Areas of Specialization and Study Plans

A M.Sc. degree requires completion of at least 48 credit hours. Students must enroll in at least eight graduate courses, that is courses at the 600 level or above, corresponding to 32 credit hours, and register for at least 16 thesis credits.

Students admitted into the Ph.D. program with a Bachelor's degree are required to complete at least 96 credits hours. There are no required courses, but students must enroll in at least eight graduate level courses, corresponding to 32 credit hours. At least four of these courses, corresponding to 16 credit hours, must be at the advanced graduate level (700 or above). Students must also register for at least 64 thesis credits.

Students admitted into the Ph.D. program with a Master's degree are required to complete at least 64 credit hours. Students must enroll in at least four advanced graduate courses (700 or above), corresponding to 16 credit hours, and must also register for at least 48 thesis credits.

A description of some of the research areas offered by the School of Mathematics with sample study plans are give below. They are for students entering with a Bachelor's degree; students entering with a Master's degree will be placed according to their background knowledge. An individual student's course of study may vary, and will depend on the student's background and interests. A student may be required to register for more than the required minimum credit hours or for more courses than listed in the study plans, and thus may need considerably more time to complete the degree than specified in the study plans.

Preliminary Year:

Students entering into the M.Sc. program with an undergraduate degree different from mathematics or with insufficient mathematical knowledge may be required to register for advanced undergraduate courses at the 500 level during their first year. Such courses can not be counted towards the degree.

Preliminary Year
term 1: 103512 Linear Algebra
103524 Mathematical Analysis I
term 2: 103522 Mathematical Analysis II
103531 Partial Differential Equations
term 3: 103521 Advanced Calculus
103522 Complex Analysis or 103511 Algebra

Core Courses:

During the first two terms, students usually register for the same core course in order to become acquainted with the basics of applied mathematics in real and functional analysis, differential equations and numerical analysis.

Core Courses
term 1: 103621 Real Analysis I
103631 Advanced Ordinary Differential Equations
103681 Computer Tools for Mathematical Research
term 2: 103531 Partial Differential Equations
103622 Applied Functional Analysis I
103654 Numerical Linear Algebra
(Note: 103621, 103622 and 103654 are required courses in the M.Sc. program. All Master's students must take these courses.)

Specialized Courses:

Beginning with the third term of studies, students will register for courses in their area of specialization.


Applied Analysis: Control Theory

In a great variety of practical situations, for example in manufacturing, robotics, disease control or economics, one needs to steer a system from an initial to a desired state within a specified amount of time, or keep the system at a fixed state, all in an optimal manner. Such systems are frequently modeled by differential equations, and control theory is the study of how to introduce external parameters into these differential equations in order to control the solutions. A M.Sc. project in this subject will most likely investigate problems involving ordinary differential equations, while a Ph.D. project will study the controllability of partial differential equations using techniques from functional analysis and operator theory.

M.Sc. in Control Theory
term 3: 103637 Introduction to Optimal Control
103729 Topics in Applied Functional Analysis
103791 Seminar
terms 4 - 6: 103791 M.Sc. Thesis

Ph.D. in Control Theory
term 3: 103637 Introduction to Optimal Control
103638 Principles of Partial Differential Equations
103721 Real Analysis II
term 4: 103722 Applied Functional Analysis II
103734 Topics in Control Theory
103790 Seminar
term 5: 103729 Topics in Functional Analysis
103731 Advanced PDEs
term 6: 103821 Advanced Topics in Analysis
103834 Advanced Topics in Control Theory
terms 7 - 12: 103891 Ph.D. Thesis


Applied Analysis: Wavelet Analysis

The purpose of traditional spectral analysis is to decompose a signal into its frequency contents by Fourier transform methods. Wavelet analysis allows the analysis of a signal not only in the frequency but also in the time domain and thus preserves information of a signal local in time. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. Wavelets were developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology, but interchanges between these fields during the last ten years have led to many new wavelet applications such as image compression, turbulence, human vision, radar, and earthquake prediction.

