Fourth Year Courses
Offered 2015-2016
MATH 4120H – Mathematical Modelling I
MATH 4510H – Mathematical Risk Management
MATH 4610H – Introduction to Graph Theory
MATH 4810H – Perspectives in Mathematics I
MATH 4850Y, 4851H, 4852H – Community-Based
Research Project
MATH 4900Y, 4903H, 4904H – Reading Course
Math 4120H - Mathematical Modelling I
This course provides an introduction to the mathematical modeling process and applies this process to simple mathematical modeling problems arising from a variety of application areas in science and engineering. Mathematical modeling techniques, such as differential equations, dimensional analysis, discrete systems and numerical methods along with computer aids will be utilized.
- Prerequisite: 60% or
higher in MATH-PHYS 2150H.
- Excludes: MATH
411, 412H.
Math 4180H - Advanced Numerical Methods
Deals with a variety of numerical methods for solving
ordinary and partial differential equations arising from
scientific and engineering applications. Topics include
finite difference, adaptive techniques, multi-step
methods, Runge-Kutta methods, direct and iterative
methods for systems, stability and convergence.
- Prerequisite: 60% or higher in MATH 2180H and
2150H.
- Excludes: MATH 403H.
Math-Cois 4215H - Mathematical Logic
An introduction to the syntax and semantics of
propositional and first-order logics through the
Soundness, Completeness, and Compactness Theorems.
- Prerequisite: 60% or higher in MATH-COIS 2200H
- Excludes: MATH-COSC 415H.
Math 4310H - Algebra IV: Galois theory
Extension fields and Galois groups; the fundamental
theorem of Galois Theory; the insolubility of the quintic
- Prerequisite: 60% or higher in MATH 3310H.
- Excludes: MATH 431H.
Math 4400H - Math Through Problem Solving
A seminar-based course in problem solving. Topics include invariants, number properties, polynomials,
functional equations, inequalities, combinatorial games.
- Prerequisites: 60% or higher in MATH 2200H.
- Excludes: MATH 4951H
Math 4450H - Voting, Bargaining, and Social Choice
Voting systems: Condorcet cycles, Borda count and
other positional systems, voting paradoxes, Arrow’s
impossibility theorem. Social choice functions: (relative)
utilitarian, egalitarian; properties and axiomatic
characterizations. Bargaining theory: Nash solution,
Kalai-Smorodinsky, utilitarian, and egalitarian solutions.
Strategic voting: Gibbard-Satterthwaite theorem;
implementation theory.
- Prerequisite: MATH 1350H, 2110H, and 2200H; or ECON
3000H and 3250H.
- Recommended: ECON 3160H and
4000H.
- Excludes: MATH 4952H.
Math 4510H - Mathematical Risk Management
This course covers the basic mathematical theory and computational techniques for how financial institutions can quantify and manage risks in portfolios of assets.
Syllabus:
- Introduction
- Portfolios
- Defining Risk
- Complete and Incomplete Markets
- Market Efficiency
- Mean-Variance Analysis
- Asset Return
- Portfolio Mean and Variance
- Efficient Frontier
- Markowitz Minimum Variance Portfolio
- One-Fund/Two-Fund Theorems
- Inclusion of Risk Free Assets
- Non-Linear Optimization
- Capital Asset Pricing Model (CAPM)
- Capital Market Line
- Beta Factor
- Security Market Line
- CAPM as a Pricing Formula
- Models and Data
- Data and Parameter Estimation
- Forecasting Volatilities and Correlations
- Risk Measurement
- Value at Risk (VaR)
- Credit Risk
- Computational Techniques
- Prerequisite: 60% or
higher in MATH 1550H and 2110H.
- Excludes: MATH 451H.
Math 4560H - Topics in Statistics
- Prerequisite: 60% or higher in MATH 2560H or
permission of instructor.
- Strongly recommended: MATH
3560H.
- Excludes: MATH 456H.
Math 4561H - Sampling and Design of Experiments
Provides background for students in applied statistics,
especially in sampling and design of experiments.
Topics in design of experiments include ANOVA,
randomized block designs, factorial designs, blocking
and confounding in factorial designs, response surface
methods. Topics in sampling include simple random,
systematic, stratified and cluster sampling, sample size
estimation, unequal probability sampling, and multistage
designs.
- Prerequisite: 60% or higher in MATH 2560H.
