# 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 course of study 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 a number of departmental requirements.

### The Written Comprehensive Examinations

Master’s students who choose the Written Comprehensive Examinations option must pass with an overall grade of B. The three-part Written Comprehensive Examination is administered by the department twice a year in early September and early January. Most Master's students tend to take the 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 who have earned a GPA of 3.7 or higher and taken at least 18 credits in the program have the option to write a Master's thesis under the supervision of a Mathematics faculty member. In certain cases involving interdisciplinary research, a second advisor outside the Department of Mathematics may be approved by the Director of Graduate Studies. All students must submit the Thesis Proposal and Advisor Approval form, outlining the research plan for the thesis which has been approved by the thesis advisor, to the Program Administrator at least four months prior to the graduation date. The completed Master's thesis must be approved by two readers -- the thesis advisor and a second reader. At least one of the readers must be a full-time Courant Mathematics faculty member. You can find more detailed information in the Thesis Guidelines FAQ.

### 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 24 course points must be taken at New York University.

### Admissions

See Mathematics for admission requirements and instructions specific to this program.

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

mathadmt@nyu.edu