M.Sc. in Wavelet Analysis
term 3: 103623 Fourier Series and Transforms
103721 Real Analysis II
103790 Seminar
term 4: 103624 Wavelet Theory
103722 Applied Functional Analysis II
terms 5-7: 103791 M.Sc. Thesis

Ph.D. in Wavelet Analysis
term 3: 103623 Fourier Series and Transforms
103721 Real Analysis II
103638 Principles of Partial Differential Equations
term 4: 103624 Wavelet Theory
103722 Applied Functional Analysis II
103790 Seminar
term 5: 103727 Topics in Applied Analysis
103729 Topics in Functional Analysis
term 6: 103711 Applied Group Representation Theory
103821 Advanced Topics in Analysis
term 7: 103822 Advanced Topics in Functional Analysis
terms 7-13: 103891 Ph.D. Thesis


Computational Mathematics: Finite Element Method

When solving a boundary value problem by the Finite Element Method, one searches for an approximate solution lying in a finite dimensional subspace of the solution space. This subspace should have very simple basis functions, usually polynomials, so that the original problem is transformed to matrix equations involving sparse matrices, which can be solved efficiently on a computer. Research may include the modelling of practical problems, devising finite element schemes and investigating their suitability, which includes proving convergence of the approximate solutions to the analytical solution, as well as numerical computations.

M.Sc. in Finite Element Method
term 3: 103651 Numerical Treatment of PDEs
103652 Finite Element Method
103790 Seminar
term 4: 103655 Numerical Treatment of ODEs
103751 Computational Numerical Methods
term 5: 103752 Topics in Numerical Analysis
terms 5-7: 103791 M.Sc. Thesis

Ph.D. in Finite Element Method
term 3: 103651 Numerical Treatment of PDEs
103652 Finite Element Method
103638 Principles of Partial Differential Equations
term 4: 103655 Numerical Treatment of ODEs
103751 Computational Numerical Methods
103790 Seminar
term 5: 103731 Advanced PDEs
103752 Topics in Numerical Analysis
erm 6: 103721 Real Analysis II
103831 Advanced Topics in Applied Math
term 7: 103722 Applied Functional Analysis II
103852 Advanced Topics in Numerical Analysis
terms 8-13: 103891 Ph.D. Thesis


Computational Mathematics: Computational Fluid Dynamics

Fluid Dynamics is the study of flows of gases and fluids, a field with many immediate applications, be it the flow of air around the wings of an airplane, convection of pollutants in the atmosphere, or water flow around a submarine, to just name a few. Work in this field will consist of modelling practical problems, analyzing the resulting boundary value problems, and finding numerical solution techniques on the computer.

M.Sc. in Computational Fluid Dynamics
term 3: 103634 Continuum Mechanics
103651 Numerical Treatment of PDEs
103790 Seminar
term 4: 103635 Classical Models of Continuum Mechanics
103736 Math. Principles of Classical Fluid Mechanics
terms 5-7: 103791 M.Sc. Thesis

Ph.D. in Computational Fluid Dynamics
term 3: 103634 Continuum Mechanics
103651 Numerical Treatment of PDEs
103638 Principles of Partial Differential Equations
term 4: 103635 Classical Models of Continuum Mechanics
103736 Math. Principles of Classical Fluid Mechanics
103790 Seminar
term 5: 103738 Topics in CFD
103752 Topics in Numerical Analysis
term 6: 103781 Mathematical Modelling
103735 Introduction to Turbulence and Flows
term 7: 103838 Advanced Topics in CFD
103821 Advanced Topics in Applied Mathematics
terms 8-13: 103891 Ph.D. Thesis


Computational Mathematics: Computer Aided Geometric Design

A common problem in CAGD or computer vision is to reconstruct a two or three dimensional object from a discrete set of data points in a way which preserves the geometric characteristics inherent in the given data. Research work in this area will consist of finding appropriate reconstruction techniques, such as modified spline interpolation methods, as well as efficient numerical algorithms for their implementation.

