeling (ICM)**

An international contest for high school students and college undergraduates (an extension of the MCM) designed to develop problem-solving skills and competence in written communication. Registration for ICM is via MCM.

**Peer Mentor Program**

The mathematics department has an active peer mentor program for mathematics majors. The program is designed to assist new students in making the transition to the mathematics major and life at NYU. If interested in becoming a mentor or mentee, please contact the department.

**S.U.R.E. Program**

Since the spring 2000 semester, the department has sponsored a number of summer research experiences (S.U.R.E.) for a selected number of undergraduate math majors. The Summer Undergraduate Research Experience is aimed at mathematics students in their junior year. The project ends with a written report and an oral presentation in the beginning of the fall semester. Funding is limited and student participants are chosen by a faculty committee based on grades, coursework, and “fit” between their research interests and those of the supervising faculty. Applications are considered more highly if students have a faculty mentor and research topic.

**Awards**

Departmental awards include the Mathematics Award, the Hollis Cooley Memorial Prize, the Perley Lenwood Thorne Medal, and the Mathematics Award for Academic Achievement. Please see descriptions under honors and awards in this bulletin.