# The Master of Science in Mathematics

### The Program

The master's degree in mathematics
encompasses the basic graduate curriculum in mathematics, and
also offers the opportunity of some more specialized training
in an area of interest. A typical
master's program will involve basic courses in real analysis,
complex analysis and linear algebra, followed by other
fundamental courses such as probability, scientific computing,
and differential equations. Depending
on their mathematical interests, students will then be able to
take more advanced graduate courses in pure and applied
mathematics.

Information about admission on a non-degree basis is ** HERE**.

A candidate for the master's degree in mathematics must fulfill the following departmental requirements:

- 36 points of credit, and the grade of at least B in the written comprehensive examinations, or,
- 32 points of credit, and a master's thesis, completed under the supervision of a faculty member and approved by the department.

Under both options, students may be able to transfer up to 8 points of credit (usually equivalent to two CIMS courses) from other academic institutions.

### Coursework

Students are required to take eight courses (24 credits) from the list below. All four courses in Group I, two courses from Group II, and two additional courses from Group II or Group III must be taken. Substitute courses are listed at the end.

**Group I**(Mandatory Preparation)

__MATH-GA 1410__Introduction to Math Analysis I (fall)

__MATH-GA 2450__Complex Variables I (fall)

__MATH-GA 2110__Linear Algebra I (fall, spring, and possibly summer)

MATH-GA 1002 Multivariable Analysis (spring)

**Group II**(Fundamentals)

__MATH-GA 1420__Introduction to Math Analysis II (spring)

__MATH-GA 2460__Complex Variables II (spring)

__MATH-GA 2120__Linear Algebra II (spring)

__MATH-GA 2901__Basic Probability (fall, spring, summer)

__MATH-GA 2043__Scientific Computing (fall, spring)

__MATH-GA 2470__Ordinary Differential Equations (spring)

**(more advanced core graduate courses)**

Group III

Group III

__MATH-GA 2010__Numerical Methods I (fall)

__MATH-GA 2020__Numerical Methods II (spring)

__MATH-GA 2130__Algebra I (fall)

__MATH-GA 2310__Topology I (fall)

MATH-GA 2210 Number Theory (spring)

__MATH-GA 2350__Differential Geometry I (fall)

MATH-GA 2490 Partial Differential Equations (fall)

__MATH-GA 2701__Methods of Applied Math (fall)

MATH-GA 2702 Fluid Dynamics (fall)

MATH-GA 2550 Functional Analysis (spring)

MATH-GA 2563 Harmonic Analysis (fall)

MATH-GA 2911 Probability: Limit Theorems I (fall)

MATH-GA 2902 Stochastic Calculus (fall, spring and summer)

MATH-GA 2962 Mathematical Statistics (spring)

or: DS-GA 1002 Statistical and Mathematical Methods (fall)

**Substitutes**

MATH-GA
2430 Real Variables
(one-term) is a substitute for MATH-GA
1410
and 1420 Introduction to Math Analysis I and II.

MATH-GA
2451 Complex Variables (one-term) is a substitute for MATH-GA
2450
and 2460 Complex Variables I and II.

MATH-GA
2111 Linear Algebra (one-term) is a substitute for MATH-GA
2110
and 2120 Linear Algebra I and II.

A substitute one-term course may be taken after the first term of its equivalent two-term course, but may not be taken after the equivalent two-term sequence is completed. Resolution of conflicts will require the approval of the Director of Graduate Studies. Only the credits of the completed courses will count toward the degree.

Please note that master's and non-degree students in their first year of study are generally not permitted to register for any Advanced Topics courses. In further semesters, students are allowed to register for such courses with permission of both the course instructor and the Director of Graduate Studies for the M.S. Program.

### The Written Comprehensive Examinations

Students choosing the first option,
must pass, with the grade of B, the three-part Written
Comprehensive Examination administered by the Department twice
a year, in early September and early January. Most master's
students tend to take exam toward the end of their graduate
studies.

Students are permitted to take the written examinations
twice, with no special permission. A third and final
attempt may be granted by the Department on a case-by-case
basis.

The examinations, in advanced calculus, complex variables and linear algebra, may include some of the following material:

**Advanced Calculus:** Real numbers. Functions of one
variable: continuity, mean-value, differentiability, maxima
and minima, integrals, fundamental theorem of calculus,
inequalities, estimation of sums and integrals, elementary
functions and their power series. Funtions of several
variables: partial derivatives, chain rule, MacLaurin
expansion, critical points, Lagrange multipliers, inverse and
implicit function theorems, jacobian, divergence and curl,
theorems of Green and Stokes.

**Complex Variables:**Complex numbers, analytical functions, Cauchy-Riemann equations, Cauchy's integral and applications, power series, maximum principle, Liouville's theorem, elementary functions and their conformal maps, bilinear transformation, classification of singularities, residue theorem and contour integration, Laurent series, Rouche's theorem, number of zeros and poles.

**Linear Algebra:** Vector spaces, linear dependence,
basis, dimension, linear transformation, inner product,
systems of linear equations, matrices, determinants, ranks,
eigenvalues, diagonalization of matrices, quadratics forms,
symmetric and orthogonal transformations.

Cooperative preparation is encouraged, as it is for all examinations. Students may also find the following books helpful:

Buck, *Advanced Calculus*; Courant and John, *Introduction
to
Calculus and Analysis*; Strang, *Linear Algebra*;
Churchill, *Complex Variables and Applications*.

### Master's Thesis

Students choosing the second option will work on a master's thesis under the supervision of a mathematics faculty member acting as their thesis advisor. In certain cases involving interdisciplinary research, a second advisor outside the Department of Mathematics may be approved by the Director of Graduate Studies. A proposal outlining the program for the thesis will be submitted to the Director of Graduate Studies for approval. Upon completion, a master's thesis must be approved by the thesis advisor and a second reader of his/her choice.

### Academic Standards

To continue registering for courses in the Department of Mathematics, a student must be in a good academic standing, fulfilling the following requirements:

- Students must maintain an average of B or better (3.0) in their first 12 credits. Students failing to achieve this will not be permitted to continue in the program. Students cannot obtain an M.S. degree unless they have maintained an overall average of at least B
- Students will be allowed no more than four no-credit grades, withdrawals, or unresolved incomplete grades during their academic tenure, and no more than two such grades in the first six courses for which they have registered.
- Credit will be given for up to two core courses taken elsewhere, subject to the normal GSAS restrictions on transfer of credit and the approval of the Program Coordinator. At least 30 course points must be taken at New York University.

An M.S. application can be downloaded at www.nyu.edu/gsas/admissions; the Graduate School of Arts and Sciences strongly prefers that you apply online at www.nyu.edu/gsas/online.

For any questions contact us at:

*Office of Admissions and Student Affairs*

*Department of Mathematics*

*Courant Institute of Mathematical Sciences*

*251 Mercer Street*

*New York, NY 10012-1185*

__Tel.__(212) 998-3238

__Fax__(212) 995-4121

E-Mail: admissions@math.nyu.edu

E-Mail: arnon@cims.nyu.edu

Web page: http://www.math.nyu.edu