Requirements for the PhD in Mathematics at the School of Engineering are primarily qualitative rather than quantitative. That being said, you must satisfy certain credit and course obligations. The number of graduate units usually associated with graduation from this PhD program equals 75 credits, with each course worth 3 credits. You must select them from a well-balanced program in 1 major and 2 minor fields. We encourage you to choose minor fields outside the Department of Mathematics, such as those in applied mechanics, financial engineering, control theory, computer science, traffic engineering, or electrical engineering.
Thirty-nine credits of coursework and at least 21 credits of thesis are required. You must earn a grade of B or better for a non-core course to satisfy degree requirements (core-courses require a grade of A). You must also be able to read mathematical text written in French, German, or Russian.
Required Core Courses (12 Credits)
You must take the following courses and receive a grade of A in each. You may repeat them, if necessary.
- Linear Algebra I MA-GY 7033
- This course covers: Basic ideas of linear algebra: Fields, vector spaces, basis, dependence, independence, dimension. Relation to solving systems of linear equations and matrices. Homomorphisms, duality, inner products, adjoints and similarity.
Prerequisites: MA-UY 2012 and MA-UY 2122 or equivalent.
- Linear Algebra II MA-GY 7043
- This course continues MA-GY 7033. Topics covered: Basic concepts of linear algebra continuing with: Range, nullity, determinants and eigenvalues of matrices and linear homomorphisms, the polar decomposition and spectral properties of linear maps, orthogonality, adjointness and its applications.
Prerequisite: MA-GY 7033.
- Elements of Real Analysis I MA-GY 6213
- This course and its sequel MA-GY 6223 rigorously treat the basic concepts and results in real analysis. Course topics include limits of sequences, topological concepts of sets for real numbers, properties of continuous functions and differentiable functions. Important concepts and theorems include supremum and infimum, Bolzano-Weierstrass theorem, Cauchy sequences, open sets, closed sets, compact sets, topological characterization of continuity, intermediate value theorem, uniform continuity, mean value theorems and inverse function theorem.
Prerequisite: MA-UY 2122 or permission of adviser.
- Elements of Real Analysis II MA-GY 6223
- This course continues MA-GY 6213. The topics are integration, series of real numbers, sequences and series of functions and Fourier series. Important concepts and theorems include Riemann and Riemann-Stieltjes integral, fundamental theorem of calculus, the mean value theorem of integrals, Dirichlet test, absolute and conditional convergence, uniform convergence, Weierstrass test, power series, orthogonal functions and Fourier series.
Prerequisite: MA-GY 6213.
You are required to pass the following:
- A Part 0 written examination covering fundamental topics
- A Part 1 written examination covering real and complex analysis, as well as linear and abstract algebra
- A Part 2 oral examination on topics chosen by you and your thesis adviser
After passing the Part 2 examination, you must write a dissertation under the supervision of a faculty adviser. You must pass a public oral exam on your dissertation as the final requirement of the PhD degree.