- Recommended: MATH 3560H.
- Excludes: MATH 456H,
4562H.
Math 4570H - Topics in Probability: A Second
Course in Stochastic Processes
This course is dedicated to the study of more advanced models in the theory of stochastic processes. These include renewal processes, martingales and Brownian motion. The course concludes with a basic introduction to stochastic calculus.
- Prerequisite: 60% or higher in MATH 3570H.
- Excludes: MATH 457H.
Math 4610H - Introduction to graph theory
An introduction to graph theory with emphasis on both theory and applications and algorithms related to computer science, operation research and management science.
- Prerequisite: 60% or higher in MATH-COIS
2600H and in MATH 2200H
- Excludes: MATH 461H.
Math 4620H - Introduction to Combinatorics
An introduction to combinatorics. The topics include
counting techniques, generating functions, and block
design.
- Prerequisite: 60% or higher in MATH 2200H.
- Excludes: MATH 460, 462H.
Math 4710H - Chaos, Symbolic Dynamics, Fractals
A (discrete time) dynamical system is a mathematical model of a physical system evolving over time. Every possible `state' of the system is represented as a point in a state space X, and the time-evolution is represented using a function f:X---> X. Thus, if the system is in state x at time zero, then it will be in state f(x) at time one, and in state f2(x)=f(f(x)) at time two, and state f3(x)=f(f(f(x))) at time three, and so on. Thus, to understand the long-term evolution of the system, we must study the behaviour of the statespace X under iterative application of the function f.
In general, it is impossible to exactly predict the result of iterating f thousands of times (except through brute-force computation), so we must use qualitative methods to understand the long-term behaviour of the dynamics. For example, the orbit of a point x is the set { fn(x)}n=0oo, which is generally an infinite scattering of points in X. By considering the distribution of this orbit (e.g. where it is more or less `densely scattered' in X) we get information about the long-term statistical behaviour of the dynamical system. In particular, a subset A in X is called an attractor if fn(x) becomes very close to A as n --> oo. Loosely speaking a dynamical system is ``chaotic'' if two points which are very close together can have orbits which rapidly diverge over time (the exact definitions are more complicated). This means that even tiny errors in measurement (which are inevitable in real life) can make long-term predictions impossible (the so-called ``butterfly effect'').
We will cover the following topics:
- Basic topological dynamics: Orbits, fixed and periodic points, attraction and repulsion, basins of attraction.
- One dimensional systems: Piecewise linear maps, logistic parametrised families, Baker maps.
- Bifurcations: Definition, examples. The Feigenbaum constant and Feigenbaum universality. The Schwarzian derivative (which enables us to theoretically explain some aspects of bifurcation diagrams).
- Chaos: Sensitivity to initial conditions; Lyapunov exponents.
- Basic symbolic dynamics: Partitions and itineraries. Conjugacy. Applications to chaos.
- Fractals: Deterministic fractals generated as fixed points of iterated function systems. Various definitions of fractal dimension.
- Prerequisite: 60% or
higher in one of MATH 3700H, 3770H, or
3790H.
- Excludes: MATH 470, 471H.
Math 4810H - Perspectives in Mathematics I
Team-taught by three instructors. Each instructor
will teach a four-week module on a special topic.
- Prerequisites: 60% or higher in 1.0 3000- or 4000-level
MATH credit.
- Excludes: MATH 491H, 481H.
Math 4820H - Perspectives in Mathematics II
Team-taught by three instructors. Each instructor
will teach a four-week module on a special topic.
- Prerequisites: 60% or higher in 1.0 3000- or 4000-level
MATH credit.
- Excludes: MATH 492H, 482H.
Math 4850H, 4851H, 4852H - Community-based Research Project
Students are placed in research projects with community organizations in the Peterborough area. Each placement is supervised jointly by a faculty member and a representative of a community organization. For details see the section on Community-based education program in the Academic Calendar.
- Prerequisites: MATH 2560H, and either MATH 3560H or MATH 4561H or MATH 4562H.
- Note: Open only to students who have a cumulative average of at least 75%.
Math 4900Y - Reading-seminar course (Full)
Details may be obtained by consulting the Department of Mathematics.
Math 4903H - Reading-seminar course (half)
Details may be obtained by consulting the Department of Mathematics.
Math 4904H - Reading-seminar course (half)
Details may be obtained by consulting the Department of Mathematics.