# M.S. in Mathematics

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**.

### Degree Requirements

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)

**Group III **(more advanced core graduate courses)

- 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.

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

Phone (212) 998-3238

Fax (212) 995-4121

admissions@math.nyu.edu