Graduate

Applied Mathematics (MSc)

Courses

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Foundation Courses:  

Given the applied nature of our program, it is highly probable that many of our students will have a background in the applied sciences/engineering fields. Thus, the purpose of the foundation courses are to fill in, as much as is necessary, whatever gaps that some of the students may have in their previous studies of mathematics. These courses will also give the students an overview of the program and an introduction to applied mathematics. Students are only required to complete one of the two foundation courses.

Core Courses:  

The core courses are to be completed by all students in the program. These courses focus on certain well-established principles and techniques that are particularly important in applied mathematics. They also provide essential background for some of the elective courses in our program.

Foundation Courses:

Analysis and Probability

Vectors and complex calculus. ODEs, PDEs, eigenfunction expansions and integral equations. Measure spaces, random variables and modes of convergence, algorithms for simulation of distributions, martingales, Brownian motion and stochastic integration, stochastic differential equations.

Algebra and Discrete Mathematical Structures

Review of basic concepts from elementary number Theory: g.c.d., Euclid’s algorithm, congruences; group theory: examples, applications to counting and coding; rings and fields: examples, polynomial rings, finite fields; linear algebra: vector spaces, basis and dimension, spaces over finite fields, linear, polynomial and BCH codes; latin squares, designs, and matroids, Steiner’s triple systems; Introduction to graph theory: isomorphism problem, paths and circuits, trees, colourings.

  

Core Courses:

Principles and Techniques in Applied Mathematics, Part I

Integral transform and applications to ODEs and PDEs; discrete Fourier transforms, FFT and applications; asymptotic expansions; perturbation methods; calculus of variations, optimizing functionals and applications.

Principles and Techniques in Applied Mathematics, Part II

Numerical methods; numerical linear algebra; numerical methods for ODEs; numerical methods for PDEs; numerical simulations.

Elective Courses:

Financial Mathematics

This course covers the fundamentals of mathematical methods in finance. Bonds, annuities, amortization, futures, profit/cost optimization. Decisions under certainty/uncertainty, capital budgeting and risk and return. Risk management, value at risk, credit risk. Modern portfolio theory. Arbitrage, utility theory, complete and incomplete markets. Modern theory of derivative pricing, Black-Scholes formulation. European, American options and credit derivatives. Numerical methods, Cox-Ross binomial models and finite difference schemes, lattice models for interest-rate derivatives.

Digital Signals and Wavelets

Digital signals; wavelets; adaptive methods; two and three dimensional transforms; applications.

Topics in Functional Analysis

Normed spaces; fundamental results; Hilbert spaces; calculus in Banach spaces; additional topics.

Topics in Discrete Mathematics

Introduction to Functions; Graphs; and Algorithms: growth of functions, O-notation, efficiency of algorithms; trees: binary and spanning tress, breadth-first and depth-first algorithms, shortest path problem; Eulerian and Hamiltonian paths and circuits: traveling salesman problem, P vs. NP; min-max algorithms: Dijkstra’s algorithm, Hungarian Method; linear programming: simplex method, duality.

Applied Statistical Methods

This course covers a wide variety of statistical methods with application in medicine, engineering, and economics. Exploratory data analysis. Parametric probability distributions. Sampling and experimental designs. Estimation, confidence intervals and tests of hypothesis. Analysis of variance. Multiple regression analysis, tests for normality. Nonparametric statistics. Statistical analysis of time series; ARMA and GARCH processes. Practical techniques for the analysis of multivariate data; principal components, factor analysis.

Partial Differential Equations

Hyperbolic equations, weak solutions, shock formation, non-linear
waves, reaction-diffusion equations, traveling wave solutions,
elliptic equations, numerical methods, applications.

Topics in Biomathematics

Discrete and continuous time processes applied to biology and chemistry.  Deterministic and stochastic descriptions for birth/death processes in chemical kinetics.  Numerical methods for spatially distributed systems including multi-species reaction-diffusion equations.  Applications will include some or all of:  chemical waves, traveling wave fronts in excitable media, spiral waves, pattern formation, blood flow and flow in chemical reactors.