M.Sc. in Computer Aided Geometric Design
term 3: 103673 Introduction to Computer Graphics
103651 Numerical Treatment of PDEs
103790 Seminar
term 4: 103671 Geometric Modelling
103655 Computer Numerical Analysis
term 5: 103672 Mathematical Methods in CAGD
103754 Computational Methods of Linear Algebra
terms 6-8: 103791 M.Sc. Thesis


Ph.D. in Computer Aided Geometric Design
term 3: 103673 Introduction to Computer Graphics
103651 Numerical Treatment of PDEs
103638 Principles of Partial Differential Equations
term 4: 103671 Geometric Modelling
103655 Computer Numerical Analysis
103790 Seminar
term 5: 103672 Mathematical Methods in CAGD
103754 Computational Methods of Linear Algebra
term 6: 103751 Computational Numerical Methods
103752 Topics in Numerical Analysis
103781 Mathematical Modelling
term 7: 103831 Advanced Topics in Applied Mathematics
103722 Applied Functional Analysis II
103641 Differential Geometry of Curves and Surfaces
terms 8-13: 103891 Ph.D. Thesis


Computational Mathematics: Numerical Optimization

Optimization is one of the vital tools in solving particular mathematically defined problems which model physical problems, by aiming at obtaining the optimal, or near optimal, of many possible solutions. The four main tasks in optimization are therefore: to develop a mathematical problem, to investigate the existence and attributes of optimal solutions, to design efficient algorithms for their computation, and to implement the optimization schemes in a particular application.

M.Sc. in Numerical Optimization
term 3: 103661 Linear Programming and Applications
one course from the following two courses:
103651 Numerical Treatment of PDEs
103652 Finite Element method
103790 Seminar
term 4: 103662 Nonlinear Programming
103751 Computational Numerical Methods
term 5-7: 103791 M.Sc. Thesis

Ph.D. in Numerical Optimization
term 3: 103661 Linear Programming and Applications
103721 Real Analysis II
103638 Principles of Partial Differential Equations
term 4: 103662 Nonlinear Programming
103722 Applied Functional Analysis II
103751 Computational Numerical Methods
term 5: 103752 Topics in Numerical Analysis
one course from the following two courses:
103831 Advanced Topics in Applied Mathematics
103861 Advanced Topics in Optimization
103790 Seminar
terms 6-12: 103891 Ph.D. Thesis


Differential Equations: Group Analysis

While various schemes have been developed to obtain approximate solutions of partial differential equations, it is much more important for both theoretical and practical purposes to obtain an exact solution of a given problem. Group Analysis aims at exploring symmetries inherent in a differential equation or a boundary value problem in order to reduce the given problem to a simpler one, or even find an exact solution.

M.Sc. in Group Analysis
term 3: 103634 Continuum Mechanics
103638 Principles of Partial Differential Equations
103790 Seminar
term 4: 103732 Group Analysis of Differential Equations
103736 Math. Principles of Classical Fluid Dynamics
term 5: 103731 Advanced PDEs
103651 Numerical Treatment of PDEs
terms 6-7 103791 M.Sc. Thesis

Ph.D. in Group Analysis
term 3: 103634 Continuum Mechanics
103638 Principles of Partial Differential Equations
103651 Numerical Treatment of PDEs
term 4: 103732 Group Analysis of Differential Equations
103736 Math. Principles of Classical Fluid Dynamics
103790 Seminar
term 5: 103731 Advanced PDEs
103733 Topics in the Theory of PDEs
term 6: 103821 Advanced Topics in Applied Mathematics
103833 Advanced Topics in the Theory of PDEs
103781 Mathematical Modelling
term 7: 103635 Classical Models of Continuum Mechanics
terms 7-13: 103891 Ph.D. Thesis